Global dynamics for a model of amyloid fibril formation in pancreatic islet beta cells subjected to a therapy

Abstract

Amyloid polypeptide (IAPP) fibril formation is the main cause of type II diabetes mellitus (T2DM) disease and cardiovascular disease (CVD). Therefore, implementing therapeutic interventions strategies to control IAPP fibril formation plays a crucial role in preventing and managing T2DM disease and CVD. For this purpose, in this paper, we make use of a mathematical model to investigate the effects of drug therapy on the dynamics of amyloid fibril formation (or amyloid plaques). The model is a hybrid system of differential equations consisting of ODEs and (hyperbolic) PDE. The (global) stability analysis is established through the construction of suitable Lyapunov functionals and application of Lasalle's invariance principle. Moreover, stability of the disease steady state is fully determined by a threshold condition, which is based on the total population of IAPP polymers.

Keywords:

• Islet amyloid polypeptide
• nucleated polymerization
• type II diabetes mellitus
• therapy
• steady-state analysis
• uniform persistence
• stability analysis

2010 Mathematics Subject Classifications:

• 58J45
• 35L50
• 35L45

1. Introduction

Amyloid plaques are widely regarded as a pathogenic factor of type II diabetes mellitus (T2DM) disease, which is initially caused by a state of insulin resistance before advancing to the state of insulin dependency. These amyloid plaques arise from misfolding of a (37 amino acid) hormone called islet amyloid polypeptide (IAPP), also known as amylin, within and around the pancreatic islet β-cells that co-secrete both insulin and IAPP [4] and it is proven in [2] that increasing insulin in the state of insulin resistance causes an increase of IAPP secretion, as a result of their co-regulation. The occurrence of amyloid deposits is done through a process known as nucleated polymerization, detailed in [16,21], by which several thousands of IAPP monomer units are attached by aggregation, in a string like manner, to produce a polymer (or an oligomer) called IAPP fibril formation also known as amyloid plaque. Since IAPP units are misfolded with the IAPP fibril formation, this results in structural and functional changes of IAPP units. As a consequence, the IAPP fibril formation becomes dysfunctional within and around the pancreatic islet β-cells.

Amyloid plaques (IAPP fibril formation) are accountable for loss in β-cell mass in patients with T2DM disease [8]. In addition to this, numerous in vitro investigations have proven that amyloid plaques occurring around (outside) β-cells can lead to their extinction through the formation of reactive oxygen species, mitochondrial dysfunction, chromatin condensation, and apoptotic mechanisms [6]. We refer to [6,9,10] and the references therein for more detailed information about the biological background for the origin and development of T2DM. It is reported in [15] that there exists a strong correlation between the degree of T2DM disease seriousness and the amount of amyloid plaque deposited.

Therapeutic interventions in form of drugs (such as beta-sheet blockers, herparin sulphate proteoglycan derivatives, serum amyloid P inhibitors) or vaccination policies (see [7,9]) can be used to block the IAPP fibril formation and therefore prevent its damaging impacts on pancreatic islet β-cells and insulin-production capacity. It is reported in [14] that amyloid plaques are also regarded as a pathogenic factor of hypertension and therefore, due to their dissimilar metabolic and structural impacts, numerous classes of anti-hypertensive drugs prevent the IAPP fibril formation, hence cannot give rise to the T2DM disease. Thus, T2DM disease and hypertension are both accountable for significant widened cardiovascular disease (CVD) morbidity and mortality [7]. With regards to anti-diabetic drugs, it also important to highlight that there is controversy related to the treatment of diabetes, and the connection, with Alzheimer's disease (AD) as reported below. In their previous study, Kim et al. exhibited evidence that the inflammation-related S100A9 gene is considerably upregulated in the brains of AD animal models and human AD patients. This was also supported by some experiments which showed that knockdown of S100A9 expression ameliorates cognition in AD model mice (Tg2576), and these animals present reduced amyloid plaque formation. In their subsequent study [12], Kim et al. developed a novel transgenic animal model of AD by crossbreeding the Tg2576 mouse with the S100A9 knockout (KO) mouse. They notice that S100A9KO/Tg2576 (KO/Tg) mice displayed an increased spatial reference memory in the Morris water maze task and Y-maze task as well as decreased amyloid beta peptide (Aβ) neuropathology because of reduced levels of Aβ, C-terminal fragments of amyloid precursor protein (APP-CT) and phosphorylated TAU and increased expression of anti-inflammatory Interleukins-10 (IL-10) and also decreased expression of inflammatory Interleukins-6 (IL-6) and tumour neurosis factor (TNF)-α when compared with age-matched S100A9WT/Tg2576 (WT/Tg) mice. Overall, these results suggest that S100A9 is responsible for the neurodegeneration and cognitive deficits in Tg2576 mice. The mechanism of S100A9 is able to coincide with the inflammatory process. The outcomes from this research study reveal that knockout of S100A9 is a potential target for the pharmacological therapy of AD. However, contrary to the previous authors, some studies have indicated that insulin resistance (IR) is a potential target for the pharmacological therapy of both AD and diabetes. The latter statement is supported by the following findings. In the review [3], the aim of the study is to discuss the existing evidence supporting the idea that brain insulin resistance and altered glucose metabolism play a crucial role in the pathogenesis of late-onset Azheimer disease (LOAD) and AD treatment approaches targeting insulin signalling using anti-diabetic drugs and mTOR inhibitors. The authors investigated some potential pharmacological approaches for LOAD treatment related with the regulation of insulin metabolism, using anti-diabetic drugs (such as pioglitazone, intranasal insulin, NMDAR antagonists such as memantine and inhibitors of mTOR activity such as rapamycin and its derivatives (rapalogs)), and the findings give support to the continued investigation on potential stimulation of insulin receptor as a therapy for LOAD for intranasal insulin treatments and to the therapeutic potentials of temsirolimus in preclinical models of AD. However, although preclinical data give support to the potential beneficial effects of pioglitazone in AD, clinical data reported until now shows conflicting results regarding efficacy due to the many limitations of these trials and thus, further clinical trials on the potential use of pioglitazone for the treatment of LOAD are necessary. In the recent published review [11], it is stated that clinical studies and preclinical data in the last decade proved that AD and diabetes mellitus share a pathophysiological pathway, indicating that insulin resistance, oxidative stress and inflammatory response would increase the risks of developing AD in diabetic patients. This review presents briefly the aetiology of AD and diabetes, discusses the feasible theories about the interplaying risk factors and the mechanism of action of anti-diabetic medications recommended for the treatment of AD. Authors used some classes of anti-diabetic drugs (including insulin, thiazolidinediones, sulfonylurea, metformin, amylin analogue) as potential treatments for AD and the findings confirm that the anti-diabetic medications can actually exert an effect on brain functions including memory, cognition and attention. Intranasal administration was considered as the best route for insulin delivery into the brain as it traverses directly into the blood-brain barrier (BBB). Metformin is considered one of the most important oral medications given in diabetic patients, except for those who suffer from lactic acidosis, and has been studied extensively. There are many clinical studies tend to support its efficacy in cognitive enhancement. Despite controversial findings, it is agreed that the earlier the stage of AD, the more chance for anti-diabetics to have a positive effect on brain functions. The authors also suggested that more clinical trials and investigations should be done in this field to limit the progression of AD, find better treatment outcomes and increase the quality of life in AD patients.

In this paper we consider an existing model, describing the nucleated polymerization's dynamics of human amyloid polypeptide (HIAPP or IAPP) within the pancreatic islet β-cells under the impacts of therapeutic interventions, as formulated in [15]. The model is made up of the rates of changes of the IAPP monomers population size, IAPP polymers of length x population size and amount drug in the system. This is a hybrid system of non-linear coupled ordinary and partial differential equations, where $M(t)$, $p(x,t)$ and $D(t)$ denote the IAPP monomers population size, the IAPP polymers of length x population density and the amount of drug in the system, respectively, at time t. The governing equations of the model are given by (1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) with initial conditions (2) $M(0)=M_0,D(0)=D_0,p(x,0)=p_0(x),(x_0<x<\mathrm{\infty }),$(2) and boundary conditions (3) $p(x_0,t)=0,t\ge 0,$(3) where $M_0$ and $p_0(x)$ are IAPP monomers population and IAPP polymers of length x initial sizes, $D_0$ is an initial amount of drug in the system and $x_0$ is the critical length of the IAPP polymer below which the IAPP polymer breaks down into normal IAPP monomers. Parameters in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) are all positive (constant or function) which account for the following:

1. constant rate of production of IAPP monomers in the pancreatic beta cells

2. constant rate of degradation of IAPP monomers due to metabolism processes

3. constant polymerization rate (also known as the conversion rate of IAPP monomers to IAPP polymers)

4. rate at which the drug is degraded from the system due to metabolic processes

5. rate at which the drug increases in the system

6. rate at which the polymer population reduces due to the presence of the drug

7. binary splitting rate of the IAPP polymers of length x

8. degradation rate of the IAPP polymers due to metabolism

9. probability density function that a polymer of length y splits into two polymers of lengths x and y−x.

On the right-hand side of the first equation in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ), the last term accounts for the number of IAPP monomers resulting from the splitting of an IAPP polymer which also produces at least one IAPP polymer of length lesser than $x_0$. It is assumed that such an IAPP polymer dissociates from IAPP monomers. The factor 2 in this term explains the binary splitting of an IAPP polymer of length x into two polymers whenever $x>x_0$. It is also assumed that the length of IAPP polymers picks up continuous values for mathematical analysis purposes. Moreover, on the right-hand side of the second equation in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ), the first term (the transport term) and the last term represent, respectively, the loss in the amount of IAPP polymers of length x as a result of lengthening and the gain of IAPP polymers of length x due to splitting of lengthy polymers.

The total amount of IAPP polymers and the total amount of IAPP monomers in IAPP polymers, of length x, will be denoted by the time functions $P(t)={\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\mathrm{and}Q(t)={\int }_{x_0}^{\mathrm{\infty }}xp(x,t)\mathrm{d}x,$ respectively. Clearly, for any $t\ge 0$, (4) $P(t)<Q(t).$(4) In [15] Murali and Raman made use of the following assumptions: $m(x)=m,b(x)=bx,\mathrm{and}f(x,y)=\{\begin{array}{ll}{\displaystyle \frac{1}{y}}& \mathrm{if}y>x_0\\ 0& \mathrm{if}y\le x_0\mathrm{or}y\le x\end{array}$ for simplifying the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ) into a model consisting of a system of ordinary differential equations given by (5) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)+b{x}_0^{2}P(t)-rM(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}P(t)& =-mP(t)+bQ(t)-2b{x}_{0}P(t)-k_{2}D(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}Q(t)& =-mQ(t)-b{x}_0^{2}P(t)-k_{2}D(t)Q(t)+rM(t)P(t)\\ \frac{d}{dt}D(t)& =-k_{0}D(t)+k_{1}D(t)P(t)\end{array}$(5) subject to initial conditions (6) $M(0)=M_0,P(0)=P_0,Q(0)=Q_0,D(0)=D_0,$(6) where $M_0,P_0,Q_0,D_0\ge 0$. An asymptotic behaviour of the system of equations (5(5) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)+b{x}_0^{2}P(t)-rM(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}P(t)& =-mP(t)+bQ(t)-2b{x}_{0}P(t)-k_{2}D(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}Q(t)& =-mQ(t)-b{x}_0^{2}P(t)-k_{2}D(t)Q(t)+rM(t)P(t)\\ \frac{d}{dt}D(t)& =-k_{0}D(t)+k_{1}D(t)P(t)\end{array}$(5) )–(6(6) $M(0)=M_0,P(0)=P_0,Q(0)=Q_0,D(0)=D_0,$(6) ) is carried out under the above-mentioned assumptions. A disease-free state equilibrium, $E_1=(\tilde{M},\tilde{P},\tilde{Q},\tilde{D})=(A/g,0,0,0)$, and a disease state equilibrium, $E_2=(\acute{M},\acute{P},\acute{Q},\acute{D})$, where $\acute{M}=\frac{(b{x}_0+m{)}^2}{br},\acute{P}=\frac{brA-g(b{x}_0+m{)}^2}{mr(2b{x}_0+m)},\acute{Q}=\frac{brA-g(b{x}_0+m{)}^2}{mbr},\acute{D}=0,$ with $brA>g(b{x}_0+m{)}^2$, of the system of equations (5(5) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)+b{x}_0^{2}P(t)-rM(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}P(t)& =-mP(t)+bQ(t)-2b{x}_{0}P(t)-k_{2}D(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}Q(t)& =-mQ(t)-b{x}_0^{2}P(t)-k_{2}D(t)Q(t)+rM(t)P(t)\\ \frac{d}{dt}D(t)& =-k_{0}D(t)+k_{1}D(t)P(t)\end{array}$(5) )–(6(6) $M(0)=M_0,P(0)=P_0,Q(0)=Q_0,D(0)=D_0,$(6) ) were found after nullifying therapeutic interventions effects; while a coexistence state equilibrium, $E_3=(M^{\ast },P^{\ast },Q^{\ast },D^{\ast })$, where $M^{\ast }=\frac{A+b{x}_0^2{P}^{\ast }}{g+r{P}^{\ast }},P^{\ast }=\frac{k_0}{k_1)},Q^{\ast }=\frac{(r{M}^{\ast }-b{x}_0^2)P^{\ast }}{m+k_2{D}^{\ast }},D^{\ast }=\frac{\sqrt{br{M}^{\ast }}-(m+b{x}_0)}{k_2},$ with $rA{M}^{\ast }>max\{b{x}_0^2,b^{-1}(b{x}_0+m{)}^2\}$, of the system of equations (5(5) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)+b{x}_0^{2}P(t)-rM(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}P(t)& =-mP(t)+bQ(t)-2b{x}_{0}P(t)-k_{2}D(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}Q(t)& =-mQ(t)-b{x}_0^{2}P(t)-k_{2}D(t)Q(t)+rM(t)P(t)\\ \frac{d}{dt}D(t)& =-k_{0}D(t)+k_{1}D(t)P(t)\end{array}$(5) )–(6(6) $M(0)=M_0,P(0)=P_0,Q(0)=Q_0,D(0)=D_0,$(6) ) is found after taking into account the therapeutic interventions. The local stability of equilibria is proven by the mean of the Routh-Hurwitz criterion and revealed that $E_1$ (respectively $E_2$) is locally asymptotically stable if and only if $brA<g(m+b{x}_0{)}^2$ (respectively $brA>g(m+b{x}_0{)}^2$ and $\acute{P}<k_0/k_1$) and $E_3$ is locally asymptotically stable if some conditions (as given in [15, p. 6]) are satisfied. Next, Murali and Raman used some numerical stimulations based on the system of equations (5(5) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)+b{x}_0^{2}P(t)-rM(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}P(t)& =-mP(t)+bQ(t)-2b{x}_{0}P(t)-k_{2}D(t)P(t)\\ \frac{\mathrm{d}}{\mathrm{d}t}Q(t)& =-mQ(t)-b{x}_0^{2}P(t)-k_{2}D(t)Q(t)+rM(t)P(t)\\ \frac{d}{dt}D(t)& =-k_{0}D(t)+k_{1}D(t)P(t)\end{array}$(5) )–(6(6) $M(0)=M_0,P(0)=P_0,Q(0)=Q_0,D(0)=D_0,$(6) ) to confirm the results from the theoretical analysis.

In contrast to the results from Murali and Raman in [15] and from other authors in references therein, here we consider arbitrary (bounded or unbounded) functions $b(\cdot )$ and $m(\cdot )$. Furthermore, for the sake of handling the hybrid system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ), we make the following assumptions:

1. $k_1=k_2$,

2. functions $b(\cdot )$ and $m(\cdot )$ satisfy $max\{(x+4)b(x),rA/g\}<m(x)$, where $1<m(x)+b(x)$, for any $x\ge x_0$,

3. $f(x,y)$ satisfies the natural constraints, see [17,19], (7) $f(x,y)=f(y-x,y),y>x_0,0<x<y,$(7) which accounts for the binary splitting of the IAPP polymers, and (8) ${\int }_0^{y}f(x,y)\mathrm{d}x=1,y>x_0.$(8) These two constraints, namely (7(7) $f(x,y)=f(y-x,y),y>x_0,0<x<y,$(7) ) and (8(8) ${\int }_0^{y}f(x,y)\mathrm{d}x=1,y>x_0.$(8) ) yields (9) $2{\int }_0^{y}xf(x,y)dx=y,y>x_0,$(9) i.e. there is the conservation of number of monomers during the splitting process.

The problem consisting of Equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ) subjected to assumptions $\mathbf{A}\mathbf{1}$–$\mathbf{A}\mathbf{3}$ is regarded as novel and have not yet been investigated to our best knowledge. Such a problem is mathematically challenging but allows to provide some relevant features of the model. We make use of the mathematical tool developed in [13] to carry out the steady states and stability analysis of Equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ). The rest of the paper is structured as follows. Section 2 deals with some preliminary results and more importantly the existence result of global attractors for the continuous semi-flow defined by the system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ). The steady states of the system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ) are computed in Section 3. Three equilibria are obtained: a disease-free state equilibrium, a disease state equilibrium and a coexistence state equilibrium. The next section, namely Section 4, is devoted to the uniform persistence of the coexistence state equilibrium. Sections 5 and 6 deal, respectively, with local and global stability analysis of the disease-free state equilibrium and disease state equilibrium.

2. Preliminaries

Let $X_1$ be defined by $X_1={\mathrm{L}}^1([x_0,\mathrm{\infty }),x\mathrm{d}x)=\{\varphi :\Vert \varphi {\Vert }_1={\int }_{x_0}^{\mathrm{\infty }}x|\varphi (x)|\mathrm{d}x\}$. We consider the Banach space $\mathbb{X}=\mathbb{R}\times {\mathrm{X}}_1\times \mathbb{R}$ endowed with the norm $\Vert (u,\varphi ,v){\Vert }_{\mathbb{X}}=|u|+\Vert \varphi {\Vert }_1+|v|$, for any $(u,\varphi ,v)\in \mathbb{X}$. Let us denote by ${\mathbb{X}}_{+}$ the positive cone of the Banach space $\mathbb{X}$ such that ${\mathbb{X}}_{+}={\mathbb{R}}_{+}\times X_{1,+}\times {\mathbb{R}}_{+}$, where $X_{1,+}={\mathrm{L}}_{+}^1([x_0,\mathrm{\infty }),x\mathrm{d}x)$

For any initial value ${\mathbf{X}}_0=(M_0,p_0(\cdot ),D_0)\in {\mathbb{X}}_{+}$ satisfying the following condition $p_0(0)=0$, the system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ) is well-posed. Indeed, the well-posedness of the system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ), without therapeutic interventions (that is, $D(\cdot )\equiv 0$), is established in [17], where the existence and uniqueness of global classical solutions for bounded functions $b(\cdot )$ and $m(\cdot )$ and the existence and uniqueness of global weak solutions for unbounded functions $b(\cdot )$ and $m(\cdot )$ are shown. So, this result can easily be extended to the solvability of the system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ) with therapeutic interventions. So, we obtain a continuous semi-flow $\mathrm{\Phi }:{\mathbb{R}}_{+}\times {\mathbb{X}}_{+}\to {\mathbb{X}}_{+}$ defined by the system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) )–(3(3) $p(x_0,t)=0,t\ge 0,$(3) ) so that (10) $\mathrm{\Phi }(t,{\mathbf{X}}_0)={\mathrm{\Phi }}_t({\mathbf{X}}_0)=(M(t),p(\cdot ,t),D(t)),t\in {\mathbb{R}}_{\mathbb{+}},{\mathbf{X}}_0\in {\mathbb{X}}_{+}.$(10)

Since the components of ${\mathrm{\Phi }}_t({\mathbf{X}}_0)$ are positive, using the above notation, we obtain (11) $\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}=M(t)+\Vert p(\cdot ,t){\Vert }_1+D(t).$(11) Differentiating (11(11) $\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}=M(t)+\Vert p(\cdot ,t){\Vert }_1+D(t).$(11) ) with respect to t yields (12) $\frac{\mathrm{d}}{\mathrm{d}t}\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}=\frac{\mathrm{d}}{\mathrm{d}t}M(t)+\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1+\frac{\mathrm{d}}{\mathrm{d}t}D(t).$(12) Next, we show that (13) $\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1=\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{x_0}^{\mathrm{\infty }}xp(x,t)\mathrm{d}x={\int }_{x_0}^{\mathrm{\infty }}x\frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)\mathrm{d}x.$(13) Substituting the second equation of (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) into (13(13) $\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1=\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{x_0}^{\mathrm{\infty }}xp(x,t)\mathrm{d}x={\int }_{x_0}^{\mathrm{\infty }}x\frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)\mathrm{d}x.$(13) ) leads to (14) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1& ={\int }_{x_0}^{\mathrm{\infty }}x(-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t))\mathrm{d}x\\ & +2{\int }_{x_0}^{\mathrm{\infty }}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x.\end{array}$(14) Expanding the integrand of the first integral, changing the order of integration of the last integral in (14(14) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1& ={\int }_{x_0}^{\mathrm{\infty }}x(-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t))\mathrm{d}x\\ & +2{\int }_{x_0}^{\mathrm{\infty }}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x.\end{array}$(14) ) and using assumption $\mathbf{A}\mathbf{1}$, we get (15) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1& =-rM(t){\int }_{x_0}^{\mathrm{\infty }}x\frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)\mathrm{d}x-{\int }_{x_0}^{\mathrm{\infty }}x(m(x)+b(x))p(x,t)\mathrm{d}x\\ & -k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}xp(x,t)\mathrm{d}x+{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}xf(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y.\end{array}$(15) Combining Equation (15(15) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1& =-rM(t){\int }_{x_0}^{\mathrm{\infty }}x\frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)\mathrm{d}x-{\int }_{x_0}^{\mathrm{\infty }}x(m(x)+b(x))p(x,t)\mathrm{d}x\\ & -k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}xp(x,t)\mathrm{d}x+{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}xf(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y.\end{array}$(15) ), the first and third equations of (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ), we get (16) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}(M(t)+\Vert p(\cdot ,t){\Vert }_1+D(t))& =A-(gM(t)+\frac{r}{g}A\Vert p(\cdot ,t){\Vert }_1+k_{0}D(t))\\ & -k_1(Q(t)-P(t))D(t).\end{array}$(16) Let η be such that (17) $\sqrt{rA}<\eta \le inf\{g,k_0,\frac{r}{g}A\}.$(17) Using (4(4) $P(t)<Q(t).$(4) ), we have (18) $\frac{\mathrm{d}}{\mathrm{d}t}(M(t)+\Vert p(\cdot ,t){\Vert }_1+D(t))\le A-\eta (M(t)+\Vert p(\cdot ,t){\Vert }_1+D(t)),$(18) that is, $(\mathrm{d}/\mathrm{d}t)\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le A-\eta \Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}$. Thus, we obtain (19) $\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le \frac{A}{\eta }-e^{-\eta t}(\frac{A}{\eta }-\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}),\mathrm{where}\Vert {\mathrm{\Phi }}_0({\mathbf{X}}_0){\Vert }_{\mathbb{X}}=\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}.$(19) Let Γ be the state space of (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ), defined by $\mathrm{\Gamma }=\{(u,\varphi ,v)\in {\mathbb{X}}_{+}:\Vert (u,\varphi ,v){\Vert }_{\mathbb{X}}\le \frac{A}{\eta }\}.$ From (19(19) $\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le \frac{A}{\eta }-e^{-\eta t}(\frac{A}{\eta }-\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}),\mathrm{where}\Vert {\mathrm{\Phi }}_0({\mathbf{X}}_0){\Vert }_{\mathbb{X}}=\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}.$(19) ), we get $\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le A/\eta$ for any $t\ge 0$, whenever ${\mathbf{X}}_0\in \mathrm{\Gamma }$. Moreover, $\underset{t\to \mathrm{\infty }}{limsup}\Vert {\mathrm{\Phi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le A/\eta$ for any ${\mathbf{X}}_0\in {\mathbb{X}}_{+}$. Then, we obtain the following result.

Lemma 2.1

The set Γ is positively invariant for Φ; that is, ${\mathrm{\Phi }}_t({\mathbf{X}}_0)\subset \mathrm{\Gamma },\mathrm{\forall }t\ge 0,$ ${\mathbf{X}}_0\in \mathrm{\Gamma },$ $\mathrm{\forall }t\ge 0,$ ${\mathbf{X}}_0\in \mathrm{\Gamma }$. Moreover, Φ is point dissipative.

As we aim to make use of the Lasalle's Invariance Principle, we are required to establish the relative compactness of the orbit $\{{\mathrm{\Phi }}_t({\mathbf{X}}_0):t\ge 0\}$ in ${\mathbb{X}}_{+}$ due to the infinite-dimensional Banach space $\mathbb{X}$. This is done as in [13] by introducing two operators Θ and Ψ, ($\mathrm{\Theta },\mathrm{\Psi }:{\mathbb{R}}_{+}\times {\mathbb{X}}_{+}\to {\mathbb{X}}_{+}$), defined by (20) ${\mathrm{\Theta }}_t({\mathbf{X}}_0)=(0,p(x,t),0)\mathrm{and}{\mathrm{\Psi }}_t({\mathbf{X}}_0)=(M(t),0,D(t)),$(20) where $\mathrm{\Theta }(t,{\mathbf{X}}_0)={\mathrm{\Theta }}_t({\mathbf{X}}_0)$ and $\mathrm{\Psi }(t,{\mathbf{X}}_0)={\mathrm{\Psi }}_t({\mathbf{X}}_0)$, so that (21) ${\mathrm{\Phi }}_t({\mathbf{X}}_0)={\mathrm{\Theta }}_t({\mathbf{X}}_0)+{\mathrm{\Psi }}_t({\mathbf{X}}_0),\mathrm{for}\mathrm{any}t\ge 0,$(21) and by obtaining, from the proof of [20, Proposition 3.13] and Lemma 2.1, the result below.

Theorem 2.2

For ${\mathbf{X}}_0\in \mathrm{\Gamma },$ the orbit $\{{\mathrm{\Phi }}_t({\mathbf{X}}_0):t\ge 0\}$ has a compact closure in ${\mathbb{X}}_{+}$ if the following conditions are satisfied.

1. There exists a function $\mathrm{\Delta }:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that for any $r>0,$ $\underset{t\to \mathrm{\infty }}{lim}\mathrm{\Delta }(t,r)=0$, and if ${\mathbf{X}}_0\in \mathrm{\Gamma }$ with $\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}\le r,$ then $\Vert {\mathrm{\Theta }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le \mathrm{\Delta }(t,r)$ for any $t\ge 0$.

2. For any $t\ge 0$, ${\mathrm{\Psi }}_t(\cdot )$ maps any bounded sets of Γ into a set with compact closure in ${\mathbb{X}}_{+}$.

Conditions (i) and (ii) of Theorem 2.2 are verified through the following lemmas.

Lemma 2.3

For $\delta >0,$ let $\mathrm{\Delta }(t,\delta )=e^{-\eta t}\delta$. Then $\underset{t\to \mathrm{\infty }}{lim}\mathrm{\Delta }(t,\delta )=0$. Then (i) of Theorem 2.2 holds.

Proof.

We have $\underset{t\to \mathrm{\infty }}{lim}\mathrm{\Delta }(t,\delta )=0$. We choose the initial condition ${\mathbf{X}}_0\in \mathrm{\Gamma }$ such that $\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}\le r$. For $t\ge 0$, we have $\Vert {\mathrm{\Theta }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}=|0|+\Vert p(\cdot ,t){\Vert }_1+|0|=\Vert p(\cdot ,t){\Vert }_1$. Using (4(4) $P(t)<Q(t).$(4) ), (9(9) $2{\int }_0^{y}xf(x,y)dx=y,y>x_0,$(9) ), (15(15) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1& =-rM(t){\int }_{x_0}^{\mathrm{\infty }}x\frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)\mathrm{d}x-{\int }_{x_0}^{\mathrm{\infty }}x(m(x)+b(x))p(x,t)\mathrm{d}x\\ & -k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}xp(x,t)\mathrm{d}x+{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}xf(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y.\end{array}$(15) ), (17(17) $\sqrt{rA}<\eta \le inf\{g,k_0,\frac{r}{g}A\}.$(17) ), assumptions $\mathbf{A}\mathbf{1}$ and $\mathbf{A}\mathbf{2}$, we have (22) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}\Vert p(\cdot ,t){\Vert }_1& \le \frac{r}{\eta }A\Vert p(\cdot ,t){\Vert }_1-\eta {\int }_{x_0}^{\mathrm{\infty }}xm(x)p(x,t)\mathrm{d}x\\ & =(\frac{r}{\eta }A-\eta )\Vert p(\cdot ,t){\Vert }_1,\end{array}$(22) and hence $\Vert p(\cdot ,t){\Vert }_1\le e^{((r/\eta )A-\eta )}\Vert p_0(\cdot ){\Vert }_1\le e^{((r/\eta )A-\eta )t}(|M_0|+\Vert p_0(\cdot ){\Vert }_1+|D_0|)$; that is, ${\displaystyle \Vert p(\cdot ,t){\Vert }_1\le e^{((r/\eta )A-\eta )t}\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}}$. Therefore, we get $\Vert {\mathrm{\Theta }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le e^{-\eta t}\delta =\mathrm{\Delta }(t,\delta )$, where $\delta =e^{(r/\eta )At}\Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}\ge \Vert {\mathbf{X}}_0{\Vert }_{\mathbb{X}}$. Moreover, since $(r/\eta )A-\eta =(\sqrt{rA}+\eta )(\sqrt{rA}-\eta )/\eta$, using (17(17) $\sqrt{rA}<\eta \le inf\{g,k_0,\frac{r}{g}A\}.$(17) ), it is easy to see that $\mathrm{\Delta }(t,\delta )\to 0$ as $t\to \mathrm{\infty }$. This completes the proof.

Lemma 2.4

For $t\ge 0,$ ${\mathrm{\Psi }}_t(\cdot )$ maps any bounded set of Γ into a set with a compact closure in ${\mathbb{X}}_{+}$.

Proof.

Since $M(t)$ and $D(t)$ remain in the compact set $[0,A/\eta ]$ by Lemma 2.1, ${\mathrm{\Psi }}_t(\cdot ):\mathrm{\Gamma }⟶{\mathrm{\Psi }}_t(\mathrm{\Gamma })$, where (23) ${\mathrm{\Psi }}_t(\mathrm{\Gamma })=\{{\mathrm{\Psi }}_t({\mathbf{X}}_0)\in {\mathbb{X}}_{+}:\Vert {\mathrm{\Psi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le \frac{A}{\eta }\}.$(23) We clearly see, from (23(23) ${\mathrm{\Psi }}_t(\mathrm{\Gamma })=\{{\mathrm{\Psi }}_t({\mathbf{X}}_0)\in {\mathbb{X}}_{+}:\Vert {\mathrm{\Psi }}_t({\mathbf{X}}_0){\Vert }_{\mathbb{X}}\le \frac{A}{\eta }\}.$(23) ), that the set ${\mathrm{\Psi }}_t(\mathrm{\Gamma })$ is bounded. To establish the closedness of the set ${\mathrm{\Psi }}_t(\mathrm{\Gamma })$, we consider α and σ such that $\alpha =\underset{t\to \mathrm{\infty }}{lim}M(t)$ and $\sigma =\underset{t\to \mathrm{\infty }}{lim}D(t)$. By the continuity of the norm $\Vert \cdot {\Vert }_{\mathbb{X}}$, we have $\Vert (\alpha ,0,\sigma ){\Vert }_{\mathbb{X}}\le A/\eta$, that is $\underset{t\to \mathrm{\infty }}{lim}{\mathrm{\Psi }}_t({\mathbf{X}}_0)\in {\mathrm{\Psi }}_t(\mathrm{\Gamma })$ for all ${\mathbf{X}}_0\in \mathrm{\Gamma }$. Hence, the set ${\mathrm{\Psi }}_t(\mathrm{\Gamma })$ is closed.

By definition, $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ is the smallest closed set that contains ${\mathrm{\Psi }}_t(\mathrm{\Gamma })$, that is ${\mathrm{\Psi }}_t(\mathrm{\Gamma })\subseteq \overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$. Moreover, $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ is also a closed set that contains ${\mathrm{\Psi }}_t(\mathrm{\Gamma })$. So, $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}\subseteq {\mathrm{\Psi }}_t(\mathrm{\Gamma })$. Therefore, $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}={\mathrm{\Psi }}_t(\mathrm{\Gamma })$.

By way of contradiction, suppose that $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ is not compact in ${\mathbb{X}}_{+}$, that is $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ is not totally bounded, since $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ is complete as a closed set of a Banach space $\mathbb{X}$, which is a complete space. This means that for some positive ε and for any finite $\mathcal{K}\subseteq \overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ we can find $\mathbf{U}\in \overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ such that $\mathbf{V}-\mathbf{U}\in {\mathbb{X}}_{+}$ and $\Vert \mathbf{V}-\mathbf{U}{\Vert }_{\mathbb{X}}>ϵ$ for every $\mathbf{V}\in \mathcal{K}$. Let $({\mathbf{V}}_n)$ be a sequence of points in $\mathcal{K}$, that is $\Vert {\mathbf{V}}_n{\Vert }_{\mathbb{X}}\le A/\eta$ for any $n\in \mathbb{N}$. Thus ${\mathbf{V}}_n-\mathbf{U}\in {\mathbb{X}}_{+}$ and $\Vert {\mathbf{V}}_n-\mathbf{U}{\Vert }_{\mathbb{X}}>ϵ$ for any $n\in \mathbb{N}$. It follows that $\Vert {\mathbf{V}}_n{\Vert }_{\mathbb{X}}\ge \Vert {\mathbf{V}}_n-\mathbf{U}{\Vert }_{\mathbb{X}}$, and hence $\Vert {\mathbf{V}}_n{\Vert }_{\mathbb{X}}>ϵ$ for any $n\in \mathbb{N}$. Choosing $ϵ=A/\eta$ yields a contradiction and therefore $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ is a totally bounded and complete set. Hence, $\overline{{\mathrm{\Psi }}_t(\mathrm{\Gamma })}$ is a compact set in ${\mathbb{X}}_{+}$.

Therefore, From Lemma 2.1 and Theorem 2.2, the existence result of global attractors (see [5]) follows.

Theorem 2.5

The semi-flow $\{{\mathrm{\Phi }}_t({\mathbf{X}}_0):t\ge 0\}$ has a global attractor in ${\mathbb{X}}_{+},$ which attracts any bounded subset of ${\mathbb{X}}_{+}$.

3. Equilibria

The equilibria of the system (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) are given by the following equations (24) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x)+k_{2}D)p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\\ 0& =(-k_0+k_1{P}^{\ast })D,\end{array}$(24) where $P^{\ast }={\int }_{x_0}^{\mathrm{\infty }}p(x)\mathrm{d}x$. From the last equation in (24(24) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x)+k_{2}D)p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\\ 0& =(-k_0+k_1{P}^{\ast })D,\end{array}$(24) ), it follows that D = 0 or $P^{\ast }=k_0/k_1$. If D = 0, the system (24(24) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x)+k_{2}D)p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\\ 0& =(-k_0+k_1{P}^{\ast })D,\end{array}$(24) ) reduces to (25) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x))p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(25) By assuming $p(x)=0$ in (25(25) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x))p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(25) ), we obtain the disease-free steady state $E^0$ of the system (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) given by $E^0=(M^0,p^0(x),D^0)$, where $M^0=A/g,p^0(x)=0,D^0=0$. However, by assuming $p(x)\ne 0$ in (25(25) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x))p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(25) ), from the last equation in (25(25) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x))p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(25) ), we have (26) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y=rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)+(m(x)+b(x))p(x)$(26) and we substitute (26(26) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y=rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)+(m(x)+b(x))p(x)$(26) ) into the first equation of (25(25) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x))p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(25) ) to obtain $A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{x}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x=0,$ that is $A-(g+r{P}^{\ast })M-2{\int }_0^{x_0}x{\int }_x^{x_0}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x=0.$ By changing the order of integration in the latter equation, we have $A-(g+r{P}^{\ast })M-{\int }_0^{x_0}(2{\int }_0^{y}xf(x,y)dx)b(y)p(y)\mathrm{d}y=0.$ Using (9(9) $2{\int }_0^{y}xf(x,y)dx=y,y>x_0,$(9) ), the latter equation yields $A-(g+r{P}^{\ast })M-{\int }_0^{x_0}yb(y)p(y)\mathrm{d}y=0,$ which implies $A-(g+r{P}^{\ast })M=0$. Thus, a disease steady state $E^{\ast }$ of the system (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) is given by $E^{\ast }=(M^{\ast },p^{\ast }(x),D^{\ast })$, where $M^{\ast }=A/(g+r{P}^{\ast }),D^{\ast }=0$ and $p^{\ast }(x)$ is the solution of the non-linear ordinary differential equation $\frac{\mathrm{d}p}{\mathrm{d}x}(x)+\frac{m(x)-b(x)}{r{M}^0}p(x)=\frac{p(x)}{A}{\int }_{x_0}^{\mathrm{\infty }}p(x)\mathrm{d}x.$ If $P^{\ast }=k_0/k_1$, the system (24(24) $\begin{array}{rl}0& =A-(g+r{P}^{\ast })M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x)+k_{2}D)p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\\ 0& =(-k_0+k_1{P}^{\ast })D,\end{array}$(24) ) reduces to (27) $\begin{array}{rl}0& =A-(g+r\frac{k_0}{k_1})M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x)+k_{2}D)p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(27)

From the last equation of (27(27) $\begin{array}{rl}0& =A-(g+r\frac{k_0}{k_1})M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x)+k_{2}D)p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(27) ), we have (28) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y=rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)+(m(x)+b(x)+k_{2}D)p(x)$(28) and the first equation of (27(27) $\begin{array}{rl}0& =A-(g+r\frac{k_0}{k_1})M+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ 0& =-rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)-(m(x)+b(x)+k_{2}D)p(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y.\end{array}$(27) ) is equivalent to (29) $\begin{array}{rl}0& =A-(g+r\frac{k_0}{k_1})M-2{\int }_0^{x_0}x{\int }_x^{x_0}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ & +2{\int }_0^{x_0}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x.\end{array}$(29) Substituting (28(28) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y=rM\frac{\mathrm{d}p}{\mathrm{d}x}(x)+(m(x)+b(x)+k_{2}D)p(x)$(28) ) into (29(29) $\begin{array}{rl}0& =A-(g+r\frac{k_0}{k_1})M-2{\int }_0^{x_0}x{\int }_x^{x_0}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x\\ & +2{\int }_0^{x_0}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x.\end{array}$(29) ) leads to $0=A-(g+r\frac{k_0}{k_1})M-2{\int }_0^{x_0}x{\int }_x^{x_0}b(y)f(x,y)p(y)\mathrm{d}y\mathrm{d}x$ and after changing the order of integration and using (9(9) $2{\int }_0^{y}xf(x,y)dx=y,y>x_0,$(9) ) in the latter equation, we get $0=A-(g+r\frac{k_0}{k_1})M-{\int }_0^{x_0}yb(y)p(y)\mathrm{d}y,$ which reduces to $0=A-(g+r\frac{k_0}{k_1})M.$ Thus, a coexistence steady state $\breve{E}$ of the system is given by $\breve{E}=(\breve{M},\breve{p}(x),\breve{D})$, where $\breve{M}=k_{1}A/(k_{0}r+k_{1}g),\breve{D}\ne 0$ and $\breve{p}(x)$ is the solution of the linear ordinary differential equation (30) $\frac{\mathrm{d}p}{\mathrm{d}x}(x)=\frac{k_{0}r+k_{1}g}{k_{1}Ar}(m(x)-b(x)+k_{2}D)p(x),$(30) where $D=\breve{D}$ and $k_2=k_1$ (see $\mathbf{A}\mathbf{1}$). Defining ${\overline{x}}_0=inf\{x|p(x)\ne 0\mathrm{for}x>x_0\}$ and integrating (30(30) $\frac{\mathrm{d}p}{\mathrm{d}x}(x)=\frac{k_{0}r+k_{1}g}{k_{1}Ar}(m(x)-b(x)+k_{2}D)p(x),$(30) ) over $[{\overline{x}}_0,x]$ leads to $p(x)=p({\overline{x}}_0)\mathrm{\Omega }(x)$, where $\mathrm{\Omega }(x)=\exp \{-\frac{k_{0}r+k_{1}g}{k_{1}rA}(k_1\breve{D}(x-{\overline{x}}_0)+{\int }_{{\overline{x}}_0}^x(m(s)-b(s))\mathrm{d}s)\}.$ Thus, the equation ${\int }_{x_0}^{\mathrm{\infty }}p(x)\mathrm{d}x=\frac{k_0}{k_1}$ yields $p({\overline{x}}_0)=k_0/k_1\cdot 1/{\int }_{x_0}^{\mathrm{\infty }}\mathrm{\Omega }(x)\mathrm{d}x$ and hence $p(x)=k_0/k_1\cdot \mathrm{\Omega }(x)/{\int }_{x_0}^{\mathrm{\infty }}\mathrm{\Omega }(x)\mathrm{d}x$, that is $\breve{p}(x)=k_0/k_1\cdot \mathrm{\Omega }(x)/{\int }_{x_0}^{\mathrm{\infty }}\mathrm{\Omega }(x)\mathrm{d}x$.

4. Uniform persistence

In this section, we investigate the uniform persistence of the system (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) at the disease state equilibrium $E^{\ast }$, that is under the condition (31) $\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}{p}^{\ast }(x)\mathrm{d}x<1.$(31) To do this, we consider a function $\rho :\mathbb{X}\to {\mathbb{R}}_{+}$ defined by (32) $\rho (u,\zeta ,v)=\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}\zeta (x)\mathrm{d}x,\mathrm{where}(u,\zeta ,v)\in {\mathbb{X}}_{+}.$(32) Let ${\mathbb{X}}_0$ be a set defined by ${\mathbb{X}}_0=\{{\mathbf{X}}_0\in {\mathbb{X}}_{+}:\rho ({\mathrm{\Phi }}_{t_0}({\mathbf{X}}_0))>0\mathrm{for}\mathrm{some}{t}_0\ge 0\}$ so that ${\mathrm{\Phi }}_t({\mathbf{X}}_0)\to E^0$ as $t\to \mathrm{\infty }$ whenever ${\mathbf{X}}_0\in {\mathbb{X}}_{+}\mathrm{\setminus }{\mathbb{X}}_0$.

Definition 4.1

see [18, p. 61]

Φ is uniformly weakly ρ-persistent (respectively, uniformly (strongly) ρ-persistent) if there exists a positive ε, not depending of initial conditions, so that $\underset{t\to \mathrm{\infty }}{limsup}\rho ({\mathrm{\Phi }}_t({\mathbf{X}}_0))>ϵ(\mathrm{respectively},\underset{t\to \mathrm{\infty }}{liminf}\rho ({\mathrm{\Phi }}_t({\mathbf{X}}_0))>ϵ)$ for any ${\mathbf{X}}_0\in {\mathbb{X}}_{+}$.

Theorem 4.2

Let (31(31) $\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}{p}^{\ast }(x)\mathrm{d}x<1.$(31) ) be satisfied. Then Φ is uniformly (strongly) ρ-persistent.

Proof.

By way of contradiction, for all $ϵ>0$ there exists ${\mathbf{X}}^{ϵ}\in {\mathbb{X}}_0$ such that $\underset{t\to \mathrm{\infty }}{liminf}\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{ϵ}))\le ϵ$. In particular, there exists ${\mathbf{X}}^{{ϵ}_0}\in {\mathbb{X}}_0$ such that (33) $\underset{t\to \mathrm{\infty }}{liminf}\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0}))\le {ϵ}_0,$(33) where $0<{ϵ}_0<1$. Choosing ${ϵ}_0=k_{1}A/k_0\eta$ and using (4(4) $P(t)<Q(t).$(4) ) yields (34) $\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}{p}^{{ϵ}_0}(x)\mathrm{d}x<\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}x{p}^{{ϵ}_0}(x)\mathrm{d}x.$(34) It follows that (35) $\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}{p}^{{ϵ}_0}(x)\mathrm{d}x<\frac{k_1}{k_0}\Vert p^{{ϵ}_0}(\cdot ,t)-p^{\ast }(\cdot ){\Vert }_1+\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}{p}^{\ast }(x)\mathrm{d}x,$(35) and hence (36) $\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0}))<\frac{k_1}{k_0}{\Vert {\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0})-E^{\ast }\Vert }_{\mathbb{X}}+\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}{p}^{\ast }(x)\mathrm{d}x.$(36) Since ${\mathrm{\Phi }}_t({\mathbf{X}}_0)-\breve{E}\in \mathrm{\Gamma }$, we can make use of (31(31) $\frac{k_1}{k_0}{\int }_{x_0}^{\mathrm{\infty }}{p}^{\ast }(x)\mathrm{d}x<1.$(31) ) so that (37) $\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0}))<\frac{k_{1}A}{k_0\eta }+1,$(37) that is, (38) $\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0}))<{ϵ}_0+1,\mathrm{for}\mathrm{all}t\ge 0.$(38) Therefore (39) $\underset{t\to \mathrm{\infty }}{liminf}\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0}))\le \underset{t\to \mathrm{\infty }}{limsup}\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0}))<{ϵ}_0+1.$(39) This contradicts (33(33) $\underset{t\to \mathrm{\infty }}{liminf}\rho ({\mathrm{\Phi }}_t({\mathbf{X}}^{{ϵ}_0}))\le {ϵ}_0,$(33) ) and thus complete the proof.

5. Local stability of equilibria

The local stability for equilibria of the system (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) will be investigated through its linearization around each equilibrium, see [13]. To achieve this, we will perturb the equilibria as follows. For the disease-free steady state $E^0$, we introduce new functions u, $v(x,\cdot )$ and w such that $u(t)=M(t)-M^0,v(x,t)=p(x,t),w(t)=D(t).$ Differentiating functions u, $v(x,\cdot )$ and w with respect to time t, making use of equations in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) and omitting terms of order higher than or equal to two, lead to the following system (40) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-gu(t)-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r{M}^0\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x))v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =-k_{0}w(t).\end{array}$(40) In order to derive the spectrum of the linear operator associated to the linearized system (40(40) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-gu(t)-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r{M}^0\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x))v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =-k_{0}w(t).\end{array}$(40) ), we consider exponential solutions of the form (41) $u(t)=\overline{u}{e}^{\lambda t},v(x,t)=\overline{v}(x)e^{\lambda t},w(t)=\overline{w}{e}^{\lambda t},$(41) where $\overline{u}$ and $\overline{w}$ are arbitrary constants, $\overline{v}(x)$ is an x-depending function and λ is a complex number. Substituting (41(41) $u(t)=\overline{u}{e}^{\lambda t},v(x,t)=\overline{v}(x)e^{\lambda t},w(t)=\overline{w}{e}^{\lambda t},$(41) ) into (40(40) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-gu(t)-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r{M}^0\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x))v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =-k_{0}w(t).\end{array}$(40) ) yields (42) $\begin{array}{rl}\lambda \overline{u}& =-g\overline{u}-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^0\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ \lambda & =-k_0.\end{array}$(42) One eigenvalue of (42(42) $\begin{array}{rl}\lambda \overline{u}& =-g\overline{u}-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^0\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ \lambda & =-k_0.\end{array}$(42) ) is $\lambda =-k_0<0$. The remaining eigenvalues will be derived later. Using the second equation in (42(42) $\begin{array}{rl}\lambda \overline{u}& =-g\overline{u}-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^0\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ \lambda & =-k_0.\end{array}$(42) ), we have (43) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y=r{M}^0\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)+(\lambda +m(x)+b(x))\overline{v}(x)$(43) and the first equation in (42(42) $\begin{array}{rl}\lambda \overline{u}& =-g\overline{u}-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^0\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ \lambda & =-k_0.\end{array}$(42) ) is equivalent to (44) $\begin{array}{rl}\lambda \overline{u}& =-g\overline{u}-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{x}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ & +2{\int }_0^{x_0}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y.\end{array}$(44) Substituting (43(43) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y=r{M}^0\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)+(\lambda +m(x)+b(x))\overline{v}(x)$(43) ) into (44(44) $\begin{array}{rl}\lambda \overline{u}& =-g\overline{u}-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{x}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ & +2{\int }_0^{x_0}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y.\end{array}$(44) ) reduces the latter to $\lambda =-(g+(r{M}^0/\overline{u}){\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x)<0$. It follows that the spectral bound of (42(42) $\begin{array}{rl}\lambda \overline{u}& =-g\overline{u}-r{M}^0{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^0\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ \lambda & =-k_0.\end{array}$(42) ) is negative and hence, the following result is stated.

Theorem 5.1

The disease-free steady state $E^0$ is stable.

Likewise, by considering the disease steady state $E^{\ast }$, we define the functions u, $v(x,\cdot )$ and w by $u(t)=M(t)-M^{\ast },v(x,t)=p(x,t)-p^{\ast }(x),w(t)=D(t)$ and we obtain the following system (45) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-(g+r{P}^{\ast })u(t)-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r{M}^{\ast }\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x))v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ & -ru(t)\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_{2}w(t)p^{\ast }(x)\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =-k_{0}w(t)+k_{1}w(t)P^{\ast }.\end{array}$(45) To derive the characteristic equations of the system (45(45) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-(g+r{P}^{\ast })u(t)-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r{M}^{\ast }\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x))v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ & -ru(t)\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_{2}w(t)p^{\ast }(x)\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =-k_{0}w(t)+k_{1}w(t)P^{\ast }.\end{array}$(45) ), the functions u, $v(x,\cdot )$ and w defined in (41(41) $u(t)=\overline{u}{e}^{\lambda t},v(x,t)=\overline{v}(x)e^{\lambda t},w(t)=\overline{w}{e}^{\lambda t},$(41) ) are used into (45(45) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-(g+r{P}^{\ast })u(t)-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r{M}^{\ast }\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x))v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ & -ru(t)\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_{2}w(t)p^{\ast }(x)\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =-k_{0}w(t)+k_{1}w(t)P^{\ast }.\end{array}$(45) ) so that (46) $\begin{array}{rl}\lambda \overline{u}& =-(g+r{P}^{\ast })\overline{u}-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^{\ast }\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_2\overline{w}{p}^{\ast }(x)\\ \lambda & =-k_0+k_1{P}^{\ast }.\end{array}$(46) One eigenvalue of (46(46) $\begin{array}{rl}\lambda \overline{u}& =-(g+r{P}^{\ast })\overline{u}-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^{\ast }\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_2\overline{w}{p}^{\ast }(x)\\ \lambda & =-k_0+k_1{P}^{\ast }.\end{array}$(46) ) is $\lambda =-k_0+k_1{P}^{\ast }$, which is negative if $P^{\ast }<k_0/k_1$ and positive if $P^{\ast }>k_0/k_1$. The remaining eigenvalues will be derived later. Using the second equation in (46(46) $\begin{array}{rl}\lambda \overline{u}& =-(g+r{P}^{\ast })\overline{u}-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^{\ast }\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_2\overline{w}{p}^{\ast }(x)\\ \lambda & =-k_0+k_1{P}^{\ast }.\end{array}$(46) ), we have (47) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y=r{M}^{\ast }\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)+(\lambda +m(x)+b(x))\overline{v}(x)+r\overline{u}\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)+k_2\overline{w}{p}^{\ast }(x)$(47) and the first equation in (46(46) $\begin{array}{rl}\lambda \overline{u}& =-(g+r{P}^{\ast })\overline{u}-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^{\ast }\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_2\overline{w}{p}^{\ast }(x)\\ \lambda & =-k_0+k_1{P}^{\ast }.\end{array}$(46) ) is equivalent to (48) $\begin{array}{rl}\lambda \overline{u}& =-(g+r{P}^{\ast })\overline{u}-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{x}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ & +2{\int }_0^{x_0}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y.\end{array}$(48) Substituting (47(47) $2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y=r{M}^{\ast }\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)+(\lambda +m(x)+b(x))\overline{v}(x)+r\overline{u}\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)+k_2\overline{w}{p}^{\ast }(x)$(47) ) into (48(48) $\begin{array}{rl}\lambda \overline{u}& =-(g+r{P}^{\ast })\overline{u}-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{x}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ & +2{\int }_0^{x_0}x{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y.\end{array}$(48) ) reduces the latter to $\lambda =-(g+r{P}^{\ast }+(r{M}^{\ast }/\overline{u}){\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x)<0$. It follows that the spectral bound of (46(46) $\begin{array}{rl}\lambda \overline{u}& =-(g+r{P}^{\ast })\overline{u}-r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r{M}^{\ast }\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x))\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)-k_2\overline{w}{p}^{\ast }(x)\\ \lambda & =-k_0+k_1{P}^{\ast }.\end{array}$(46) ) is negative if $P^{\ast }<k_0/k_1$. Hence, we have the following result.

Theorem 5.2

The disease steady state $E^{\ast }$ is locally asymptotically stable if $P^{\ast }<k_0/k_1$ and unstable if $P^{\ast }>k_0/k_1$.

In a similar way, by considering the coexistence steady state $\breve{E}$, we define the functions u, $v(x,\cdot )$ and w by $u(t)=M(t)-\breve{M},v(x,t)=p(x,t)-\breve{p}(x),w(t)=D(t)-\breve{D}$ and we obtain the following system (49) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-(g+r\frac{k_0}{k_1})u(t)-r\breve{M}{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r\breve{M}\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x)+k_2\breve{D})v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ & -ru(t)\frac{\mathrm{d}\breve{p}}{\mathrm{d}x}(x)-k_{2}w(t)\breve{p}(x)\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =k_1\breve{D}{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x.\end{array}$(49) To derive the characteristic equations of the system (49(49) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-(g+r\frac{k_0}{k_1})u(t)-r\breve{M}{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r\breve{M}\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x)+k_2\breve{D})v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ & -ru(t)\frac{\mathrm{d}\breve{p}}{\mathrm{d}x}(x)-k_{2}w(t)\breve{p}(x)\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =k_1\breve{D}{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x.\end{array}$(49) ), the functions u, $v(x,\cdot )$ and w defined in (41(41) $u(t)=\overline{u}{e}^{\lambda t},v(x,t)=\overline{v}(x)e^{\lambda t},w(t)=\overline{w}{e}^{\lambda t},$(41) ) are used into (49(49) $\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}t}(t)& =-(g+r\frac{k_0}{k_1})u(t)-r\breve{M}{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }v}{\mathrm{\partial }t}(x,t)& =-r\breve{M}\frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,t)-(m(x)+b(x)+k_2\breve{D})v(x,t)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)v(y,t)\mathrm{d}y\\ & -ru(t)\frac{\mathrm{d}\breve{p}}{\mathrm{d}x}(x)-k_{2}w(t)\breve{p}(x)\\ \frac{\mathrm{d}w}{\mathrm{d}t}(t)& =k_1\breve{D}{\int }_{x_0}^{\mathrm{\infty }}v(x,t)\mathrm{d}x.\end{array}$(49) ) so that (50) $\begin{array}{rl}\lambda \overline{u}& =-(g+r\frac{k_0}{k_1})\overline{u}-r\breve{M}{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r\breve{M}\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x)+k_2\breve{D})\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}\breve{p}}{\mathrm{d}x}(x)-k_2\overline{w}\breve{p}(x)\\ \lambda & =\frac{k_1}{\overline{w}}{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x.\end{array}$(50) From the last equation in (50(50) $\begin{array}{rl}\lambda \overline{u}& =-(g+r\frac{k_0}{k_1})\overline{u}-r\breve{M}{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r\breve{M}\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x)+k_2\breve{D})\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}\breve{p}}{\mathrm{d}x}(x)-k_2\overline{w}\breve{p}(x)\\ \lambda & =\frac{k_1}{\overline{w}}{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x.\end{array}$(50) ), it follows that the spectral bound of (50(50) $\begin{array}{rl}\lambda \overline{u}& =-(g+r\frac{k_0}{k_1})\overline{u}-r\breve{M}{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\mathrm{d}x\\ \lambda \overline{v}(x)& =-r\breve{M}\frac{\mathrm{d}\overline{v}}{\mathrm{d}x}(x)-(m(x)+b(x)+k_2\breve{D})\overline{v}(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)\overline{v}(y)\mathrm{d}y\\ & -r\overline{u}\frac{\mathrm{d}\breve{p}}{\mathrm{d}x}(x)-k_2\overline{w}\breve{p}(x)\\ \lambda & =\frac{k_1}{\overline{w}}{\int }_{x_0}^{\mathrm{\infty }}\overline{v}(x)\mathrm{d}x.\end{array}$(50) ) is positive and hence, and then we have the result below.

Theorem 5.3

The coexistence steady state $\breve{E}$ is unstable if it exists.

6. Global stability of equilibria

This section focuses on the global asymptotic stability analysis of the system (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) at the equilibria $E^0$ and $E^{\ast }$. This is achieved by means of appropriate Volterra-type functions of the form $G(X)=X-1-\ln X$ and the following result are obtained.

Theorem 6.1

The disease-free steady state $E^0$ is globally asymptotically stable.

Proof.

We consider the Lyapunov function $L^0(t)=L_1^0(t)+L_2^0(t)$, where $L_1^0(t)=M^{0}G(M(t)/M^0),L_2^0(t)=D(t)+Q(t)$. Differentiating $L_1^0(t)$, making use of the first equation in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) and $A=g{M}^0$ and changing the order of integration, we have (51) $\begin{array}{rl}\frac{\mathrm{d}{L}_1^0}{\mathrm{d}t}(t)& =g{M}^0(2-\frac{M^0}{M(t)}-\frac{M(t)}{M^0})+r{M}^0(1-\frac{M(t)}{M^0})P(t)\\ & +(1-\frac{M^0}{M(t)}){\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y.\end{array}$(51) We now differentiate $L_2^0(t)$, make use of the second and last equations in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) and change the order of integration so that (52) $\begin{array}{rl}\frac{\mathrm{d}{L}_2^0}{\mathrm{d}t}& =-k_{0}D+(k_{1}D+rM)P-{\int }_{x_0}^{\mathrm{\infty }}x(m(x)+b(x)+k_{2}D)p(x,\cdot )\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}xf(x,y)\mathrm{d}x)b(y)p(y,\cdot )\mathrm{d}y.\end{array}$(52) Combining (51(51) $\begin{array}{rl}\frac{\mathrm{d}{L}_1^0}{\mathrm{d}t}(t)& =g{M}^0(2-\frac{M^0}{M(t)}-\frac{M(t)}{M^0})+r{M}^0(1-\frac{M(t)}{M^0})P(t)\\ & +(1-\frac{M^0}{M(t)}){\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y.\end{array}$(51) ) and (52(52) $\begin{array}{rl}\frac{\mathrm{d}{L}_2^0}{\mathrm{d}t}& =-k_{0}D+(k_{1}D+rM)P-{\int }_{x_0}^{\mathrm{\infty }}x(m(x)+b(x)+k_{2}D)p(x,\cdot )\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}xf(x,y)\mathrm{d}x)b(y)p(y,\cdot )\mathrm{d}y.\end{array}$(52) ) and using (9(9) $2{\int }_0^{y}xf(x,y)dx=y,y>x_0,$(9) ), of $\mathbf{A}\mathbf{3}$, we get (53) $\begin{array}{rl}\frac{\mathrm{d}{L}^0}{\mathrm{d}t}& =-k_{0}D+g{M}^0(2-\frac{M^0}{M}-\frac{M}{M^0})+(r{M}^{0}P-{\int }_{x_0}^{\mathrm{\infty }}x(m(x)p(x,\cdot )\mathrm{d}x)\\ & +(k_{1}P-k_{2}Q)D-\frac{M^0}{M}{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)b(y)p(y,\cdot )\mathrm{d}y.\end{array}$(53) Clearly, the first, second and last terms in (53(53) $\begin{array}{rl}\frac{\mathrm{d}{L}^0}{\mathrm{d}t}& =-k_{0}D+g{M}^0(2-\frac{M^0}{M}-\frac{M}{M^0})+(r{M}^{0}P-{\int }_{x_0}^{\mathrm{\infty }}x(m(x)p(x,\cdot )\mathrm{d}x)\\ & +(k_{1}P-k_{2}Q)D-\frac{M^0}{M}{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)b(y)p(y,\cdot )\mathrm{d}y.\end{array}$(53) ) are negative. Using assumption $\mathbf{A}\mathbf{1}$ and condition (4(4) $P(t)<Q(t).$(4) ), we get (54) $(k_{1}P(t)-k_{2}Q(t))D(t)=k_{1}D(t)Q(t)(\frac{P(t)}{Q(t)}-1)<0$(54) and by assumption $\mathbf{A}\mathbf{2}$ and condition (4(4) $P(t)<Q(t).$(4) ), we obtain (55) $r{M}^{0}P(t)-{\int }_{x_0}^{\mathrm{\infty }}xm(x)p(x,t)\mathrm{d}x\le r{M}^{0}Q(t)(\frac{P(t)}{Q(t)}-1)<0.$(55) Hence, $\mathrm{d}{L}^0/\mathrm{d}t<0$ is satisfied for any $M(t),p(x,t),D(t)\in {\mathbb{R}}_{+}$ and $\mathrm{d}{L}^0/\mathrm{d}t=0$ for $(M(t),p(x,t),D(t))=(M^0,p^0(x),D^0)$. Furthermore, it can be verified that $\{(M(t),p(x,t),D(t)):{\dot{L}}^0(t)=0\}=\{E^0\}$. Therefore, it follows from Lasalle's Invariance Theorem [1, p. 200] that $E^0$ is globally asymptotically stable.

Theorem 6.2

The disease steady state $E^{\ast }$ is globally asymptotically stable if $P^{\ast }<k_0/k_1$.

Proof.

We consider the Lyapunov function $L^{\ast }(t)=L_1^{\ast }+L_2^{\ast }(t)$, where $L_1^{\ast }(t)=M^{\ast }G(M(t)/M^{\ast }),L_2^{\ast }(t)=D(t)+{\int }_{x_0}^{\mathrm{\infty }}{p}^{\ast }(x)G(p(x,t)/p^{\ast }(x))\mathrm{d}x$. Differentiating $L_1^{\ast }(t)$ and then making use of the first equation in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) and substituting A by the following expression $A=(g+r{P}^{\ast })M^{\ast }-2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x$ and changing the order of integration, we obtain $\begin{array}{rl}\frac{\mathrm{d}{L}_1^{\ast }}{\mathrm{d}t}& =-\frac{(M-M^{\ast }{)}^2}{M}(g+r{P}^{\ast })-rM{\int }_{x_0}^{\mathrm{\infty }}{p}^{\ast }(x)(1-\frac{M^{\ast }}{M})(1-\frac{p^{\ast }(x)}{p(x,\cdot )})\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M})(1-\frac{p^{\ast }(y)}{p(y,\cdot )})b(y)p(y,t)\mathrm{d}y\end{array}$ We now differentiate $L_2^{\ast }(t)$, make use of the second and the last equations in (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ) and $-r{M}^{\ast }(\mathrm{d}{p}^{\ast }/dx)(x)-(m(x)+b(x))p^{\ast }(x)+2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p^{\ast }(y)\mathrm{d}y=0$ and change the order of integration so that $\begin{array}{rl}\frac{\mathrm{d}{L}_2^{\ast }}{\mathrm{d}t}(t)=& -(k_0-k_1{P}^{\ast })D(t)-(k_2-k_1)D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & +r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})\frac{\mathrm{\partial }p}{\mathrm{\partial }x}(x,t)\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))p(x,t){(1-\frac{p^{\ast }(x)}{p(x,t)})}^2\mathrm{d}x\\ & +2{\int }_{x_0}^{\mathrm{\infty }}({\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$ Expanding terms in the latter equation and making use of conditions $p(x_0,t)=p^{\ast }(x_0)=0$ and $\underset{x\to \mathrm{\infty }}{lim}p(x,t)=\underset{x\to \mathrm{\infty }}{lim}{p}^{\ast }(x)=0$, we get (56) $\begin{array}{rl}\frac{\mathrm{d}{L}_2^{\ast }}{\mathrm{d}t}& =-k_1(\frac{k_0}{k_1}-P^{\ast })D-k_2(1-\frac{k_1}{k_2})D{\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,\cdot )})p(x,\cdot )\mathrm{d}x\\ & {\int }_{x_0}^{\mathrm{\infty }}(\frac{M}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,\cdot )})(-r{M}^{\ast }\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x))\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))p(x,\cdot ){(1-\frac{p^{\ast }(x)}{p(x,t)})}^2\mathrm{d}x\\ & +2{\int }_{x_0}^{\mathrm{\infty }}({\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,\cdot )})f(x,y)dx)(1-\frac{p^{\ast }(y)}{p(y,\cdot )})b(y)p(y,\cdot )\mathrm{d}y.\end{array}$(56) Using $-r{M}^{\ast }(\mathrm{d}{p}^{\ast }/\mathrm{d}x)(x)=(m(x)+b(x))p^{\ast }(x)-2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p^{\ast }(y)\mathrm{d}y$ in Equation (56(56) $\begin{array}{rl}\frac{\mathrm{d}{L}_2^{\ast }}{\mathrm{d}t}& =-k_1(\frac{k_0}{k_1}-P^{\ast })D-k_2(1-\frac{k_1}{k_2})D{\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,\cdot )})p(x,\cdot )\mathrm{d}x\\ & {\int }_{x_0}^{\mathrm{\infty }}(\frac{M}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,\cdot )})(-r{M}^{\ast }\frac{\mathrm{d}{p}^{\ast }}{\mathrm{d}x}(x))\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))p(x,\cdot ){(1-\frac{p^{\ast }(x)}{p(x,t)})}^2\mathrm{d}x\\ & +2{\int }_{x_0}^{\mathrm{\infty }}({\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,\cdot )})f(x,y)dx)(1-\frac{p^{\ast }(y)}{p(y,\cdot )})b(y)p(y,\cdot )\mathrm{d}y.\end{array}$(56) ) yields $\begin{array}{rl}\frac{\mathrm{d}{L}_2^{\ast }}{\mathrm{d}t}(t)& =-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)-k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))p(x,t){(1-\frac{p^{\ast }(x)}{p(x,t)})}^2\mathrm{d}x\\ & +2{\int }_{x_0}^{\mathrm{\infty }}({\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\end{array}$ and therefore, we obtain (57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) From the right-hand side in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ), we clearly see that the first term is negative, the second term is also negative since $P^{\ast }<k_0/k_1$ and the third term vanishes by assumption $\mathbf{A}\mathbf{1}$. Most of the remaining terms on the right-hand side in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ) will be expanded in the following manner. The fourth term in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ) becomes (58) $\begin{array}{rl}& -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =-rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})p^{\ast }(x)\mathrm{d}x\\ & +rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)}){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x.\end{array}$(58) The fifth term in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ) is expanded as (59) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =\frac{M(t)}{M^{\ast }}{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\ln p(x,t)-p(x,t))p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x,\end{array}$(59) where the first term on the right-hand side in (59(59) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =\frac{M(t)}{M^{\ast }}{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\ln p(x,t)-p(x,t))p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x,\end{array}$(59) ) is negative. Next, we transform the sixth term in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ) into (60) $\begin{array}{rl}& -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & =-\frac{M(t)}{M^{\ast }}{\int }_{x_0}^{\mathrm{\infty }}2(\ln p(x,t)+p(x,t)){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}2({\int }_{x_0}^y(\frac{M(t)}{M^{\ast }}p(x,t)-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y,\end{array}$(60) where the first term on the right-hand side in (60(60) $\begin{array}{rl}& -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & =-\frac{M(t)}{M^{\ast }}{\int }_{x_0}^{\mathrm{\infty }}2(\ln p(x,t)+p(x,t)){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}2({\int }_{x_0}^y(\frac{M(t)}{M^{\ast }}p(x,t)-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y,\end{array}$(60) ) is negative. We also expand the seventh term in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ) as (61) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & ={\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+2\frac{p^{\ast }(x)}{p(x,t)}-{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^2]p(x,t)\mathrm{d}x.\end{array}$(61) The eighth term in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ) yields (62) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(62) Finally, we transform the last term in (57(57) $\begin{array}{rl}\frac{\mathrm{d}{L}^{\ast }}{\mathrm{d}t}(t)=& -\frac{(M(t)-M^{\ast }{)}^2}{M(t)}(g+r{P}^{\ast })-k_1(\frac{k_0}{k_1}-P^{\ast })D(t)\\ & -k_2(1-\frac{k_1}{k_2})D(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x\\ & -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y.\end{array}$(57) ) into (63) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(63) By adding up the last terms in Equations (60(60) $\begin{array}{rl}& -2{\int }_{x_0}^{\mathrm{\infty }}(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)}){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & =-\frac{M(t)}{M^{\ast }}{\int }_{x_0}^{\mathrm{\infty }}2(\ln p(x,t)+p(x,t)){\int }_x^{\mathrm{\infty }}f(x,y)b(y)p^{\ast }(y)\mathrm{d}y\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}2({\int }_{x_0}^y(\frac{M(t)}{M^{\ast }}p(x,t)-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y,\end{array}$(60) ) and (63(63) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(63) ), it follows that (64) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}2({\int }_{x_0}^y(\frac{M(t)}{M^{\ast }}p(x,t)-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y\\ & =-2{\int }_{x_0}^{\mathrm{\infty }}({\int }_{x_0}^y(1-\frac{M(t)}{M^{\ast }}p(x,t))f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y,\end{array}$(64) which is negative provided that $p(x,t)<M^{\ast }/M(t)$ for any $t\ge 0$ and $x>x_0$. In the sequel, we shall investigate the sign of the remaing terms under the conditions $p(x,t)<M^{\ast }/M(t)<1$ and $1<p(x,t)<M^{\ast }/M(t)$.

First, we assume that $p(x,t)<M^{\ast }/M(t)<1$. We clearly see in Equation (58(58) $\begin{array}{rl}& -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =-rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})p^{\ast }(x)\mathrm{d}x\\ & +rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)}){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x.\end{array}$(58) ) that the first term on right-hand side is negative, while the second term can be expanded as (65) $\begin{array}{rl}& rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)}){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x\\ & =rM(t){\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x\\ & -r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x,\end{array}$(65) where the second term on the right-hand side is negative, and the first term on the right-hand side in Equation (63(63) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(63) ) can be expanded as (66) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y\frac{p^{\ast }(x)}{p(x,t)}f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y,\end{array}$(66) where the second term on the right-hand side is negative. Adding up the second term on the right hand side in (65(65) $\begin{array}{rl}& rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)}){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x\\ & =rM(t){\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x\\ & -r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x,\end{array}$(65) ) and the first term on the right-hand side in (66(66) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y\frac{p^{\ast }(x)}{p(x,t)}f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y,\end{array}$(66) ), we have $\begin{array}{rl}& -r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x+{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^{y}f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & \le {\int }_{x_0}^{\mathrm{\infty }}m(x)(1-\frac{\sqrt{r{M}^{\ast }}}{\sqrt{m(x)}}\frac{p^{\ast }(x)}{p(x,t)})(1+\frac{\sqrt{r{M}^{\ast }}}{\sqrt{m(x)}}\frac{p^{\ast }(x)}{p(x,t)})p(x,t)\mathrm{d}x<0,\end{array}$ provided that $1<\sqrt{m(x)}/\sqrt{r{M}^{\ast }}<p^{\ast }(x)/p(x,t)$ for any $t\ge 0$ and $x>x_0$. Moreover, the second term on the right-hand side in (59(59) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =\frac{M(t)}{M^{\ast }}{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\ln p(x,t)-p(x,t))p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x,\end{array}$(59) ), Equation (61(61) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & ={\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+2\frac{p^{\ast }(x)}{p(x,t)}-{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^2]p(x,t)\mathrm{d}x.\end{array}$(61) ), the first term on the right-hand side in (62(62) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(62) ) and the first term on the right-hand side in (65(65) $\begin{array}{rl}& rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)}){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x\\ & =rM(t){\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x\\ & -r{M}^{\ast }{\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x,\end{array}$(65) ) yields (67) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(-1+2\frac{p^{\ast }(x)}{p(x,t)}-{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^2)p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p(y,t)\mathrm{d}y\\ & +rM(t){\int }_{x_0}^{\mathrm{\infty }}{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x\\ & \le {\int }_{x_0}^{\mathrm{\infty }}[(m(x)+b(x))(3\frac{p^{\ast }(x)}{p(x,t)}-\frac{M^{\ast }}{M(t)})+rM(t){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^2]p(x,t)\mathrm{d}x,\end{array}$(67) since $p(x,t)<M^{\ast }/M(t)$. It is not difficult to show that $rM(t)(\frac{p^{\ast }(x)}{p(x,t)}{)}^2+3(m(x)+b(x))\frac{p^{\ast }(x)}{p(x,t)}-(m(x)+b(x))\frac{M^{\ast }}{M(t)}<0$ if $p^{\ast }(x)/p(x,t)<\sqrt{m(x)+b(x)}\sqrt{rM}/rM$, that is $p^{\ast }(x)/p(x,t)<\sqrt{m(x)+b(x)}$ for any $t\ge 0$ and $x>x_0$. Therefore $\sqrt{m(x)}/\sqrt{r{M}^{\ast }}<p^{\ast }(x)/p(x,t)<\sqrt{m(x)+b(x)}$ and hence, $\mathrm{d}{L}^{\ast }/\mathrm{d}t<0$ is satisfied for $(M(t),p(x,t),D(t))$ such that $p(x,t)<\frac{M^{\ast }}{M(t)}<1,1<\frac{\sqrt{m(x)}}{\sqrt{r{M}^{\ast }}}<\frac{p^{\ast }(x)}{p(x,t)}<\sqrt{m(x)+b(x)},D(t)\in {\mathbb{R}}_{+}$ and $\mathrm{d}{L}^{\ast }/\mathrm{d}t=0$ for $(M(t),p(x,t),D(t))=(M^{\ast },p^{\ast }(x),D^{\ast })$.

Finally, we assume that $1<p(x,t)<M^{\ast }/M(t)$. It follows that the second term on the right-hand side in (58(58) $\begin{array}{rl}& -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =-rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})p^{\ast }(x)\mathrm{d}x\\ & +rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)}){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x.\end{array}$(58) ) and the first term on the right-hand side in (62(62) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(62) ) are both negative, while the remaining terms on the right-hand side in (58(58) $\begin{array}{rl}& -rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =-rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})p^{\ast }(x)\mathrm{d}x\\ & +rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)}){\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^{2}p(x,t)\mathrm{d}x.\end{array}$(58) ) and (62(62) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(62) ) can be transformed, respectively, as follows (68) $-rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})p^{\ast }(x)\mathrm{d}x=-r{P}^{\ast }M(t)+{\int }_{x_0}^{\mathrm{\infty }}r{M}^{\ast }\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x$(68) and (69) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p^{\ast }(y)\mathrm{d}y\\ & \le -{\int }_{x_0}^{\mathrm{\infty }}yb(y)p^{\ast }(y)\mathrm{d}y+{\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y,\end{array}$(69) where the first terms on the right-hand side in (68(68) $-rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})p^{\ast }(x)\mathrm{d}x=-r{P}^{\ast }M(t)+{\int }_{x_0}^{\mathrm{\infty }}r{M}^{\ast }\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x$(68) ) and (69(69) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p^{\ast }(y)\mathrm{d}y\\ & \le -{\int }_{x_0}^{\mathrm{\infty }}yb(y)p^{\ast }(y)\mathrm{d}y+{\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y,\end{array}$(69) ) are negative. After using $\mathbf{A}\mathbf{2}$, the second terms in (68(68) $-rM(t){\int }_{x_0}^{\mathrm{\infty }}(1-\frac{M^{\ast }}{M(t)})p^{\ast }(x)\mathrm{d}x=-r{P}^{\ast }M(t)+{\int }_{x_0}^{\mathrm{\infty }}r{M}^{\ast }\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x$(68) ) and (69(69) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_0^{x_0}xf(x,y)\mathrm{d}x)(1-\frac{M^{\ast }}{M(t)})b(y)p^{\ast }(y)\mathrm{d}y\\ & \le -{\int }_{x_0}^{\mathrm{\infty }}yb(y)p^{\ast }(y)\mathrm{d}y+{\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y,\end{array}$(69) ) yield, respectively, (70) ${\int }_{x_0}^{\mathrm{\infty }}r{M}^{\ast }\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x<{\int }_{x_0}^{\mathrm{\infty }}m(x)\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x$(70) and (71) ${\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y<{\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}m(y)p^{\ast }(y)\mathrm{d}y.$(71) The first term on the right-hand side in (63(63) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)(1-\frac{p^{\ast }(y)}{p(y,t)})b(y)p(y,t)\mathrm{d}y\\ & ={\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & -{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p^{\ast }(y)\mathrm{d}y.\end{array}$(63) ) can also be transformed into (72) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y(1-\frac{p^{\ast }(x)}{p(x,t)})f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y\\ & \le {\int }_{x_0}^{\mathrm{\infty }}2b(y)p(y,t)\mathrm{d}y-{\int }_{x_0}^{\mathrm{\infty }}(2{\int }_{x_0}^y\left(\frac{p^{\ast }(x)}{p(x,t)}\right)f(x,y)\mathrm{d}x)b(y)p(y,t)\mathrm{d}y,\end{array}$(72) where the second term on the right-hand side is negative, while the first term on the right-hand side yields (73) ${\int }_{x_0}^{\mathrm{\infty }}2b(y)p(y,t)\mathrm{d}y<{\int }_{x_0}^{\mathrm{\infty }}m(y)p(y,t)\mathrm{d}y.$(73) From (71(71) ${\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y<{\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}m(y)p^{\ast }(y)\mathrm{d}y.$(71) ) and (73(73) ${\int }_{x_0}^{\mathrm{\infty }}2b(y)p(y,t)\mathrm{d}y<{\int }_{x_0}^{\mathrm{\infty }}m(y)p(y,t)\mathrm{d}y.$(73) ), we get (74) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y+{\int }_{x_0}^{\mathrm{\infty }}2b(y)p(y,t)\mathrm{d}y\\ & <{\int }_{x_0}^{\mathrm{\infty }}[\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2+\frac{1}{2}{\left(\frac{p^{\ast }(y)}{p(y,t)}\right)}^2]m(y)p(y,t)\mathrm{d}y.\end{array}$(74) Combining the second term on the right-hand side in (59(59) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}\ln p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & =\frac{M(t)}{M^{\ast }}{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\ln p(x,t)-p(x,t))p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x,\end{array}$(59) )–(61(61) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & ={\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+2\frac{p^{\ast }(x)}{p(x,t)}-{\left(\frac{p^{\ast }(x)}{p(x,t)}\right)}^2]p(x,t)\mathrm{d}x.\end{array}$(61) ), (70(70) ${\int }_{x_0}^{\mathrm{\infty }}r{M}^{\ast }\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x<{\int }_{x_0}^{\mathrm{\infty }}m(x)\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x$(70) ) and (74(74) $\begin{array}{rl}& {\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y+{\int }_{x_0}^{\mathrm{\infty }}2b(y)p(y,t)\mathrm{d}y\\ & <{\int }_{x_0}^{\mathrm{\infty }}[\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2+\frac{1}{2}{\left(\frac{p^{\ast }(y)}{p(y,t)}\right)}^2]m(y)p(y,t)\mathrm{d}y.\end{array}$(74) ), we obtain (75) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y+{\int }_{x_0}^{\mathrm{\infty }}2b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}r{M}^{\ast }\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x\\ & <{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+3\frac{p^{\ast }(x)}{p(x,t)}+\frac{M(t)}{M^{\ast }}{p}^{\ast }(x)+\frac{M^{\ast }}{M(t)}\\ & +\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2+\frac{1}{2}{\left(\frac{p^{\ast }(y)}{p(y,t)}\right)}^2]p(x,t)\mathrm{d}x\\ & <{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+4p^{\ast }(x)+\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2+\frac{1}{2}{\left(\frac{p^{\ast }(y)}{p(y,t)}\right)}^2]\\ & \times p(x,t)\mathrm{d}x,\\ & <{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+4p^{\ast }(x)+\frac{p^{\ast }(x)}{2}+\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2]p(x,t)\mathrm{d}x,\end{array}$(75) since $1<p(x,t)<M^{\ast }/M(t)$. Writing the integrand in (75(75) $\begin{array}{rl}& -{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x)){(1-\frac{p^{\ast }(x)}{p(x,t)})}^{2}p(x,t)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))(\frac{M(t)}{M^{\ast }}p(x,t)+\frac{p^{\ast }(x)}{p(x,t)})p^{\ast }(x)\mathrm{d}x\\ & +{\int }_{x_0}^{\mathrm{\infty }}\frac{M^{\ast }}{M(t)}yb(y)p^{\ast }(y)\mathrm{d}y+{\int }_{x_0}^{\mathrm{\infty }}2b(y)p(y,t)\mathrm{d}y\\ & +{\int }_{x_0}^{\mathrm{\infty }}r{M}^{\ast }\frac{p^{\ast }(x)}{p(x,t)}p(x,t)\mathrm{d}x\\ & <{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+3\frac{p^{\ast }(x)}{p(x,t)}+\frac{M(t)}{M^{\ast }}{p}^{\ast }(x)+\frac{M^{\ast }}{M(t)}\\ & +\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2+\frac{1}{2}{\left(\frac{p^{\ast }(y)}{p(y,t)}\right)}^2]p(x,t)\mathrm{d}x\\ & <{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+4p^{\ast }(x)+\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2+\frac{1}{2}{\left(\frac{p^{\ast }(y)}{p(y,t)}\right)}^2]\\ & \times p(x,t)\mathrm{d}x,\\ & <{\int }_{x_0}^{\mathrm{\infty }}(m(x)+b(x))[-1+4p^{\ast }(x)+\frac{p^{\ast }(x)}{2}+\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2]p(x,t)\mathrm{d}x,\end{array}$(75) ) as (76) $\begin{array}{rl}& -1+4p^{\ast }(x)+\frac{p^{\ast }(x)}{2}+\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2\\ & =\frac{1}{2}[{(p^{\ast }(x)+4)}^2+{(\frac{M^{\ast }}{M(t)}+1)}^2-19],\end{array}$(76) it is not difficult to see that (76(76) $\begin{array}{rl}& -1+4p^{\ast }(x)+\frac{p^{\ast }(x)}{2}+\frac{M^{\ast }}{M(t)}+\frac{1}{2}{\left(\frac{M^{\ast }}{M(t)}\right)}^2\\ & =\frac{1}{2}[{(p^{\ast }(x)+4)}^2+{(\frac{M^{\ast }}{M(t)}+1)}^2-19],\end{array}$(76) ) is negative provided that (77) ${(p^{\ast }(x)+4)}^2+{(\frac{M^{\ast }}{M(t)}+1)}^2<{\left(\sqrt{19}\right)}^2,$(77) for any $t\ge 0$ and $x>x_0$, that is $p^{\ast }(x)+4<\sqrt{19}$ and $M^{\ast }/M(t)+1<\sqrt{19}$, and then (78) $p^{\ast }(x)<\frac{9}{25}<1\mathrm{and}\frac{M^{\ast }}{M(t)}<\frac{17}{5}<4.$(78) Since $1<p(x,t)<M^{\ast }/M(t)$, thus $0<p^{\ast }x)/p(x,t)<1$ and $1<M^{\ast }/M(t)<4$. Hence, $\mathrm{d}{L}^{\ast }/\mathrm{d}t<0$ is satisfied for $(M(t),p(x,t),D(t))$ such that $1<p(x,t)<\frac{M^{\ast }}{M(t)}<\frac{17}{5}<4,0<\frac{p^{\ast }x)}{p(x,t)}<\frac{9}{25}<1,D(t)\in {\mathbb{R}}_{+}$ and $\mathrm{d}{L}^{\ast }/\mathrm{d}t=0$ for $(M(t),p(x,t),D(t))=(M^{\ast },p^{\ast }(x),D^{\ast })$.

In summary, $d{L}^{\ast }/dt<0$ if $(M(t),p(x,t),D(t))$ satisfies either of the following cases:

1. $p(x,t)<M^{\ast }/M(t)<1,1<\sqrt{m(x)}/\sqrt{r{M}^{\ast }}<p^{\ast }(x)/p(x,t)<\sqrt{m(x)+b(x)},D(t)\in {\mathbb{R}}_{+}$

2. $1<p(x,t)<M^{\ast }/M(t)<\frac{17}{5}<4,0<p^{\ast }x)/p(x,t)<\frac{9}{25}<1,D(t)\in {\mathbb{R}}_{+}$

and $\mathrm{d}{L}^{\ast }/\mathrm{d}t=0$ for $(M(t),p(x,t),D(t))=(M^{\ast },p^{\ast }(x),D^{\ast })$. Furthermore, it can be verified that $\{(M(t),p(x,t),D(t)):{\dot{L}}^{\ast }(t)=0\}=\{E^{\ast }\}$. Therefore, it follows from Lasalle's Invariance Theorem [1, p. 200] that $E^{\ast }$ is globally asymptotically stable if $P^{\ast }<k_0/k_1$.

7. Discussions

Let ${\mathfrak{C}}_0$ be a parameter defined by ${\mathfrak{C}}_0=(k_1/k_0)P^{\ast }$. Since, from the results of Theorems 5.2 and 6.2, the disease state equilibrium $E^{\ast }$ is stable if ${\mathfrak{C}}_0<1$ and unstable if ${\mathfrak{C}}_0>1$ and the (unstable) coexistence state equilibrium $\breve{E}$ exists if ${\mathfrak{C}}_0=1$, thus ${\mathfrak{C}}_0$ can be regarded as an essential threshold parameter in the steady state and stability analysis for the dynamics of amyloid fibril formation defined by the system of equations (1(1) $\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}M(t)& =A-gM(t)-rM(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x+2{\int }_0^{x_0}x{\int }_{x_0}^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\mathrm{d}x\\ \frac{\mathrm{\partial }}{\mathrm{\partial }t}p(x,t)& =-rM(t)\frac{\mathrm{\partial }}{\mathrm{\partial }x}p(x,t)-(m(x)+b(x)+k_{2}D(t))p(x,t)\\ & +2{\int }_x^{\mathrm{\infty }}b(y)f(x,y)p(y,t)\mathrm{d}y\\ \frac{\mathrm{d}}{\mathrm{d}t}D(t)& =-k_{0}D(t)+k_{1}D(t){\int }_{x_0}^{\mathrm{\infty }}p(x,t)\mathrm{d}x,\end{array}$(1) ). Often, such a parameter can play a crucial role in implementing therapeutic interventions strategies. Clearly, ${\mathfrak{C}}_0\to 0$ as $P^{\ast }\to 0$, that is the development of T2DM disease is stable if the total population of polymers, $P^{\ast }$, is unsignificant, and ${\mathfrak{C}}_0\to \mathrm{\infty }$ as $P^{\ast }\to \mathrm{\infty }$, that is the development of T2DM is unstable if the total population of polymers, $P^{\ast }$, is extremely high. Moreover, ${\mathfrak{C}}_0\to \mathrm{\infty }$ as $k_0\to 0$, that is the development of T2DM disease cannot stabilize if $k_0$ is negligible, and ${\mathfrak{C}}_0\to 0$ as $k_0\to \mathrm{\infty }$, that is development of T2DM disease stabilizes if $k_0$ is extremely large. However, the following $\frac{\mathrm{\partial }{\mathfrak{C}}_0}{\mathrm{\partial }{k}_0}=-\frac{k_1}{k_0^2}{P}^{\ast }=-\frac{{\mathfrak{C}}_0}{k_0}<0$ shows that increasing the rate at which the drug is degraded from the system due to metabolic processes, $k_0$, result in a beneficial impact on the control of amyloid fibril formation, that is, rising of $k_0$ brakes the development of T2DM disease. Therefore, $k_0$ is substantial parameter in elaborating an efficient therapeutic strategy aiming at controlling and stopping the development of T2DM disease.

Disclosure statement

No potential conflict of interest was reported by the author(s).