Global stability analysis for a model with carriers and non-linear incidence rate

Abstract

We analysed a epidemiological model with varying populations of susceptible, carriers, infectious and recovered (SCIR) and a general non-linear incidence rate of the form $f(S)[g(C)+h(I)]$. We show that this model exhibits two positive equilibriums: the disease-free and disease equilibrium. We proved using the Lyapunov direct method that these two equilibriums are globally asymptotically stable under some sufficient conditions over the functions f, g, h.

Keywords:

• Non-linear incidence rate
• carriers
• global stability
• population dynamics
• asymptomatic

2010 Mathematics Subject Classification:

• 34D23

1. Introduction

Non-linear incidence rate has been used in mathematical models describing population dynamics of infectious diseases. For example, in [16], it is used a Lyapunov functions to demonstrate global stability of the SEIRS and SIRS models with non-linear incidence rate of the form $h(S)g(I)$. In [22], it is used the function $\beta f(S)g(I)$ for an SIRS model with vaccination in susceptible population. In [14], it is considered a non-linear function of the form $f(S,I)$ for the SIR and SIRS models and in [15], for the SEIR model with a constant population. In [1], an SIRS model with varying population and transfer from infectious to susceptible is studied with incidence rate $f(S,I)$. In [8], a SVEIR model with non-linear incidence rate is studied to get the global stability. It should be noted, though, that those works do not include the infectious state of persons who are infected but do not show symptoms, which are frequently called carriers or asymptomatic carriers, they are potential spreader of the disease and play a fundamental role in the transmission of disease. For example, in Hepatitis B infection, the 90% of infected infants are asymptomatic [11] and since their daily routines are unchanged, they can disseminate the disease and in the influenza disease, the asymptomatic carriers children can be above 66% and probably becoming a silent transmitter of influenza [9]. In the Norovirus infection, which is involved in the acute phase of gastroenteritis, about 32% in Norovirus infection is asymptomatic and they can be potential transmitters of the disease [21]. This work is based on the following model (1) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-S[\beta C+\gamma I]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pS[\beta C+\gamma I]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)S[\beta C+\gamma I]-(d_3+\pi )I+\alpha C\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I-d_{4}R,\end{array}$(1) where S is the susceptible, C is the asymptomatic carrier, I is the infected and R the recovered. The model (1(1) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-S[\beta C+\gamma I]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pS[\beta C+\gamma I]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)S[\beta C+\gamma I]-(d_3+\pi )I+\alpha C\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I-d_{4}R,\end{array}$(1) ), published in [12], was proposed to investigate the infectious diseases that can be transmitted by carriers and was obtained the global stability of disease-free and disease equilibrium, using the bilinear incidence rate $S[\beta C+\gamma I]$ for infected and carriers. Inspired by this, we propose a mathematical model describe by the next systems of ordinary differential equations: (2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) The infection process can occur when susceptible persons come into contact with infected individuals or asymptomatic carriers. We describe this process with the non-linear incidence rate $f(S)[g(C)+h(I)]$, where $f(S)$ is the contact function at concentration S, $h(I)$ is the force of infection by infected individuals at concentration I and $g(C)$ the force of infection by carriers at concentration C. A newly infected individual can become a carrier with probability p or shows disease symptoms with probability 1−p, ( $0\le p<1$). The parameters b, θ and $d_1$ are the recruitment rate, the vaccination rate and natural death rate of susceptibles, respectively. Besides, $d_2,d_3,d_4$ are the death rates of asymptomatic carriers, infected and recovered, respectively, while π is the recovery rate of infected and α is the recovery rate of asymptomatic carriers. The model parameters of system (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ) are described in Table 1.

Table 1. Parameters of the model (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ).

Symbol	Meaning
b	The recruitment rate of susceptibles.
$d_1$	Natural death rate.
θ	Vaccination rate.
$d_2,d_3,d_4$	Death rates for C, I and R compartments, respectively.
$\alpha ,\pi$	Natural recovery rate of C and I population, respectively.
p	The probability of a infected person become asymptomatic carrier.

There are a variety of models considering asymptomatic carriers, for example: The models that consider that asymptomatic carriers show symptoms after some time as in [12,19] or models that after got infected some proportion could become asymptomatic carriers as in [2,5,7,17,20], or like in our model, where we do not make any of this assumption (i.e. transition of asymptomatic carriers to infected or from infected to asymptomatic carriers) as in [3,6,10,13,23,24], these particular models were used to describe or analyse mostly the dynamical transmission of influenza or transmission with super-spreaders, for another models considering asymptomatic carriers see [4]. Besides, the model (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ) consider the non-linear incidence rate for the following reasons. First, permits to incorporate a population structure and heterogeneous mixing (in the traditional bilinear incidence rate $S[\beta C+\gamma I]$ homogeneous mixing is implicit, which is not always true). Second, the possibility of incorporate a saturation effects with respect to number of infectious individuals [16,18]. Finally, the general incidence rate allows to have different scenarios for a wide range of infection forces for both infected and asymptomatic.

The aim of this paper is to prove the global stability of equilibriums of the model (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ), for this purpose, we follow the ideas of the paper [16], which use a Lyapunov direct method and propose a Lyapunov function for non-linear incidence rate $h(S)g(I)$ to study the stability of the SEIRS and SIRS models.

2. Equilibrium points

We omit the equation for recovery state R because it does not appear in the other equations, as was done in [12]. In this way, we can study the model in the next positively invariant region $\mathrm{\Omega }=\{(S,C.I)\in {\mathbb{R}}_{+}^3:S\le b/(d_1+\theta ),S+C+I\le b/\overline{d}\},$ where $\overline{d}=\min \{d_1,d_2,d_3,d_4\}$. To study the dynamics of the model (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ), we enumerate some conditions over the functions $f(S),g(C),h(I)$, as follows:

Let f, g, h be a continuously differentiable functions in ${\mathbb{R}}_{+}$ and

1. $f(S)>0$, $g(C)>0$, $h(I)>0$ for all S, C, I>0, respectively.

2. $f(0)=0$, $g(0)=0$ and $h(0)=0$.

3. f, g and h are non-decreasing functions in $[0,+\mathrm{\infty }]$.

4. $\frac{g(C)}{C}$ and $\frac{h(I)}{I}$ are non-increasing functions in $(0,+\mathrm{\infty })$.

2.1. Existence of equilibrium points

Proposition 2.1

The system (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ), has a disease-free equilibrium (DFE) $(S_0,0,0)$, where $S_0=\frac{b}{d_1+\theta }$ and if $R_0=\frac{p}{d_2+\alpha }f(S_0)g^{\prime }(0)+\frac{(1-p)}{d_3+\pi }f(S_0)h^{\prime }(0)>1$, there exists a disease equilibrium point (DE) $(S^{\ast },C^{\ast },I^{\ast })$.

Proof.

Initially, we find the solutions of the equations (3) $\begin{array}{rl}b-\theta S-f(S)[g(C)+h(I)]-d_{1}S& =0,\end{array}$(3) (4) $\begin{array}{rl}pf(S)[g(C)+h(I)]-(d_2+\alpha )C& =0,\end{array}$(4) (5) $\begin{array}{rl}(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I& =0.\end{array}$(5) From these equations, we have (6) $f(S)[g(C)+h(I)]=b-(\theta +d_1)S=\frac{(d_2+\alpha )C}{p}=\frac{(d_3+\pi )I}{(1-p)}.$(6) Then, we can write C and I as functions of S, as follows: $C=\frac{[b-(\theta +d_1)S]p}{(d_2+\alpha )}$ and $I=\frac{[b-(\theta +d_1)S](1-p)}{(d_3+\pi )}$, and to find the solutions of Equations (3(3) $\begin{array}{rl}b-\theta S-f(S)[g(C)+h(I)]-d_{1}S& =0,\end{array}$(3) )–(5(5) $\begin{array}{rl}(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I& =0.\end{array}$(5) ), it is enough to solve the equation (7) $b-(\theta +d_1)S-f(S)[g(C)+h(I)]=0,$(7) which just depends on S. To solve this equation, let $\phi (S)$ be a function defined by $\phi (S)=b-(\theta +d_1)S-f(S)[g(C)+h(I)].$ Clearly, $\phi (S_0)=0$, then a first solutions of Equation (7(7) $b-(\theta +d_1)S-f(S)[g(C)+h(I)]=0,$(7) ) is $(S_0,0,0)$, this point will be called disease-free equilibrium (DFE). Besides, we have that $\phi (0)=b$. Now, to show that Equation (7(7) $b-(\theta +d_1)S-f(S)[g(C)+h(I)]=0,$(7) ) has a solution in the interval $(0,S_0)$, it is enough to show that ${\phi }^{\prime }(S_0)>0$. The derivative of ϕ is ${\phi }^{\prime }(S)=-(\theta +d_1)-f^{\prime }(S)[g(C)+h(I)]-f(S)[g^{\prime }(C)\frac{\mathrm{d}C}{\mathrm{d}S}+h^{\prime }(I)\frac{\mathrm{d}I}{\mathrm{d}S}],$ and its value at $S=S_0$ is $\begin{array}{rl}{\phi }^{\prime }(S_0)& =-(\theta +d_1)-f(S_0)[g^{\prime }(0)\frac{[-(\theta +d_1)]p}{(d_2+\alpha )}+h^{\prime }(0)\frac{[-(\theta +d_1)](1-p)}{(d_3+\pi )}]\\ {\phi }^{\prime }(S_0)& =(\theta +d_1)[\frac{p}{(d_2+\alpha )}f(S_0)g^{\prime }(0)+\frac{(1-p)}{(d_3+\pi )}f(S_0)h^{\prime }(0)-1]\\ {\phi }^{\prime }(S_0)& =(\theta +d_1)(R_0-1).\end{array}$ Which implies that ${\phi }^{\prime }(S_0)>0$, if $R_0>1$. Then there exists at least a solution $S^{\ast }$ of Equation (7(7) $b-(\theta +d_1)S-f(S)[g(C)+h(I)]=0,$(7) ) in the interval $(0,S_0)$ and therefore a equilibrium point $(S^{\ast },C^{\ast },I^{\ast })$ which we call disease equilibrium point (DE).

3. Global stability of the equilibrium points

Theorem 3.1

The DFE $(S_0,0,0)$ is globally asymptotically stable if $R_0<1$.

Proof.

We follow the approach used in [16], therefore, let $L_1$ be a function defined by $L_1=k_1(S-f(S_0){\int }_a^S\frac{\mathrm{d}\tau }{f(\tau )})+k_{2}C+k_{3}I,$ where $k_1=\frac{g^{\prime }(0)p}{d_2+\alpha }+\frac{h^{\prime }(0)(1-p)}{d_3+\pi }$, $k_2=\frac{g^{\prime }(0)}{d_2+\alpha }$, $k_3=\frac{h^{\prime }(0)}{d_3+\pi }$ and a is a positive constant that can be sufficiently small. The orbital derivative of this function is ${\dot{L}}_1=k_1(1-\frac{f(S_0)}{f(S)})\dot{S}+k_2\dot{C}+k_3\dot{I},$ from the system (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ) we have $\begin{array}{rl}{\dot{L}}_1& =k_{1}b-k_1(d_1+\theta )S-k_{1}f(S)[g(C)+h(I)]-k_{1}b\frac{f(S_0)}{f(S)}+k_1(d_1+\theta )S\frac{f(S_0)}{f(S)}\\ & +k_{1}f(S_0)[g(C)+h(I)]+k_{2}pf(S)[g(C)+h(I)]-k_2(d_2+\alpha )C\\ & +k_3(1-p)f(S)[g(C)+h(I)]-k_3(d_3+\pi )I,\end{array}$ and from the equations of $k_1,k_2$ and $k_3$, we obtain $k_1=k_{2}p+k_3(1-p)$ and $\begin{array}{rl}{\dot{L}}_1& =k_{1}b-k_1(d_1+\theta )S-k_{1}b\frac{f(S_0)}{f(S)}+k_1(d_1+\theta )S\frac{f(S_0)}{f(S)}\\ & +k_{1}f(S_0)[g(C)+h(I)]-g^{\prime }(0)C-h^{\prime }(0)I,\\ {\dot{L}}_1& =k_1(d_1+\theta )(S_0-S)-k_1(d_1+\theta )\frac{f(S_0)}{f(S)}(S_0-S)\\ & +k_{1}f(S_0)g(C)-g^{\prime }(0)C+k_{1}f(S_0)h(I)-h^{\prime }(0)I,\\ {\dot{L}}_1& =k_1(d_1+\theta )(S_0-S)(1-\frac{f(S_0)}{f(S)})+g^{\prime }(0)C(\frac{k_1}{g^{\prime }(0)}f(S_0)\frac{g(C)}{C}-1)\\ & +h^{\prime }(0)I(\frac{k_1}{h^{\prime }(0)}f(S_0)\frac{h(I)}{I}-1),\end{array}$ since the functions $\frac{g(C)}{C}$ and $\frac{h(I)}{I}$ are non-increasing (by condition (IV)) then $\frac{g(C)}{C}\le \underset{C\to 0^{+}}{lim}\frac{g(C)}{C}=g^{\prime }(0)$, and $\frac{h(I)}{I}\le \underset{I\to 0^{+}}{lim}\frac{h(I)}{I}=h^{\prime }(0)$ and $k_{1}f(S_0)=R_0$, therefore, ${\dot{L}}_1\le k_1(d_1+\theta )(S_0-S)(1-\frac{f(S_0)}{f(S)})+g^{\prime }(0)C(R_0-1)+h^{\prime }(0)I(R_0-1).$ The function f is non-decreasing (by condition (III)) then $(S_0-S)(1-\frac{f(S_0)}{f(S)})\le 0$ for all S>0, therefore, ${\dot{L}}_1\le 0$ if $R_0<1$, then the DFE equilibrium is globally asymptotically stable.

Theorem 3.2

The DE equilibrium is globally asymptotically stable if $R_0>1$.

Proof.

Let $L_2$ be a function defined by $L_2=a_1(S-f(S^{\ast }){\int }_a^S\frac{\mathrm{d}\tau }{f(\tau )})+a_2(C-g(C^{\ast }){\int }_a^C\frac{\mathrm{d}\tau }{g(\tau )})+(I-h(I^{\ast }){\int }_a^I\frac{\mathrm{d}\tau }{h(\tau )}),$ where $a_1=(1-p)\frac{g(C^{\ast })+h(I^{\ast })}{h(I^{\ast })}$ and $a_2=\frac{1-p}{p}\frac{g(C^{\ast })}{h(I^{\ast })}$. The orbital derivative with respect to the system (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ), is $\begin{array}{rl}{\dot{L}}_2& =a_1(1-\frac{f(S^{\ast })}{f(S)})\dot{S}+a_2(1-\frac{g(C^{\ast })}{g(C)})\dot{C}+(1-\frac{h(I^{\ast })}{h(I)})\dot{I}\\ {\dot{L}}_2& =a_{1}b-a_1(d_1+\theta )S-a_{1}f(S)[g(C)+h(I)]-a_{1}b\frac{f(S^{\ast })}{f(S)}+a_1(d_1+\theta )S\frac{f(S^{\ast })}{f(S)}\\ & +a_{1}f(S^{\ast })[g(C)+h(I)]+a_{2}pf(S)[g(C)+h(I)]-a_2(d_2+\alpha )C-a_{2}pf(S)g(C^{\ast })\\ & -a_{2}pf(S)h(I)\frac{g(C^{\ast })}{g(C)}+a_2(d_2+\alpha )C\frac{g(C^{\ast })}{g(C)}+(1-p)f(S)[g(C)+h(I)]\\ & -(d_3+\pi )I-(1-p)f(S)g(C)\frac{h(I^{\ast })}{h(I)}-(1-p)f(S)h(I^{\ast })+(d_3+\pi )I\frac{h(I^{\ast })}{h(I)}.\end{array}$ From the equilibrium Equation (3(3) $\begin{array}{rl}b-\theta S-f(S)[g(C)+h(I)]-d_{1}S& =0,\end{array}$(3) ), we have $b=(d_1+\theta )S^{\ast }+f(S^{\ast })[g(C^{\ast })+h(I^{\ast })]$, and the values of $a_1,a_2$ satisfy $a_1=a_{2}p+(1-p)$, then $\begin{array}{rl}{\dot{L}}_2& =a_1(d_1+\theta )S^{\ast }+a_{1}f(S^{\ast })[g(C^{\ast })+h(I^{\ast })]-a_1(d_1+\theta )S-a_1(d_1+\theta )S^{\ast }\frac{f(S^{\ast })}{f(S)}\\ & -a_1\frac{f^2(S^{\ast })}{f(S)}[g(C^{\ast })+h(I^{\ast })]+a_1(d_1+\theta )S\frac{f(S^{\ast })}{f(S)}+a_{1}f(S^{\ast })g(C)+a_{1}f(S^{\ast })h(I)\\ & -a_2(d_2+\alpha )C-a_{2}pf(S)g(C^{\ast })-a_{2}pf(S)h(I)\frac{g(C^{\ast })}{g(C)}+a_2(d_2+\alpha )C\frac{g(C^{\ast })}{g(C)}\\ & -(d_3+\pi )I-(1-p)f(S)g(C)\frac{h(I^{\ast })}{h(I)}-(1-p)f(S)h(I^{\ast })+(d_3+\pi )I\frac{h(I^{\ast })}{h(I)},\end{array}$ grouping some terms, we have $\begin{array}{rl}{\dot{L}}_2& =a_1(d_1+\theta )S^{\ast }(1-\frac{S}{S^{\ast }}-\frac{f(S^{\ast })}{f(S)}+\frac{S}{S^{\ast }}\frac{f(S^{\ast })}{f(S)})-a_1\frac{f^2(S^{\ast })}{f(S)}[g(C^{\ast })+h(I^{\ast })]\\ & -a_{2}pf(S)g(C^{\ast })-a_{2}pf(S)h(I)\frac{g(C^{\ast })}{g(C)}-(1-p)f(S)g(C)\frac{h(I^{\ast })}{h(I)}\\ & -(1-p)f(S)h(I^{\ast })+a_{1}f(S^{\ast })[g(C^{\ast })+h(I^{\ast })]\\ & +[a_{1}f(S^{\ast })h(I)-(d_3+\pi )I+(d_3+\pi )I\frac{h(I^{\ast })}{h(I)}]\\ & +[a_{1}f(S^{\ast })g(C)-a_2(d_2+\alpha )C+a_2(d_2+\alpha )C\frac{g(C^{\ast })}{g(C)}],\end{array}$ adding and subtracting the term $a_{1}f(S^{\ast })[g(C^{\ast })+h(I^{\ast })]$, we obtain $\begin{array}{rl}{\dot{L}}_2& =a_1(d_1+\theta )S^{\ast }(1-\frac{S}{S^{\ast }})(1-\frac{f(S^{\ast })}{f(S)})-a_1\frac{f^2(S^{\ast })}{f(S)}[g(C^{\ast })+h(I^{\ast })]\\ & -a_{2}pf(S)g(C^{\ast })-a_{2}pf(S)h(I)\frac{g(C^{\ast })}{g(C)}-(1-p)f(S)g(C)\frac{h(I^{\ast })}{h(I)}\\ & -(1-p)f(S)h(I^{\ast })+2a_{1}f(S^{\ast })[g(C^{\ast })+h(I^{\ast })]+V_1+V_2,\end{array}$ where $V_1=a_{1}f(S^{\ast })h(I)-(d_3+\pi )I+(d_3+\pi )I\frac{h(I^{\ast })}{h(I)}-a_{1}f(S^{\ast })h(I^{\ast })$ and $V_2=a_{1}f(S^{\ast })g(C)-a_2(d_2+\alpha )C+a_2(d_2+\alpha )C\frac{g(C^{\ast })}{g(C)}-a_{1}f(S^{\ast })g(C^{\ast })$, these two expressions can be written as (8) $\begin{array}{rl}{V}_1& =a_{1}f(S^{\ast })h(I^{\ast })(\frac{h(I)}{h(I^{\ast })}-\frac{I}{I^{\ast }})(1-\frac{h(I^{\ast })}{h(I)})\end{array}$(8) (9) $\begin{array}{rl}{V}_2& =a_{1}f(S^{\ast })g(C^{\ast })(\frac{g(C)}{g(C^{\ast })}-\frac{C}{C^{\ast }})(1-\frac{g(C^{\ast })}{g(C)}).\end{array}$(9) In fact, the left side of Equation (8(8) $\begin{array}{rl}{V}_1& =a_{1}f(S^{\ast })h(I^{\ast })(\frac{h(I)}{h(I^{\ast })}-\frac{I}{I^{\ast }})(1-\frac{h(I^{\ast })}{h(I)})\end{array}$(8) ) is equivalent to $\begin{array}{rl}& a_{1}f(S^{\ast })h(I)(1-\frac{h(I^{\ast })}{h(I)})-(d_3+\pi )I(1-\frac{h(I^{\ast })}{h(I)})\\ & =a_{1}f(S^{\ast })(h(I)-\frac{(d_3+\pi )I}{a_{1}f(S^{\ast })})(1-\frac{h(I^{\ast })}{h(I)})\end{array}$ from the equilibrium Equation (5(5) $\begin{array}{rl}(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I& =0.\end{array}$(5) ), we have $(d_3+\pi )=(1-p)\frac{f(S^{\ast })}{I^{\ast }}[g(C^{\ast })+h(I^{\ast })]$, it implies that $\frac{(d_3+\pi )}{a_{1}f(S^{\ast })}=\frac{h(I^{\ast })}{I^{\ast }}$, so (10) $\begin{array}{r}{a}_{1}f(S^{\ast })(h(I)-\frac{(d_3+\pi )I}{a_{1}f(S^{\ast })})(1-\frac{h(I^{\ast })}{h(I)})=a_{1}f(S^{\ast })h(I^{\ast })(\frac{h(I)}{h(I^{\ast })}-\frac{I}{I^{\ast }})(1-\frac{h(I^{\ast })}{h(I)});\end{array}$(10) similarly, we can proof the Equation (9(9) $\begin{array}{rl}{V}_2& =a_{1}f(S^{\ast })g(C^{\ast })(\frac{g(C)}{g(C^{\ast })}-\frac{C}{C^{\ast }})(1-\frac{g(C^{\ast })}{g(C)}).\end{array}$(9) ), then ${\dot{L}}_2$ is $\begin{array}{rl}{\dot{L}}_2& =a_1(d_1+\theta )S^{\ast }(1-\frac{S}{S^{\ast }})(1-\frac{f(S^{\ast })}{f(S)})\\ & +[-a_1\frac{f^2(S^{\ast })}{f(S)}[g(C^{\ast })+h(I^{\ast })]-a_{2}pf(S)g(C^{\ast })-a_{2}pf(S)h(I)\frac{g(C^{\ast })}{g(C)}\\ & -(1-p)f(S)g(C)\frac{h(I^{\ast })}{h(I)}-(1-p)f(S)h(I^{\ast })]+2a_{1}f(S^{\ast })[g(C^{\ast })+h(I^{\ast })]\\ & +a_{1}f(S^{\ast })h(I^{\ast })(\frac{h(I)}{h(I^{\ast })}-\frac{I}{I^{\ast }})(1-\frac{h(I^{\ast })}{h(I)})\\ & +a_{1}f(S^{\ast })g(C^{\ast })(\frac{g(C)}{g(C^{\ast })}-\frac{C}{C^{\ast }})(1-\frac{g(C^{\ast })}{g(C)}),\end{array}$ from $a_1=a_{2}p+(1-p)$, we have $\begin{array}{rl}{\dot{L}}_2& =a_1(d_1+\theta )S^{\ast }(1-\frac{S}{S^{\ast }})(1-\frac{f(S^{\ast })}{f(S)})\\ & +[-a_{2}p\frac{f^2(S^{\ast })}{f(S)}g(C^{\ast })-a_{2}p\frac{f^2(S^{\ast })}{f(S)}h(I^{\ast })-(1-p)\frac{f^2(S^{\ast })}{f(S)}g(C^{\ast })\\ & -(1-p)\frac{f^2(S^{\ast })}{f(S)}h(I^{\ast })-a_{2}pf(S)g(C^{\ast })-a_{2}pf(S)h(I)\frac{g(C^{\ast })}{g(C)}\\ & -(1-p)f(S)g(C)\frac{h(I^{\ast })}{h(I)}-(1-p)f(S)h(I^{\ast })\phantom{\frac{f^2(S^{\ast })}{f(S)}}]\\ & +2a_{2}pf(S^{\ast })g(C^{\ast })+2a_{2}pf(S^{\ast })h(I^{\ast })+2(1-p)f(S^{\ast })g(C^{\ast })+2(1-p)f(S^{\ast })h(I^{\ast })\\ & +a_{1}f(S^{\ast })h(I^{\ast })(\frac{h(I)}{h(I^{\ast })}-\frac{I}{I^{\ast }})(1-\frac{h(I^{\ast })}{h(I)})\\ & +a_{1}f(S^{\ast })g(C^{\ast })(\frac{g(C)}{g(C^{\ast })}-\frac{C}{C^{\ast }})(1-\frac{g(C^{\ast })}{g(C)}).\end{array}$ The constant $a_2$, satisfies, $a_{2}ph(I^{\ast })=(1-p)g(C^{\ast })$ then, $\begin{array}{rl}{\dot{L}}_2& =a_1(d_1+\theta )S^{\ast }(1-\frac{S}{S^{\ast }})(1-\frac{f(S^{\ast })}{f(S)})\\ & +[a_{2}pf(S^{\ast })g(C^{\ast })+(1-p)f(S^{\ast })h(I^{\ast })][2-\frac{f(S^{\ast })}{f(S)}-\frac{f(S)}{f(S^{\ast })}]\\ & +(1-p)f(S^{\ast })g(C^{\ast })[4-2\frac{f(S^{\ast })}{f(S)}-\frac{f(S)}{f(S^{\ast })}\frac{h(I)}{h(I^{\ast })}\frac{g(C^{\ast })}{g(C)}-\frac{f(S)}{f(S^{\ast })}\frac{h(I^{\ast })}{h(I)}\frac{g(C)}{g(C^{\ast })}]\\ & +a_{1}f(S^{\ast })h(I^{\ast })(\frac{h(I)}{h(I^{\ast })}-\frac{I}{I^{\ast }})(1-\frac{h(I^{\ast })}{h(I)})\\ & +a_{1}f(S^{\ast })g(C^{\ast })(\frac{g(C)}{g(C^{\ast })}-\frac{C}{C^{\ast }})(1-\frac{g(C^{\ast })}{g(C)}).\end{array}$ Now remembering that $\sum _{i=1}^n\frac{x_i}{n}\ge \sqrt[n]{\prod _{i=1}^n{x}_i}$, we have $\begin{array}{rl}2-\frac{f(S^{\ast })}{f(S)}-\frac{f(S)}{f(S^{\ast })}& \le 0\\ 4-2\frac{f(S^{\ast })}{f(S)}-\frac{f(S)}{f(S^{\ast })}\frac{h(I)}{h(I^{\ast })}\frac{g(C^{\ast })}{g(C)}-\frac{f(S)}{f(S^{\ast })}\frac{h(I^{\ast })}{h(I)}\frac{g(C)}{g(C^{\ast })}& \le 0,\end{array}$ and by conditions (III) and (IV), we have $\begin{array}{rl}{a}_1(d_1+\theta )S^{\ast }(1-\frac{S}{S^{\ast }})(1-\frac{f(S^{\ast })}{f(S)})& \le 0\\ a_{1}f(S^{\ast })h(I^{\ast })(\frac{h(I)}{h(I^{\ast })}-\frac{I}{I^{\ast }})(1-\frac{h(I^{\ast })}{h(I)})& \le 0\\ a_{1}f(S^{\ast })g(C^{\ast })(\frac{g(C)}{g(C^{\ast })}-\frac{C}{C^{\ast }})(1-\frac{g(C^{\ast })}{g(C)})& \le 0.\end{array}$ Therefore, ${\dot{L}}_2\le 0$, for all $(S,C.I)$ and ${\dot{L}}_2=0$ if and only if $(S,C.I)=(S^{\ast },C^{\ast }.I^{\ast })$. Then the DE equilibrium is globally asymptotically stable.

4. Application

The aim of this section is to give some particular non-linear incidences rates to illustrate our main result. Consider the system (11) $\begin{array}{rl}\frac{\mathrm{d}S}{dt}& =b-\theta S-S[\frac{\beta C}{1+a_1{C}^{p_1}}+{\displaystyle \frac{\gamma I^{p_2}}{1+a_2{I}^{p_2}}}]-d_{1}S\\ \frac{\mathrm{d}C}{dt}& =pS[{\displaystyle \frac{\beta C}{1+a_1{C}^{p_1}}}+{\displaystyle \frac{\gamma I^{p_2}}{1+a_2{I}^{p_2}}}]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{dt}& =(1-p)S[{\displaystyle \frac{\beta C}{1+a_1{C}^{p_1}}}+{\displaystyle \frac{\gamma I^{p_2}}{1+a_2{I}^{p_2}}}]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{dt}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(11) which is a particular case of system (2(2) $\begin{array}{rl}\frac{\mathrm{d}S}{\mathrm{d}t}& =b-\theta S-f(S)[g(C)+h(I)]-d_{1}S\\ \frac{\mathrm{d}C}{\mathrm{d}t}& =pf(S)[g(C)+h(I)]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{\mathrm{d}t}& =(1-p)f(S)[g(C)+h(I)]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{\mathrm{d}t}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(2) ) by letting $f(S)=S$, $g(C)=\frac{\beta C}{1+a_1{C}^{p_1}}$ and $h(I)=\frac{\gamma I^{p_2}}{1+a_2{I}^{p_2}}$, where $p_1,p_2\in (0,1]$, $\beta >\gamma$ and β, γ, $a_1$, $a_2$ are positive. It is easy to check that the functions f, g, h, $\frac{g(C)}{C}=\frac{\beta }{1+a_1{C}^{p_1}}$, and $\frac{h(I)}{I}=\frac{\gamma I^{p_2-1}}{1+a_2{I}^{p_2}}$ satisfies the conditions (I)–(IV). The basic reproduction number is $R_0=\{\begin{array}{ll}{\displaystyle \frac{p}{d_2+\alpha }}{\displaystyle \frac{\beta b}{d_1+\theta }}+{\displaystyle \frac{(1-p)}{d_3+\pi }}\frac{\gamma b}{d_1+\theta }& \mathrm{if}{p}_1\in (0,1]\mathrm{and}{p}_2=1,\\ {\displaystyle \frac{p}{d_2+\alpha }}{\displaystyle \frac{b}{d_1+\theta }}& \mathrm{if}{p}_1\in (0,1]\mathrm{and}{p}_2\in (0,1).\end{array}$ Besides we have that by Proposition 2.1, the system has two equilibrium points, a disease-free equilibrium which is globally asymptotic stable if $R_0<1$ by Theorem 3.1 and a disease equilibrium which is globally asymptotic stable if $R_0>1$ by Theorem 3.2. In Figure 1, we show the force of infection $h(I)$ and $g(C)$ for some values of $p_1$ and $p_2$, these particular functions show a saturation effect while the number of asymptomatic carriers and infected individuals is increased. Besides, we assume that the force of infection g by carriers increases more than the force of infection h by infected individuals, since it is believed that carriers do not modify their behaviour in the course of an infection and their contribution to the spread of the disease will be greater.

Figure 1. Forces of infection for h and g in the model (11(11) $\begin{array}{rl}\frac{\mathrm{d}S}{dt}& =b-\theta S-S[\frac{\beta C}{1+a_1{C}^{p_1}}+{\displaystyle \frac{\gamma I^{p_2}}{1+a_2{I}^{p_2}}}]-d_{1}S\\ \frac{\mathrm{d}C}{dt}& =pS[{\displaystyle \frac{\beta C}{1+a_1{C}^{p_1}}}+{\displaystyle \frac{\gamma I^{p_2}}{1+a_2{I}^{p_2}}}]-(d_2+\alpha )C\\ \frac{\mathrm{d}I}{dt}& =(1-p)S[{\displaystyle \frac{\beta C}{1+a_1{C}^{p_1}}}+{\displaystyle \frac{\gamma I^{p_2}}{1+a_2{I}^{p_2}}}]-(d_3+\pi )I\\ \frac{\mathrm{d}R}{dt}& =\theta S+\pi I+\alpha C-d_{4}R.\end{array}$(11) ). (a) The graph represent the force of infection by carriers $g_i(C)=\frac{2.1C}{1+C^{\frac{1}{i}}}$, where i = 1, 2, 3, 4 and (b) The graph represent the force of infection by infected individuals $h_i(I)=\frac{1.8I^{\frac{1}{i}}}{1+I^{\frac{1}{i}}}$, where i = 1, 2, 3, 4.

The functions $f(S)=S^p$, $g(C)=\beta C^{q_1}$, $h(I)=\gamma I^{q_2}$, where $p\in [1,\mathrm{\infty })$, $q_1,q_2\in (0,1]$, and $\beta >\gamma >0$ satisfy the conditions (I) –(IV) ; however, the functions g and h are not differentiable in 0, this condition is necessary in the definition of $R_0$ and in the proof of Theorem 3.1. Then another incidence rate could be $S^p[\beta C+\gamma I]$.

5. Discussion

In this paper, we consider a mathematical model of infectious disease with varying population, where asymptomatic carriers can infect to susceptible population and it is used a general incidence rate of the form $f(S)g(C)+f(S)h(I)$. We showed the global stability of disease-free and disease equilibrium, using the Lyapunov direct method and assuming some sufficient conditions in the functions f, g, h. One of these conditions is that the functions $\frac{g(C)}{C}$ and $\frac{h(I)}{I}$ have to be non-increasing functions (condition (IV)); we can write a weaker condition, as follow

1. $\frac{g(C)}{C}\le g^{\prime }(0)$, $\frac{h(I)}{I}\le h^{\prime }(0)$, for all C>0 and I>0

2. $\frac{h(I)}{h(I^{\ast })}\ge \frac{I}{I^{\ast }}$ for $0<I\le I^{\ast }$ and $\frac{h(I)}{h(I^{\ast })}\le \frac{I}{I^{\ast }}$ for $I\ge I^{\ast }$

3. $\frac{g(C)}{g(C^{\ast })}\ge \frac{C}{C^{\ast }}$ for $0<C\le C^{\ast }$ and $\frac{g(C)}{g(c^{\ast })}\le \frac{C}{C^{\ast }}$ for $C\ge C^{\ast }$

The condition (a) is necessary in the global stability analysis for the disease-free equilibrium, while the conditions (b,c) are necessary to get global stability for the disease equilibrium. It is easy to check that a non-increasing functions $\frac{g(C)}{C}$ and $\frac{h(I)}{I}$ satisfies all the conditions (a–c). In the model, we consider that the carriers do not show symptoms, which allowed us to apply the Lyapunov direct method to obtain global stability, while the model with general incidence rate where the carriers show symptoms (i.e. the transition of carriers to infected) is a challenge to consider.

Acknowledgments

M. C. G. thanks financial support, project No. 1669/09/2018 from Vicerrectoria de Investigaciones y relaciones internacionales (Vipri-Udenar).

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Miller Cerón Gómez http://orcid.org/0000-0002-2689-495X

Additional information

Funding

M. C. G. thanks financial support, (Universidad de Nariño) project No. 1669/09/2018 from Vicerrectoria de Investigaciones y relaciones internacionales (Vipri-Udenar).