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

Figures & data

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.

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.