Transmission Dynamics of an SIS Model with Age Structure on Heterogeneous Networks

Abstract

Infection age is often an important factor in epidemic dynamics. In order to realistically analyze the spreading mechanism and dynamical behavior of epidemic diseases, in this paper, a generalized disease transmission model of SIS type with age-dependent infection and birth and death on a heterogeneous network is discussed. The model allows the infection and recovery rates to vary and depend on the age of infection, the time since an individual becomes infected. We address uniform persistence and find that the model has the sharp threshold property, that is, for the basic reproduction number less than one, the disease-free equilibrium is globally asymptotically stable, while for the basic reproduction number is above one, a Lyapunov functional is used to show that the endemic equilibrium is globally stable. Finally, some numerical simulations are carried out to illustrate and complement the main results. The disease dynamics rely not only on the network structure, but also on an age-dependent factor (for some key functions concerned in the model).

Keywords: Basic reproduction number; Global stability; Infection age; Scale-free network.

Fig. 1

(Color figure online) The influence of $\phi (k)$ on $R_0$, here $\phi (k)=\frac{a{k}^{\alpha }}{1+b{k}^{\alpha }}$

Fig. 2

(Color figure online) Here, $\phi (k)=k,\beta (\tau )=\frac{\tau (200-\tau )}{15,000},\gamma (\tau )=\frac{1}{1+\tau },\mu =0.06,b=0.07,R_0=0.6066<1$. a Dynamics of $I_k(t)$ subject to time $t,k=10,20,30,40$. b Dynamics of $I_k(t)$ subject to time $t,k=1,2,\dots ,40$

Fig. 3

(Color figure online) Here $\phi (k)=k,\beta (\tau )=\frac{\tau (200-\tau )}{15000},\gamma (\tau )=\frac{1}{1+10\tau },\mu =0.06,b=0.07,R_0=3.4798>1$. a Dynamics of $I_k(t)$ subject to time $t,k=10,20,30,40$. b Dynamics of $I_k(t)$ subject to time $t,k=1,2,\dots ,40$

Fig. 4

(Color figure online) a Transmission rate $\beta (\tau )$; b diagnosis rate $\delta (\tau ).$ The symptoms appear at day 4, which coincides with the beginning of infectious period. Infected individuals are quarantined at a rate of 30% per day after day 4

Fig. 5

(Color figure online) Here is the comparison of dynamics of $I_k(t)$ and $I(t,\tau )$ of system (1) and system (23). a and c represent the dynamics in system (1), b and d show dynamics of system (23). a and b are the evolution of $I_k(t)$, while c and d are the evolution of $I(t,\tau )$. Each line with the same initial conditions and parameters. a Dynamics of $I_k(t)$ subject to time t of system (1), $k=10,20,30,40$. b Dynamics of $I_k(t)$ subject to time t of system (23). c Dynamics of $I(t,\tau )$ subject to time t of system (1), $\tau =5,10,20,50$. d Dynamics of $I(t,\tau )$ subject to time t of system (23)