Global dynamics of an epidemiological model with age of infection and disease relapse

Figures & data

Table 1. The definition of the parameters in system (2(2) $\begin{array}{rl}& \dot{S}\left(t\right)=A-\mu S\left(t\right)-S\left(t\right){\int }_0^{\mathrm{\infty }}\beta \left(a\right)i(a,t)\mathrm{d}a,\\ & \frac{\mathrm{\partial }i(a,t)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }i(a,t)}{\mathrm{\partial }a}=-(\mu +\gamma (a\left)\right)i(a,t),a>0,\\ & \dot{R}\left(t\right)={\int }_0^{\mathrm{\infty }}\gamma \left(a\right)i(a,t)\mathrm{d}a-(\mu +\delta )R\left(t\right),\end{array}$(2) ).

Parameters	Description
A	Recruitment rate
a	Age of infection, i.e. the time that has lapsed since the individual became infected
μ	Per capita natural death rate
$\beta \left(a\right)$	Transmission coefficient of the infected individuals at age of infection a
$\gamma \left(a\right)$	The rate that an infected individual with age of infection a recovers from the disease
δ	The rate at which an individual in the recovered class reverts to the infective class

Figure 1. The temporal solution found by numerical integration of system (2(2) $\begin{array}{rl}& \dot{S}\left(t\right)=A-\mu S\left(t\right)-S\left(t\right){\int }_0^{\mathrm{\infty }}\beta \left(a\right)i(a,t)\mathrm{d}a,\\ & \frac{\mathrm{\partial }i(a,t)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }i(a,t)}{\mathrm{\partial }a}=-(\mu +\gamma (a\left)\right)i(a,t),a>0,\\ & \dot{R}\left(t\right)={\int }_0^{\mathrm{\infty }}\gamma \left(a\right)i(a,t)\mathrm{d}a-(\mu +\delta )R\left(t\right),\end{array}$(2) ) with the boundary condition (3(3) $i(0,t)=S\left(t\right){\int }_0^{\mathrm{\infty }}\beta \left(a\right)i(a,t)\mathrm{d}a+\delta R\left(t\right),$(3) ) and the initial condition $S\left(0\right)=0.9,i(\cdot ,0)=0.1,R\left(0\right)=0$.