Theoretical and numerical results for an age-structured SIVS model with a general nonlinear incidence rate

Figures & data

Table 1. List of parameter values.

Param.	Value	Units	Param.	Expression
α	0.5	year^−1	$\gamma \left(a\right)$	$0.01[1+\sin (\frac{\pi (a-5)}{10}+\frac{1}{50}\left)\right]$
Λ	40	year^−1	$\beta \left(a\right)$	varied
φ	0.2	people/year	$ϵ\left(a\right)$	$\psi [1+\sin (\frac{(a-5)\pi }{10}\left)\right]$
μ	0.01	year^−1	$\delta \left(a\right)$	0.8

Figure 1. Time evolution of the infective population $i(t,a)$, $0\le t\le 100$, $0\le a\le 20$ for system (2(2) $\begin{array}{rl}& \frac{\mathrm{d}S\left(t\right)}{\mathrm{d}t}=\mathrm{\Lambda }-\mu S\left(t\right)-f\left(S\right(t),J(t\left)\right)-\phi S\left(t\right)+{\int }_0^{\mathrm{\infty }}\epsilon \left(a\right)v(t,a)\mathrm{d}a,\\ & \frac{\mathrm{\partial }i(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }i(t,a)}{\mathrm{\partial }a}=-(\mu +\gamma (a)+\delta (a\left)\right)i(t,a),\\ & i(t,0)=f\left(S\right(t),J(t\left)\right),J\left(t\right)={\int }_0^{\mathrm{\infty }}\beta \left(a\right)i(t,a)\mathrm{d}a,\\ & \frac{\mathrm{\partial }v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }v(t,a)}{\mathrm{\partial }a}=-(\mu +\epsilon (a\left)\right)v(t,a),\\ & v(t,0)=\phi S\left(t\right),\end{array}$(2) ) with initial value $i_0\left(a\right)=1+\cos (-2a)$ and $v_0\left(a\right)={\mathrm{e}}^{-a}$. (a) and (b) with ${\beta }^{\ast }=1.527$, (c) and (d) with ${\beta }^{\ast }=6.667$.

Figure 2. Time evolution of the infective population $i(t,a)$, $0\le t\le 100$, $0\le a\le 20$ for system (2(2) $\begin{array}{rl}& \frac{\mathrm{d}S\left(t\right)}{\mathrm{d}t}=\mathrm{\Lambda }-\mu S\left(t\right)-f\left(S\right(t),J(t\left)\right)-\phi S\left(t\right)+{\int }_0^{\mathrm{\infty }}\epsilon \left(a\right)v(t,a)\mathrm{d}a,\\ & \frac{\mathrm{\partial }i(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }i(t,a)}{\mathrm{\partial }a}=-(\mu +\gamma (a)+\delta (a\left)\right)i(t,a),\\ & i(t,0)=f\left(S\right(t),J(t\left)\right),J\left(t\right)={\int }_0^{\mathrm{\infty }}\beta \left(a\right)i(t,a)\mathrm{d}a,\\ & \frac{\mathrm{\partial }v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }v(t,a)}{\mathrm{\partial }a}=-(\mu +\epsilon (a\left)\right)v(t,a),\\ & v(t,0)=\phi S\left(t\right),\end{array}$(2) ) with different vaccinated rates and input rate. (a) with $\phi =0.3,0.4,\mathrm{0.5.}$ (b) with $\mathrm{\Lambda }=10,20,40$.

Figure 3. Time evolution of the infective population $i(t,a)$, $0\le t\le 100$, $0\le a\le 20$ for system (2(2) $\begin{array}{rl}& \frac{\mathrm{d}S\left(t\right)}{\mathrm{d}t}=\mathrm{\Lambda }-\mu S\left(t\right)-f\left(S\right(t),J(t\left)\right)-\phi S\left(t\right)+{\int }_0^{\mathrm{\infty }}\epsilon \left(a\right)v(t,a)\mathrm{d}a,\\ & \frac{\mathrm{\partial }i(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }i(t,a)}{\mathrm{\partial }a}=-(\mu +\gamma (a)+\delta (a\left)\right)i(t,a),\\ & i(t,0)=f\left(S\right(t),J(t\left)\right),J\left(t\right)={\int }_0^{\mathrm{\infty }}\beta \left(a\right)i(t,a)\mathrm{d}a,\\ & \frac{\mathrm{\partial }v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }v(t,a)}{\mathrm{\partial }a}=-(\mu +\epsilon (a\left)\right)v(t,a),\\ & v(t,0)=\phi S\left(t\right),\end{array}$(2) ) with different transmission rates and different immunity waning rates. (a) with different values $\alpha =0,0.5$, (b) with different immunity waning rates $\psi =0.06,0.09,0.12$.

Table 2. Incidence rates.

Models	Incidence Rates	References
SIVS	$S\left(t\right){\int }_0^{\mathrm{\infty }}\beta \left(a\right)i(t,a)da$	[18,22,31]
SIR(PDE)	$\begin{array}{c}f(S,J)\\ S{\int }_0^{\mathrm{\infty }}\beta \left(a\right)f\left(i\right(t,a\left)\right)\mathrm{d}a\end{array}$	[24]
		[4]
SIR(ODE,DDE)	$\begin{array}{c}f(S,I)\\ F\left(S\right)G\left(I\right(t-\tau \left)\right)\end{array}$	[16]
		[3]

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