A mathematical model for tilapia lake virus transmission with waning immunity

Figures & data

Table 1. Baseline values of the model parameters.

	Description	Dimension	Values	Sources
b	birth rate	day ${}^{-1}$	$\frac{4}{365}$	[19,26]
θ	Normalization param. for shedding rate	copies.day ${}^{-3}$.fish${}^{-1}$	${10}^4$	[19,35,39]
${\alpha }_1$	Mean duration of the latency period	day	2	[10,13,19]
${\alpha }_2$	Mean duration of the infectious period	day	13	[10,13,19]
${\mu }_0$	Density non-dependent mortality rate	day ${}^{-1}$	$\frac{1}{365}$	[1,19,38]
${\mu }_1$	Density-dependent mortality rate	day ${}^{-1}$.fish${}^{-1}$	$8.22\times {10}^{-8}$	fixed
${\mu }_V$	Mortality rate of virus	day ${}^{-1}$	0.5	[19,28,32]
${\nu }_0$	Normalization param. for death rate	day ${}^{-3}$	$1.63\times {10}^{-2}$	[13,19]
$a_1$	Mean delay between infection and onset of mortality	day	2	[10,13,19]
$a_2$	Mean duration of disease related mortality	day	7	[10,13,19]
${\sigma }^{dir}$	Horizontal direct transmission rate	copies${}^{-1}$	$6.8\times {10}^{-9}$	estimated
${\sigma }^{ind}$	Horizontal indirect transmission rate	copies${}^{-1}$.day${}^{-1}$	$3.6\times {10}^{-11}$	estimated
${\pi }_0$	Normalization param.	day ${}^{-2}$	$\frac{1}{86}$	[11,19]
${\pi }_1$	Mean delay between infection and laying being affected	day	2	[13,19,35]
${\pi }_2$	Mean duration of the laying being affected	day	13	[13,19,33]

Figure 1. Results of simulations achieved with parameters in Table 1. The waning parameter γ is given in (28(28) $\begin{array}{r}\gamma (\tau )=\{\begin{array}{ll}{\displaystyle \frac{1}{\varsigma -\frac{\varsigma -1}{\varsigma }\tau }}& \mathrm{if}\tau \in [0,\varsigma ],\\ 1& \mathrm{if}\tau \in [\varsigma ,+\mathrm{\infty }[.\end{array}\end{array}$(28) ) with the bifurcation parameter ς that varies. We show to simplify the graphical representations the quantities (33(33) $\begin{array}{r}I(t)={\int }_0^{\mathrm{\infty }}i(a,t)\mathrm{d}a\text{and}R(t)={\int }_0^{\mathrm{\infty }}r(\tau ,t)\mathrm{d}\tau .\end{array}$(33) ) in (a,b). In 3D figures (c,d), we present the density of pathogen. We show the density of infected tilapias in 3D figures (e,f). (a) $\varsigma =2000$. (b) $\varsigma =180$. (c) $\varsigma =2000$. (d) $\varsigma =180$. (e) $\varsigma =2000$. (f) $\varsigma =180$.

Figure 2. In 3D figures (a,b), we present the density of recovered tilapias. (a) $\varsigma =2000$. (b) $\varsigma =180$.

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