# Mathematical Modeling of Japanese Encephalitis under Aquatic Environmental Effects

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

- we do not consider immigration of infected humans;
- the human population is not constant (we consider a disease-induced death rate, due to fatality, of 25%);
- we assume that the coefficient of transmission of the virus is constant and does not vary with seasons, which is reasonable due to the short course of the disease;
- mosquitoes are assumed to be born susceptible.

## 3. Mathematical Analysis of the JE Model

**Theorem**

**1**

**Proof.**

**Theorem**

**2**

## 4. Numerical Simulations

`integrate.odeint`of library SciPy. The following values of the parameters, borrowed from [7,9], are considered:

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Python Code for Figure 1–3

`"""" Numerical simulations for Japanese Encephalitis disease~"""`

`# import modules for solving`

`import scipy`

`import scipy.integrate`

`import numpy as~np`

`# import module for plotting`

`import pylab as~pl`

`# System with substitutions`

`#E=X[0], I_r = X[1]; A_m=X[2]; N_m=X[3]; I_m=[4]; N=X[5]; I=X[6].`

`def JEmodel(X, t, Q0, theta, theta0, betamr, mu1r, mu2r, dr, delta0, psi, K, muA, nuA,`

`delta, mum, B, betarm, Lambdah, muh, nuh, dh, betamh ):`

`z1= Q0 + theta*X[5] - theta0*X[0]`

`z2=betamr*X[1]*X[4]/X[3] - (mu1r +mu2r*X[1] + dr)*X[1] + delta0*X[1]*X[0]`

`z3= psi*(1- X[2]/K)*X[3] - (muA + nuA)*X[2] + delta*X[0]*X[2]`

`z4= nuA*X[2]-mum*X[3]`

`z5= B*betarm*X[1]*(X[3]-X[4])-mum*X[4]`

`z6= Lambdah - muh*X[5]-dh*X[6]`

`z7= (B*betamh*X[4]/X[3])*(X[5]-X[6])-nuh*X[6] - muh*X[6] - dh*X[6]`

`return (z1, z2, z3, z4, z5, z6, z7)`

`if __name__== "__main__":`

`X0= [40000, 500, 12000, 10000, 9000, 7000, 1000];`

`X1= [45000, 700, 15000, 12000, 11000, 10000, 1200];`

`X2= [35000, 300, 10000, 7000, 6000, 5000, 800];`

`t = np.arange(0, 20, 0.1)`

`Q0= 50`

`theta=0.01`

`theta0=0.0001`

`betamr= 0.0001`

`mu1r=0.1`

`dr=1/15.0`

`delta0=0.000001`

`psi=0.6`

`K=1000`

`muA=0.25`

`nuA=0.5`

`delta=0.0001`

`mum=0.3`

`B=1; mu2r= 0.001`

`betarm=0.00021`

`Lambdah=150`

`muh=1.0/65`

`dh=1.0/45`

`nuh=0.45`

`betamh=0.0003`

`r=scipy.integrate.odeint(JEmodel, X0, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r,`

`dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh))`

`r1=scipy.integrate.odeint(JEmodel, X1, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r,`

`dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh))`

`r2=scipy.integrate.odeint(JEmodel, X2, t, args=(Q0, theta, theta0, betamr, mu1r, mu2r,`

`dr, delta0, psi, K, muA, nuA, delta, mum, B, betarm, Lambdah, muh, dh, nuh, betamh))`

`pl.plot(t,r[:,1], t,r1[:,1], t,r2[:,1])`

`pl.legend([’$X_1(0)$’, ’$X_2(0)$’, ’$X_3(0)$’],loc=’upper right)`

`pl.xlabel(’Time (weeks))`

`pl.ylabel(’Reservoir population)`

`#pl.title(’Japaneese model)`

`pl.savefig(’reservoir.eps)`

`pl.show();`

`pl.plot(t,r[:,4], t,r1[:,4],t,r2[:,4])`

`pl.xlabel(’Time (weeks))`

`pl.ylabel(’Infected mosquitoes)`

`pl.legend([’$X_1(0)$’, ’$X_2(0)$’, ’$X_3(0)$’],loc=’upper right)`

`pl.savefig(’mosquitoes.eps)`

`pl.show();`

`pl.plot(t,r[:,6], t,r1[:,6],t,r2[:,6])`

`pl.xlabel(’Time (weeks))`

`pl.ylabel(’Infected humans)`

`pl.legend([’$X_1(0)$’, ’$X_2(0)$’, ’$X_3(0)$’],loc=’upper right)`

`pl.savefig(’infected_human.eps)`

`pl.show()`

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**Figure 1.**The solution of Model (5) tends toward the disease-free equilibrium. In this figure, we show the evolution of the infected animals population for different initial conditions.

**Figure 2.**The solution of Model (5) tends toward the disease-free equilibrium. In this figure, we show the evolution of the infected mosquitoes population for different initial conditions.

**Figure 3.**The solution of Model (5) tends toward the disease-free equilibrium. In this figure, we show the evolution of the infected humans population for different initial conditions.

$\mathit{E}\left(0\right)$ | ${\mathit{I}}_{\mathit{r}}\left(0\right)$ | ${\mathit{A}}_{\mathit{m}}\left(0\right)$ | ${\mathit{N}}_{\mathit{m}}\left(0\right)$ | ${\mathit{I}}_{\mathit{m}}\left(0\right)$ | $\mathit{N}\left(0\right)$ | $\mathit{I}\left(0\right)$ | |
---|---|---|---|---|---|---|---|

${X}_{1}\left(0\right)$ | 40,000 | 500 | 12,000 | 10,000 | 9000 | 7000 | 1000 |

${X}_{2}\left(0\right)$ | 45,000 | 700 | 15,000 | 12,000 | 11,000 | 10,000 | 12,000 |

${X}_{3}\left(0\right)$ | 35,000 | 300 | 10,000 | 7000 | 6000 | 5000 | 800 |

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**MDPI and ACS Style**

Ndaïrou, F.; Area, I.; Torres, D.F.M.
Mathematical Modeling of Japanese Encephalitis under Aquatic Environmental Effects. *Mathematics* **2020**, *8*, 1880.
https://doi.org/10.3390/math8111880

**AMA Style**

Ndaïrou F, Area I, Torres DFM.
Mathematical Modeling of Japanese Encephalitis under Aquatic Environmental Effects. *Mathematics*. 2020; 8(11):1880.
https://doi.org/10.3390/math8111880

**Chicago/Turabian Style**

Ndaïrou, Faïçal, Iván Area, and Delfim F. M. Torres.
2020. "Mathematical Modeling of Japanese Encephalitis under Aquatic Environmental Effects" *Mathematics* 8, no. 11: 1880.
https://doi.org/10.3390/math8111880