Mathematical study for Zika virus transmission with general incidence rate

Ahmed Alshehri; Miled El Hajji; Ahmed Alshehri; Miled El Hajji

doi:10.3934/math.2022397

AIMS Mathematics

2022, Volume 7, Issue 4: 7117-7142. doi: 10.3934/math.2022397

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Mathematical study for Zika virus transmission with general incidence rate

Ahmed Alshehri ¹,
Miled El Hajji ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia

Received: 11 December 2021 Revised: 13 January 2022 Accepted: 23 January 2022 Published: 09 February 2022
MSC : 34K20, 34D23, 37B25, 49K40, 92D25

An appropriate mathematical model for describing the Zika virus transmission with nonlinear general incidence rate was proposed. The basic reproduction number $ \mathcal{R}_0 $ was calculated using the next generation matrix method. Analysis of the local and the global stability of the equilibrium points was detailed using Jacobian linearisation method and Lyapunov theory, respectively. We proved that the disease-free equilibrium is locally and globally asymptotically stable when $ \mathcal{R}_0 $ is small than 1, and the infected equilibrium point is locally and globally asymptotically stable when $ \mathcal{R}_0 $ is greater than 1. The overall sensitivity analysis is based on statistical tools. This method consists of varying the parameters of the model to study one by one and then observe the effect of this variation on the model output. Sensitivity indices quantifying the influence of parameters on the output, always depend on the sample list of parameters. Later, we used optimal control to examine the effect of treatment where the purpose is to minimize the number of infected individuals with optimal treatment cost by applying Pontryagin's maximum principle. Therefore, we formulated an optimal control problem using the most parameter that influences the model output as a control parameter. The existence of the solution was proved and characterized using adjointt variables. Finally, a numerical scheme was applied to solve the coupled systems. Obtained results are validated numerically.
- Zika virus,
- global stability,
- Lyapunov function,
- LaSalle's invariance principle,
- sensitivity analysis,
- nonlinear incidence Rate,
- optimal control.
Citation: Ahmed Alshehri, Miled El Hajji. Mathematical study for Zika virus transmission with general incidence rate[J]. AIMS Mathematics, 2022, 7(4): 7117-7142. doi: 10.3934/math.2022397

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Abstract

An appropriate mathematical model for describing the Zika virus transmission with nonlinear general incidence rate was proposed. The basic reproduction number $ \mathcal{R}_0 $ was calculated using the next generation matrix method. Analysis of the local and the global stability of the equilibrium points was detailed using Jacobian linearisation method and Lyapunov theory, respectively. We proved that the disease-free equilibrium is locally and globally asymptotically stable when $ \mathcal{R}_0 $ is small than 1, and the infected equilibrium point is locally and globally asymptotically stable when $ \mathcal{R}_0 $ is greater than 1. The overall sensitivity analysis is based on statistical tools. This method consists of varying the parameters of the model to study one by one and then observe the effect of this variation on the model output. Sensitivity indices quantifying the influence of parameters on the output, always depend on the sample list of parameters. Later, we used optimal control to examine the effect of treatment where the purpose is to minimize the number of infected individuals with optimal treatment cost by applying Pontryagin's maximum principle. Therefore, we formulated an optimal control problem using the most parameter that influences the model output as a control parameter. The existence of the solution was proved and characterized using adjointt variables. Finally, a numerical scheme was applied to solve the coupled systems. Obtained results are validated numerically.

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