The stochastic extinction and stability conditions for nonlinear malaria epidemics

Divine Wanduku; Divine Wanduku

doi:10.3934/mbe.2019187

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3771-3806. doi: 10.3934/mbe.2019187

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The stochastic extinction and stability conditions for nonlinear malaria epidemics

Divine Wanduku ^,

Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave, Room 3042, Statesboro, Georgia, 30460, USA

Received: 15 February 2019 Accepted: 11 April 2019 Published: 28 April 2019

The stochastic extinction and stability in the mean of a family of SEIRS malaria models with a general nonlinear incidence rate is presented. The dynamics is driven by independent white noise processes from the disease transmission and natural death rates. The basic reproduction number $R^{*}_{0}$, the expected survival probability of the plasmodium $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$, and other threshold values are calculated, where $\mu_{v}$ and $\mu$ are the natural death rates of mosquitoes and humans, respectively, and $T_{1}$ and $T_{2}$ are the incubation periods of the plasmodium inside the mosquitoes and humans, respectively. A sample Lyapunov exponential analysis for the system is utilized to obtain extinction results. Moreover, the rate of extinction of malaria is estimated, and innovative local Martingale and Lyapunov functional techniques are applied to establish the strong persistence, and asymptotic stability in the mean of the malaria-free steady population. %The extinction of malaria depends on $R^{*}_{0}$, and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$. Moreover, for either $R^{*}_{0} < 1$, or $E(e^{-(\mu_{v}T_{1}+\mu T_{2})}) < \frac{1}{R^{*}_{0}}$, whenever $R^{*}_{0}\geq 1$, respectively, extinction of malaria occurs. Furthermore, the robustness of these threshold conditions to the intensity of noise from the disease transmission rate is exhibited. Numerical simulation results are presented.
- disease-free steady state,
- stability in the mean,
- basic reproduction number,
- sample lyapunov exponent,
- survival probability
Citation: Divine Wanduku. The stochastic extinction and stability conditions for nonlinear malaria epidemics[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3771-3806. doi: 10.3934/mbe.2019187

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Abstract

The stochastic extinction and stability in the mean of a family of SEIRS malaria models with a general nonlinear incidence rate is presented. The dynamics is driven by independent white noise processes from the disease transmission and natural death rates. The basic reproduction number $R^{*}_{0}$, the expected survival probability of the plasmodium $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$, and other threshold values are calculated, where $\mu_{v}$ and $\mu$ are the natural death rates of mosquitoes and humans, respectively, and $T_{1}$ and $T_{2}$ are the incubation periods of the plasmodium inside the mosquitoes and humans, respectively. A sample Lyapunov exponential analysis for the system is utilized to obtain extinction results. Moreover, the rate of extinction of malaria is estimated, and innovative local Martingale and Lyapunov functional techniques are applied to establish the strong persistence, and asymptotic stability in the mean of the malaria-free steady population. %The extinction of malaria depends on $R^{*}_{0}$, and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$. Moreover, for either $R^{*}_{0} < 1$, or $E(e^{-(\mu_{v}T_{1}+\mu T_{2})}) < \frac{1}{R^{*}_{0}}$, whenever $R^{*}_{0}\geq 1$, respectively, extinction of malaria occurs. Furthermore, the robustness of these threshold conditions to the intensity of noise from the disease transmission rate is exhibited. Numerical simulation results are presented.

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