The Stochastic Extinction and Stability Conditions for Nonlinear Malaria Epidemics

Document Type

Article

Publication Date

2019

Publication Title

Mathematical Biosciences and Engineering

DOI

10.3934/mbe.2019187

ISSN

1551-0018

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 R0∗" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">R∗0R0∗, the expected survival probability of the plasmodium E(e−(μvT1+μT2))" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">E(e−(μvT1+μT2))E(e−(μvT1+μT2)), and other threshold values are calculated, where μv" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">μvμv and μ" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">μμ are the natural death rates of mosquitoes and humans, respectively, and T1" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">T1T1 and T2" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">T2T2 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. Moreover, for either R0∗<1" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">R∗0<1R0∗<1, or E(e−(μvT1+μT2))<1R0∗" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">E(e−(μvT1+μT2))<1R∗0E(e−(μvT1+μT2))<1R0∗, whenever R0∗≥1" role="presentation" style="margin: 0px; padding: 0px; list-style: none; border: 0px; display: inline; line-height: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(102, 102, 102); font-family: Arial; background-color: rgb(240, 240, 242); position: relative;">R∗0≥1R0∗≥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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