Characterizing the permanence and stationary distribution for a family of malaria stochastic models

Presented at Biology and Medicine Through Mathematics Conference (BAMM) According to WHO estimates released in December 2016, about 212 million cases of malaria occurred in 2015 resulting in about 429 thousand deaths. The highest mortality rates were recorded for sub-Saharan African countries, where nearly 90% of the global malaria cases occurred, and approximately 75% of the global malaria deaths. Several studies suggest the existence of temporal and spatial variations in malaria transmission rates, where climatic drivers such as temperature, rainfall, and vegetation indices etc. are culprits for the observed variability. This talk presents the stochastic permanence of malaria and the existence of a stationary distribution for a malaria SEIRS system of stochastic differential equation model. Malaria spreads in a very noisy environment with variability of white noise type in the disease transmission and death rates. A general nonlinear incidence rate defines a family for the malaria models. The mosquito and human dynamics are presented. Improved analytical techniques and local martingale characterizations are applied to describe the character of the sample paths of the solution process of the system in the neighborhood of an endemic equilibrium. Emphasis is laid on examination of the impacts of the noises in the system on the stochastic permanence of malaria, and on the existence of a stationary distribution for the solution process over sufficiently long time. The model is applied to P. vivax malaria, and attempt is made to numerically approximate the stationary distribution, and the statistical properties of the states of the solution process over sufficiently ling time

A nonlinear multi-population behavioral model to assess the roles of education campaigns, random supply of aids, and delayed ART treatment in HIV/AIDS epidemics

The successful reduction in prevalence rates of HIV in many countries is attributed to control measures such as information and education campaigns (IEC), antiretroviral therapy (ART), and national, multinational and multilateral support providing offcial developmental assistance (ODAs) to combat HIV. However, control of HIV epidemics can be interrupted by limited random supply of ODAs, high poverty rates and low living standards. This study presents a stochastic HIV/AIDS model with treatment assessing the roles of IEC, the supply of ODAs and early treatment in HIV epidemics. The supply of ODAs is assessed via the availability of medical and financial resources leading more people to get tested and begin early ART. The basic reproduction number (R0) for the dynamics is obtained, and other results for HIV control are obtained by conducting stability analysis for the stochastic SITRZ disease dynamics. Moreover, the model is applied to Uganda HIV/AIDS data, wherein linear regression is applied to predict the R0 over time, and to determine the importance of ART treatment in the dynamics

The Stochastic Permanence of Disease and the Stationary Behavior for a Class of Nonlinear SEIRS Epidemic Models

An interesting topic for investigation in the study of stochastic differential equation epidemic models involving Brownian motion perturbations concerns the permanence of disease and existence of a stationary behavior for the state of the stochastic process over time. Conditions for the permanence of the disease hold the key to understand the endemic behavior of the disease; a stationary distribution leads to knowing the statistical properties of the disease over long time. This talk discusses a class of Ito stochastic differential equation SEIRS epidemic models for vector-borne diseases e.g. malaria. Lyapunov functional techniques and some local martingale characterizations are applied to find persistence conditions for the disease by examining the average behavior of all sample paths of the system over time. Moreover, the conditions for the existence of a stationary distribution for the SEIRS system are presented. Furthermore, the stationary distribution is explored numerically

The statistical estimation of the basic reproduction number and other parameters of SEIR stochastic epidemic models. Case study- influenza

Presentation given at Biology and Medicine Through Mathematics Conference (BAMM). Conference was originally scheduled for May 2020 but was rescheduled to May 2021 due to the covid-19 pandemic

The stochastic extinction and stability conditions for a class of malaria epidemic models

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. 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.Comment: arXiv admin note: substantial text overlap with arXiv:1808.09842, arXiv:1809.03866, arXiv:1809.0389