Witbooi et al. Advances in Difference Equations ( 2020) 2020:347 https://doi.org/10.1186/s13662-020-02803-w RESEARCH Open Access Stochastic modeling of a mosquito-borne disease Peter J. Witbooi 1* , Gbenga J. Abiodun 1 , Garth J. van Schalkwyk 1 and Ibrahim H.I. Ahmed 2 * Correspondence: pwitbooi@uwc.ac.za 1 Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, 7535, South Africa Full list of author information is available at the end of the article Abstract We present and analyze a stochastic differential equation (SDE) model for the population dynamics of a mosquito-borne infectious disease. We prove the solutions to be almost surely positive and global. We introduce a numerical invariant R of the model with R < 1 being a condition guaranteeing the almost sure stability of the disease-free equilibrium. We show that stochastic perturbations enhance the stability of the disease-free equilibrium of the underlying deterministic model. We illustrate the main stability theorem through simulations and show how to obtain interval estimates when making forward projections. We consulted a wide range of literature to find relevant numerical parameter values. MSC: 92D30; 43F05 Keywords: SDE model; Basic reproduction number; Exponential stability; Malaria; Extinction 1 Introduction A variety of mosquito-borne infectious diseases are the cause of millions of illnesses and deaths. In particular, the malaria disease is responsible for millions of fatalities in Africa each year and is a serious burden of the disease worldwide. The report [27] by South- ern African Development Community (SADC) gives a summary of malaria statistics in Southern Africa. A map of the region clearly shows the intensity of the malaria problem in different areas. In Mozambique, for instance, malaria is the leading cause of death [12], whereas the biggest part of South Africa is malaria-free [27, Malaria Map]. Mathematical modeling is useful in the planning of interventions to curb the spread of malaria and other diseases. Indeed, different models have been proposed for different situations. A popular type of models is the compartmental model in terms of ODEs. Recent models of this type include, for instance, [4, 22, 23, 25]. Malaria prevalence numbers have been proved to be influenced by climatic factors; see, for example, [1, 2, 6, 14, 28]. Malaria population dynamics is also correlated with other variables such as altitude and topography, land use, land cover, human behavior, and living conditions. Some regions have partial protection against malaria through indoor residual spraying or bednets [5]. In some regions, people may use traditional plant remedies that are effective against malaria; see, for example, [8]. Consequently, modeling malaria pop- © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.