Journal of Applied Mathematics

Volume 2018, Article ID 2439520, 13 pages

https://doi.org/10.1155/2018/2439520

## A Stochastic Model for Malaria Transmission Dynamics

Institute of Mathematical Sciences (IMS), Strathmore University, Box 59857 00200, Nairobi, Kenya

Correspondence should be addressed to Rachel Waema Mbogo; ude.eromhtarts@ogobmr

Received 14 September 2017; Accepted 26 December 2017; Published 11 February 2018

Academic Editor: Zhidong Teng

Copyright © 2018 Rachel Waema Mbogo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Malaria is one of the three most dangerous infectious diseases worldwide (along with HIV/AIDS and tuberculosis). In this paper we compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in malaria transmission dynamics. Relationships between the basic reproduction number for malaria transmission dynamics between humans and mosquitoes and the extinction thresholds of corresponding continuous-time Markov chain models are derived under certain assumptions. The stochastic model is formulated using the continuous-time discrete state Galton-Watson branching process (CTDSGWbp). The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction from stochastic models contribute crucial knowledge on disease control and elimination and mitigation of infectious diseases. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that malaria outbreak is more likely if the disease is introduced by infected mosquitoes as opposed to infected humans. These insights demonstrate the importance of a policy or intervention focusing on controlling the infected mosquito population if the control of malaria is to be realized.

#### 1. Introduction

Malaria is an infectious disease caused by the Plasmodium parasite and transmitted between humans through bites of female* Anopheles* mosquitoes. Approximately half of the world’s population is at risk of malaria. It remains one of the most prevalent and lethal human infections throughout the world. An estimated 40% of the world’s population lives in malaria endemic areas. Most cases and deaths occur in sub-Saharan Africa. It causes an estimated to million cases and 1.5 to 2.7 million deaths each year worldwide. Africa shares 80% of the cases and 90% of deaths [1]. According to the website of the World Health Organization [2] there were approximately 214 million new cases of malaria and 438,000 deaths worldwide in 2015. Most cases were reported in the African region.

Recently, the incidence of malaria has been rising due to drug resistance. Various control strategies have been taken to reduce malaria transmissions. Since the first mathematical model of malaria transmission was introduced by Ross [3], quite a number of mathematical models have been formulated to investigate the transmission dynamics of malaria. Xiao and Zou [4] used mathematical models to explore a natural concern of possible epidemics caused by multiple species of malaria parasites in one region. They found that epidemics involving both species in a single region are possible.

Li and others [5] considered fast and slow dynamics of malaria model with relapse and analyzed the global dynamics by using the geometric singular perturbation theory. They suggested that a treatment should be given to symptomatic patients completely and adequately rather than to asymptomatic patients. On the other hand, for the asymptomatic patients, their results strongly suggested that, to control and eradicate the malaria, it is very necessary for governments to control the relapse rate strictly. Relapse is when symptoms reappear after the parasites had been eliminated from blood but persist as dormant hypnozoites in liver cells [6]. This commonly occurs between 8 and 24 weeks and is commonly seen with* P. vivax* and* P. ovale* infections. Other papers also consider the influence of relapse in giving up smoking or quitting drinking; please see [7].

Chitnis et al. [8] and Li et al. [5] assumed that the recovered humans have some immunity to the disease and do not get clinically ill but they still harbour low levels of parasite in their blood streams and can pass the infection to mosquitoes. After some period of time, they lose their immunity and return to the susceptible class. Unfortunately, Li and others did not consider that the recovered humans will return to their infectious state because of incomplete treatment.

Stochasticity is fundamental to biological systems. In some situations the system can be treated as a large number of similar agents interacting in a homogeneously mixing environment, and so the dynamics are well-captured by deterministic ordinary differential equations. However, in many situations, the system can be driven by a small number of agents or strongly influenced by an environment fluctuating in space and time [9–12].

Stochastic models incorporate discrete movements of individuals between epidemiological classes and not average rates at which individuals move between classes [13–15]. In stochastic epidemic models, numbers in each class are integers and not continuously varying quantities [13]. A significant possibility is that the last infected individual can recover before the disease is transmitted and the infection can only reoccur if it is reintroduced from outside the population [16]. In contrast, most deterministic models have the flaw that infections can fall to very low levels—well below the point at which there is only one infected individual only to rise up later [17]. In addition, the variability introduced in stochastic models may result in dynamics that differ from the predictions made by deterministic models [16].

For a large population size and a large number of infectious individuals, the deterministic threshold provides a good prediction of a disease outbreak. However, this prediction breaks down when the outbreak is initiated by a small number of infectious individuals. In this setting, Markov chain (MC) models with a discrete number of individuals are more realistic than deterministic models where the number of individuals is assumed to be continuous-valued [18].

Motivated by these works, in this paper, we propose a model which is an extension of the model formulated by Huo and Qui (2014), who assumed that the pseudorecovered humans can recover and return to the susceptible class or relapse and become infectious again. Using the extended model, we will formulate the basic reproductive number and use it to compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in malaria transmission dynamics.

This paper is organized as follows; in Section 2, we present a malaria transmission deterministic model with relapse, which is an extension of the model in [6]. We compute the basic reproduction number, , of the malaria transmission deterministic model using the next-generation matrix approach. The stochastic version of the deterministic model and its underlying assumptions necessary for model formulation are presented and discussed in Section 3. In this section, we also compute the stochastic threshold for disease extinction or invasion by applying the multitype Galton-Watson branching process. In Section 4, we show the relationship between reproductive number of the deterministic model and the thresholds for disease extinction of the stochastic version; we also illustrate our results using numerical simulations. We conclude with a discussion of the results in Section 5.

#### 2. The Malaria Deterministic Model

The interaction of the Mosquito-host is shown in Figure 1.