Journal of Applied Mathematics

Volume 2017 (2017), Article ID 6754097, 15 pages

https://doi.org/10.1155/2017/6754097

## A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality

Department of Mathematics, Polytechnic University of Bobo Dioulasso, 01 BP 1091, Bobo-Dioulasso 01, Burkina Faso

Correspondence should be addressed to Boureima Sangaré

Received 22 January 2017; Accepted 26 April 2017; Published 4 June 2017

Academic Editor: Sabri Arik

Copyright © 2017 Bakary Traoré 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

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than , then the disease-free equilibrium is globally asymptotically stable and if it is greater than , then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.

#### 1. Introduction

Malaria is an infectious disease caused by plasmodium parasite which is transmitted to humans through the bites of infectious female mosquitoes. According to the estimations of World Health Organization (WHO) in 2015, 3.2 billion persons were at risk of infection and 2.4 million new cases were detected with 438,000 cases of deaths. However sub-Saharan Africa remains the most vulnerable region with high rate of deaths due to malaria.

To reduce the impact of malaria in the world, many scientific efforts were done including mathematical models construction. The first model of malaria transmission was developed by Ross [1]. According to Ross, if the mosquito population can be reduced to below a certain threshold, then malaria can be eradicated. Later, Macdonald did some modifications to the model and included superinfection. He showed that reducing the number of mosquitoes has little effect on the epidemiology of malaria in areas of intense transmission [2]. Nowadays, several mathematical models have been developed in order to reduce the malaria death rate in the world [3, 4]. In spite of the efforts made, it is still difficult to predict future malaria intensity, particularly in view of climate change.

It must be noticed that transmission and distribution of vector-borne diseases are greatly influenced by environmental and climatic factors. Seasonality and circadian rhythm of mosquito population, as well as other ecological and behavioural features, are strongly influenced by climatic factors such as temperature, rainfall, humidity, wind, and duration of daylight [5]. Moreover, in most mathematical models, the mosquito life cycle is generally ignored because eggs, larvae, and pupae are not involved in the transmission cycle. That is a useful simplification of the system but unfortunately the results of these models do not predict malaria intensity in most endemic regions. Thus, it is necessary to consider the life cycle of mosquitoes and the seasonality effect, which are very important aspects of the dynamics of malaria transmission.

Recently, Moulay et al. [6] have formulated a mathematical model describing the mosquito population dynamics which takes into account autoregulation phenomena of eggs and larvae stages. They have defined a threshold and proved that the growth of the mosquito population is governed by that threshold. Considering the climatic factors and the mosquitoes life cycle, we formulate a mathematical model describing the dynamics of malaria transmission. We analyze the impact of the model describing the mosquito population dynamics on the model of malaria transmission. Besides, by using the comparison theorem and the theory of uniform persistence, we, respectively, study the global stability of the nontrivial disease-free equilibrium [7–10] and the existence of positive periodic solutions.

This paper is organized as follows. In Section 2, we formulate the mathematical model of our problem. Section 3 provides the mathematical analysis of the model. Computational simulations are performed in Section 4 in order to illustrate our mathematical results. In the last section, Section 5, we conclude and give some remarks and future works.

#### 2. Model Formulation

Motivated by the compartmental models in [6, 11], we derive an age-structured malaria model with seasonality to account for the cross infection between mosquitoes and humans. The human population is divided into four epidemiological categories representing the state variables:* the susceptible* class ,* exposed* class ,* infectious* class , and* recovered* class (immune and asymptomatic, but slightly infectious humans). In the life cycle of anopheles, there are mainly two major stages: mature stage and aquatic stage. Therefore, we divide the mosquitoes population into these stages: immature and mature. The immature stage is divided in two compartments:* eggs* class , larvae and pupae class . In the mature stage, we have three compartments:* the susceptible* class ,* exposed* class , and* infectious* class . At any time, the total number of humans and mature mosquitoes is given, respectively, byIt is assumed throughout this paper that (**H1**)all vector population measures refer to densities of female mosquitoes,(**H2**)the mosquitoes bite only humans,(**H3**)there is no vertical transmission of malaria,(**H4**)all the new recruits are susceptibles.

##### 2.1. Interactions between Humans and Mosquitoes

When an infectious mosquito bites a susceptible human, the parasite enters the body of the human with a probability and the human moves into the exposed class . Some time after, he leaves from class to class with rate . Infectious humans migrate into the class after acquisition of their immunity with rate . The immunized lose their immunity with rate if they do not have continuous exposure to infection. Humans leave the total population through natural death rate and malaria death rate .

Similarly, when a susceptible mosquito bites an infectious human, it enters the class with a probability . Some time after, it leaves from class to infective class with rate where it remains for life. Mature mosquitoes leave the population through natural mortality .

Using the standard incidence as in the model of Ngwa and Shu [4], we define, respectively, the infection incidence from mosquitoes to humans, , and from humans to mosquitoes, :Furthermore, using the above assumptions, we obtain the transfer diagram (Figure 1) of the model.