International Scholarly Research Notices

Volume 2017 (2017), Article ID 6182492, 16 pages

https://doi.org/10.1155/2017/6182492

## Modeling the Impact of Bed-Net Use and Treatment on Malaria Transmission Dynamics

^{1}Department of General Studies, School of Health Technology, Jahun, Nigeria^{2}Department of Mathematical Sciences, Bayero University Kano, Kano, Nigeria

Correspondence should be addressed to Saminu Iliyasu Bala

Received 19 March 2017; Revised 23 May 2017; Accepted 8 June 2017; Published 1 August 2017

Academic Editor: Marcos Amaku

Copyright © 2017 Bello Gimba and Saminu Iliyasu Bala. 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

We modeled the impact of bed-net use and insecticide treated nets (ITNs), temperature, and treatment on malaria transmission dynamics using ordinary differential equations. To achieve this we formulated a simple model of mosquito biting rate that depends on temperature and usage of insecticides treated bed nets. We conducted global uncertainty and sensitivity analysis using Latin Hypercube Sampling (LHC) and Partial Rank Correlation Coefficient (PRCC) in order to find the most effective parameters that affect malaria transmission dynamics. We established the existence of the region where the model is epidemiologically feasible. We conducted the stability analysis of the disease-free equilibrium by the threshold parameter. We found the condition for the existence of the endemic equilibrium and provided necessary condition for its stability. Our results show that the peak of mosquitoes biting rate occurs at a range of temperature values not on a single value as previously reported in literature. The results also show that the combination of treatment and ITNs usage is the most effective intervention strategy towards control and eradication of malaria transmissions. Sensitivity analysis results indicate that the biting rate and the mosquitoes death rates are the most important parameters in the dynamics of malaria transmission.

#### 1. Introduction

Malaria is one of the most devastating infectious diseases in the world and is caused by* Plasmodium* parasite, which is transmitted via the bites of infected mosquitoes. Among the high risk groups are pregnant women, nonimmune travelers, and children [1]. In pregnant women, malaria has adverse effect on birth outcome which includes low birth weight, abortion, and still born [1]. Apart from health related problems, it also imposes huge socioeconomic burden in malaria-endemic nations. As discussed in Forouzannia and Gumel [2] the annual economic burden of malaria in Africa alone was estimated to be around US $8 billion. These necessitated the formation of several intervention strategies in many countries to mitigate the impact of malaria disease. This includes the use of insecticide treated bed nets (ITNs), intermittent preventive treatment (IPT) especially, for pregnant women during antenatal period, reducing mosquitoes population through the destruction of breeding sites or killing of the larva stage at breading sites that cannot be destroyed [3–5]. Other interventions strategies are the use of indoor residual spraying (IRS) in killing infected mosquitoes resting indoors after blood meal and the use of sterile insect technique [6]. Despite broad efforts for eradication, malaria remains a significant problem resulting in the death of millions of people [2, 7–9]. Most malaria cases and deaths occur in sub-Saharan Africa with Nigeria and Democratic Republic of Congo accounting for about 40% of malaria mortality worldwide [9]. There are a number of characteristics of malaria disease that complicates control efforts. Typical among them is clinical immunity, which is a situation where protection against the clinical symptoms of the disease is developed despite the presence of the parasites [7, 10, 11]. Others include seasonality [12, 13] and treatment failure that might occur due to wrong dosage of medication; see [2] and the references therein. In the context of malaria transmission, seasonality encapsulates complex phenomenon whose definition varies in many studies. Temperature variations have been reported by many to play significant role in the dynamics of malaria transmissions. For example, the report of Roll Back Malaria 2015 indicates that a rise in temperature by 2-3°C will increase the number of people at climatic risk of malaria by 3–5%. Furthermore, the abundance of mosquitoes and the transmission risk have been reported to be influenced by temperature [13–16]. At high temperature, studies have indicated that people are unlikely to use ITNs much [17].

Mathematical models of malaria transmission have been developed by several researchers to gain insight into the dynamics of the disease transmission so as to contribute towards its eradication. Some of these models can be found in [18–23]. These models are in varying degree of complexity. For example, the model of [2] is an age-structured model with several compartments. The model of [7] is made up of four compartments comprising of only the human population. In the model of [5, 17] the authors introduced explicit equation for the proportion of ITNs use as a function of mosquito biting rates.

Some of the problems that complicate malaria control include (1) the presence of individuals who are clinically immune to the disease but can transmit it through bite from susceptible mosquitoes and (2) hot weather which can lead to reduction in the use of ITN. For these reasons, it is important to study the qualitative impact of treatment, immunity, and seasonality on the dynamics of malaria transmission. In this work, we present a vector-host model of malaria transmission dynamics of immune and nonimmune human populations that accounts for the impact of ITN usage and seasonality on the disease. We propose a model of mosquito biting rate as a nonlinear function of temperature and ITN usage to mimic seasonality. This will help in devising optimal intervention strategies that will offer more realistic predictions to control malaria spread. To the best of our knowledge, this is the first vector-host mathematical model for malaria transmission, which explores the impact of daily temperature variations and ITNs usage on control of malaria transmissions. The current study extends the work of [7] by designing a vector-host model for malaria transmission dynamics. The study also extends the work of [17] by modeling the mosquito biting rate as a function of temperature to mimic seasonality. The paper is organized as follows. We formulated the model in Section 2 and analyzed it qualitatively in Section 3, in Section 4 we conducted global uncertainty and sensitivity analysis, Section 5 is the discussion part, and in Section 6 we present our conclusions.

#### 2. Model Formulation

In this section, we modify an existing mathematical model for malaria transmission dynamics developed by [7]. The process of the modification is presented below. Following [7] we defined naive individuals as those who have never been infected with malaria, or those who have been infected but have not developed clinical immunity, or those who have lost all immunity. Similarly, clinically immune individuals are those with immunity to clinical symptoms. The total human population denoted by is divided into mutually exclusive subpopulations of susceptible naive , susceptible clinically immune , infected naive , and infected clinically immune , so that .

The total mosquitoes population denoted by is divided into compartments of susceptible and infected mosquitoes, so that .

All recruitment is assumed to be into the susceptible naive human population generated via birth and/or immigration at a rate The population of naive susceptible individuals () is increased by naive infected individuals that recovered without immunity at a rate , treated naive individuals that recovered without immunity at a rate , and clinically immune individuals that lost immunity at a rate The population of naive susceptible individuals () is decreased by natural death rate and force of infection (, following effective contacts with infected mosquito. Here and represent the density dependent and density independent part of human death rate and emigration, respectively. We model the force of infection from mosquitoes to human as .

Here is the probability of infection of susceptible human per bite by an infected mosquito and is the biting rate of mosquitoes on susceptible human; represents the proportion of ITN usage and depends on environmental temperature See (7) for the functional form of We assumed that temperature is a parameter that is time independent to make the analysis easier. Thus,The clinically susceptible population is generated by the treated naive individuals that become clinically immune, infected naive individuals that recovered with clinical immunity at a rate , infected clinically immune individuals that recovered with clinical immunity at a rate , and treated clinically immune infected individuals that recover at a rate It is decreased by susceptible clinically immune individuals that lose immunity at rate , natural death at rate , and the force of infection that pushed out susceptible clinically immune human into infected clinically immune population as a result of contact with infected mosquitoes at rate . We assume that clinically infected individuals recover into the clinically susceptible compartment only. Thus,The population of infected naive humans is generated by the population of the infectious susceptible naive human that become infected. It is decreased by the treated infected naive individuals at a rate , infected naive individuals that recover without immunity at a rate , infected naive individuals that recover with immunity at a rate , the natural death , and disease induced death rate Thus,The population of infected clinically immune human is generated by the population of susceptible clinically immune humans that become infected. It is decreased by the treated infected clinically immune individuals at a rate , population of clinically immune individuals that recover with immunity at a rate , the natural death , and disease induced death at a rate Thus,The population of susceptible mosquitoes is generated by birth at a rate . It is reduced by natural death , contact with ITNs at the rate , and infection when in contact with infected human at rate The parameters and represent the density independent and density dependent parts of the mosquitoes death rate, respectively. Here , , and and are the probabilities that susceptible mosquitoes become infectious after biting an infected naive or clinically immune human, respectively. Thus,The population of infected mosquitoes is increased by infected susceptible mosquitoes at rate It is decreased by the natural death and death when they come into contact with ITN Thus,In the report of [17], the authors model the mosquitoes biting rate as a linear function of ITN usage while [24] considers a more general form. None of these authors consider the impact of temperature on bed-net use despite its significance. In this work we model the biting rate aswhere represents the proportion of ITN usage and the parameters and are location and scale parameters measured in °C, respectively. The choice of temperature as a parameter in the biting rate is to mimic seasonality. The justification of this novel approach is due to many reports in literature on the relative importance of temperature in malaria transmission dynamics as outlined in the introduction. In Figure 1, we study impact of shape and scale parameters on the biting rate. From Figure 1, we observe that the optimum temperature for the biting rate is not a single temperature value as discussed in [25] but a range of values. In [15], the authors review some calibrated models of temperature variations in terms of mosquitoes biting rates. One of the findings is that biting rates are optimal at certain temperature values. Typical values reported are 24.4°C, 25.0°C, 26.3°C, and 27.5°C. In this work we are reporting a range of values that encapsulates individual values from several reports. From Figure 1(a), it can be seen that before the maximum biting rate is attained, high values of location parameter will predict relatively lower biting rates. This finding is in contrast to the result of increasing the scale parameter as depicted on Figure 1(b).