Computational and Mathematical Methods in Medicine

Volume 2018, Article ID 5293568, 13 pages

https://doi.org/10.1155/2018/5293568

## Mathematical Modelling of Human African Trypanosomiasis Using Control Measures

^{1}African Institute for Mathematical Sciences, Biriwa, Cape Coast, Ghana^{2}University of Dar es Salaam, Dar es Salaam, Tanzania^{3}University of Dodoma, Dodoma, Tanzania^{4}University of Cape Coast, Cape Coast, Ghana

Correspondence should be addressed to Hamenyimana Emanuel Gervas; hg.ude.smia@anamiynemah

Received 10 July 2018; Revised 12 September 2018; Accepted 22 October 2018; Published 22 November 2018

Academic Editor: Konstantin Blyuss

Copyright © 2018 Hamenyimana Emanuel Gervas 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

Human African trypanosomiasis (HAT), commonly known as sleeping sickness, is a neglected tropical vector-borne disease caused by trypanosome protozoa. It is transmitted by bites of infected tsetse fly. In this paper, we first present the vector-host model which describes the general transmission dynamics of HAT. In the tsetse fly population, the HAT is modelled by three compartments, while in the human population, the HAT is modelled by four compartments. The next-generation matrix approach is used to derive the basic reproduction number, , and it is also proved that if , the disease-free equilibrium is globally asymptotically stable, which means the disease dies out. The disease persists in the population if the value of . Furthermore, the optimal control model is determined by using the Pontryagin’s maximum principle, with control measures such as education, treatment, and insecticides used to optimize the objective function. The model simulations confirm that the use of the three control measures is very efficient and effective to eliminate HAT in Africa.

#### 1. Introduction

Human African trypanosomiasis (HAT), commonly known as sleeping sickness, is a vector-borne tropical disease which is caused by *Trypanosoma brucei* protozoa species. It is one of the neglected tropical diseases which affect people in sub-Saharan Africa, specifically those living in rural areas. HAT is caused by two species of protozoa which are *Trypanosoma brucei gambiense* (TBG), which causes the chronic form of HAT in central and western Africa, and *Trypanosoma brucei rhodesiense* (TBR), which causes the acute form of the disease in eastern and southern Africa [1]. The HAT disease has killed millions of people since the beginning of century and it is transmitted from one individual to another by tsetse flies (genus ); TBG is transmitted by riverine tsetse species, while TBR is transmitted by savanna tsetse species [1]. Rhodesiense HAT is an acute disease that can lead to death if not treated within 6 months, while gambiense HAT is a slow chronic progressive disease which causes death with an average duration of 3 years [2]. The signs and symptoms for both forms of HAT are not specific and their appearances vary from one person to another; at the first stage of HAT, the disease is not severe and the signs and symptoms such as intermittent fever, headache, pruritus, lymphadenopathies, asthenia, anemia, cardiac disorders, endocrine disturbances, musculoskeletal pains, and hepatosplenomegaly may be observed, while in the second stage of HAT, sleep disorders and neuropsychiatric disorders are likely to dominate. The HAT disease can be treated by using drugs such as suramin, eflornithine, melarsoprol, and pentamidine.

The disease is reported to affect about 37 sub-Saharan African countries; it affects much rural areas where there are suitable environments for the tsetse flies to live and reproduce, and the periurban areas can also be affected. The transmission of HAT can occur during human activities such as hunting, farming, as well as fishing [3]. The transmission of HAT needs the reservoir; *reservoir* is a species that can permanently maintain the pathogen and from which the pathogen can be transmitted to the target population [4]. Rhodesiense HAT is zoonotic which requires a nonhuman reservoir (animals) for maintaining its population, while in gambiense HAT, humans act as key reservoir [4].

Mathematical models have been used to study the transmission and effective control of diseases simply and cheaply with no need of expensive and complicated experiments [5]. So far, different models have been developed and formulated by different researchers. One of the important modelling work on HAT was done by Rogers [6]; the model explained the mathematical framework on transmission of HAT in multiple host populations [6]. Rogers’ model was generalized by Hargrove et al [7], and a new parameter which allows the tsetse flies to feed off multiple hosts was introduced. The model compared the effectiveness of two methods used to control HAT: insecticide-treated cattle and the use of trypanocide drugs to treat cattle. They found out that treating cattle with insecticides is more effective and a cheaper approach to control HAT than using trypanocide drugs. Kajunguri [8] developed a model which was based on a constant population with a fixed number of domestic animals, human, and tsetse flies in one of the villages in West Africa. The major findings of their model estimated that the cattle population contributes to about of the total TBR transmission, while the rest is the contribution of human population in transmission of the disease. The study by Kajunguri [8], which also formulated a multihost model, was used to study the control of tsetse flies and TBR in southern Uganda. They found out that the effective application of insecticides brings about a cost-effective method of control and eliminating the disease. They realized that using insecticides for controlling HAT is more effective and efficient in the area where there are few wild hosts.

Due to low mortality rate of the disease and poverty of its sufferers, the efforts toward the control of HAT has reduced. Most attention is given to popular diseases such as HIV/AIDS, tuberculosis, malaria, and ebola, although the disease is still a threat to the lives of sub-Saharan African people. Moreover, very few studies have been carried out on applying optimal control theory to HAT transmission models. In this paper, we use optimal control theory to study the transmission dynamics of HAT diseases by using education, treatment, and insecticides as the control measures.

The rest of this paper is outlined as follows: Section 2 represents the vector-host model and the underlying assumptions. In section 3, the model equilibria and stabilities are determined, whereas in Section 4, the optimal control model is analyzed by modifying the previous one to control the HAT by using control measures (education, insecticides, and treatment). In addition, the numerical simulations for the optimal control model are done in this section and we use the results obtained to compare the efforts of each control measure to control the HAT in Africa. Finally, we provide the conclusion in Section 5.

#### 2. Model Formulation

In this section, the vector-host model as well as the necessary differential equations to describe the transmission of HAT from tsetse fly to human and vice versa are developed. The transmission of HAT in the human population is modelled using four subclasses: Susceptible , Exposed , Infectious , and Recovered . The total human population, , is thus defined by

The transmission of HAT in the vector (tsetse flies) population is also divided into Susceptible (), Exposed (), and Infectious (). The total population of the tsetse flies, , is also defined by

We assume a constant population for both host and vector. It is also assumed that the tsetse fly cannot recover from the disease and the infected tsetse fly remains infectious throughout the rest of its life; there is no disease-induced death rate for tsetse flies and the recruitment rates are assumed to be constant due to birth and immigration.

In our model, the recruitment rate of hosts and vectors are represented by and , respectively. The susceptible host gets the disease when bitten by infectious tsetse fly, and susceptible tsetse fly gets the disease when it bites an infectious human at the rate *a*. The natural mortality rate for humans and vectors are represented by and , respectively. The parameter *ω* represents the disease-induced death rate for humans, while and are the force of infection for humans and vectors, respectively. The parameter *σ* represents per capita rate of a vector becoming infectious, and the rest of the parameters are explained in Table 1. Assuming that the transmission per bite from infectious tsetse fly to human is *a*, then the rate of infection per susceptible human is given byand also if we further assume that *a* is the tsetse-fly biting rate, that is, the average number of bites per tsetse fly per unit, then the rate of infection per susceptible tsetse fly can be represented by