Mathematical Modelling of Human African Trypanosomiasis Using Control Measures

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, R 0, and it is also proved that if R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, which means the disease dies out. The disease persists in the population if the value of R 0 > 1. 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.

Modelling malaria dynamics with partial immunity and protected travellers: optimal control and cost-effectiveness analysis

“…The malaria transmission dynamics given by system (2) undergoes backward bifurcation as R 0 crosses unity whenever the inequality, given by (14), holds.…”

Estimating the basic reproduction number for the 2015 bubonic plague outbreak in Nyimba district of Eastern Zambia

“…By estimating R 0 , this current study was able to attach a quantitative aspect to describing and understanding the possible transmission dynamics of bubonic plague in a local setup in Zambia. With this new added knowledge of R 0 , we can now use this estimate to predict the size and progression of future outbreaks and quantitatively assess the effects of various control measures required to prevent the spread of infection through further mathematical modelling techniques [13,14,[24][25][26][27][28]37,55]. For example, one simple and classical demonstration of the invaluable use of R 0 is that it is used to determine the minimum mass treatment or vaccination coverage rate required to prevent spread of the infectious disease in a population [13,37].…”

“…As the infection was assumed to be spread through vectors, the Ross Macdonald Susceptible-Infectious-Removed-Susceptible-Infectious model, denoted as SIR-SI, was initially created as a first step in the modelling process [ 38 ]. This model type, originally developed for malaria, is a standard mathematical model for vector-borne pathogens that tracks the infections in both the human and the vector populations [ 6 , 25 – 27 , 36 , 37 , 38 , 44 – 46 ]. The parameter values used for creating the model were obtained from various literature and where such information could not be readily found the parameter values were estimated based on reasonable assumption and biological plausibility.…”

“…To give credence to the results of this primary method, a second alternative method was also used to estimate R 0 for the same bubonic plague outbreak in Nyimba by means of creating a complete compartmental mathematical model for the infection. Both these methods used however relied on the following two underlying assumptions made about the Nyimba bubonic plague outbreak: (1) Since various disease interventions were applied during the course of this outbreak (i.e antibiotic treatment of patients, indoor spraying, and sensitization), it was assumed in this study that the disease transmission must have occurred naturally in the community only during the early stages of the outbreak which caused the initial peak in cases around the 7 th of April 2015 [11,[24][25][26][27][28]. For this reason, both methods of R 0 estimation used in this study relied only on analysis of the data from the initial growth phase of the Nyimba bubonic plague outbreak leading up to its peak; i.e before the provided interventions took full effect [20,29].…”

The Role of Control Measures and the Environment in the Transmission Dynamics of Cholera

“…After some algebraic calculations, we obtain Abstract and Applied Analysis 7 Proof. Based on the approach by [31,32], the theorem is proved from the EE defined in equation (15). Since all the values are positive, the value of B * is expressed in terms of R 0 which gives…”