Abstract
In this paper, we formulate an optimal control problem based on a tuberculosis model with multiple infectious compartments and time delays. In order to have a more realistic model that allows highlighting the role of detection, loss to follow-up and treatment in TB transmission, we propose an extension of the classical SEIR model by dividing infectious patients in the compartment (I) into three categories: undiagnosed infected (I), diagnosed patients who are under treatment (T) and diagnosed patients who are lost to follow-up (L). We incorporate in our model delays representing the incubation period and the time needed for treatment. We also introduce three control variables in our delayed system which represent prevention, detection and the efforts that prevent the failure of treatment. The purpose of our control strategies is to minimize the number of infected individuals and the cost of intervention. The existence of the optimal controls is investigated, and a characterization of the three controls is given using the Pontryagin's maximum principle with delays. To solve numerically the optimality system with delays, we present an adapted iterative method based on the iterative Forward-Backward Sweep Method (FBSM). Numerical simulations performed using Matlab are also provided. They indicate that the prevention control is the most effective one. To the best of our knowledge, it is the first work to apply optimal control theory to a TB model which considers infectious patients diagnosis, loss to follow-up phenomenon and multiple time delays.
Highlights
Infectious diseases, the scourge of humanity, have marked the history of human societies
Taking into consideration the works mentionned above and as an extension of a previous work [43] in which we have discussed the global dynamic of an SIR model with two stages of infection, we propose here an optimal control problem based on a more realistic SEIR model for TB that includes two infectious levels, multiple time delays and control terms
The paper is organised as follows: in Section 2, we present our mathematical model with time delays and control terms
Summary
Infectious diseases, the scourge of humanity, have marked the history of human societies. Since Daniel Bernoulli’s famous work [1], mathematical models have become one of the most important tools used in fighting against epidemics They help to understand the dynamics of infectious diseases and to estimate the effect of different control and prevention strategies. Taking into consideration the works mentionned above and as an extension of a previous work [43] in which we have discussed the global dynamic of an SIR model with two stages of infection, we propose here an optimal control problem based on a more realistic SEIR model for TB that includes two infectious levels, multiple time delays and control terms. This work is the first to apply optimal control theory on a TB model which considers the loss to follow-up phenomenon, detection of infectious individuals and multiple time delays.
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