Abstract
The main goal of this article is to devise the spatial-temporal spread of TB, in multiple neighboring domains, taking into account the epidemiological diversity of their populations. However, since both the environment and any population are spatially heterogeneous, it is obviously desirable to include spatial structure into an epidemic model. Individuals with tuberculosis can spread the disease by moving from one area to another. In addition, people travel by air between cities, so diseases can be spread quickly between very distant places (as was the case with the COVID-19). In our model, each region’s studied population is divided into five compartments S, L1, I, L2, and R. Further, we introduce in our discrete systems three control variables which represent the effectiveness rates of vaccination, travel-blocking operation, and treatment. We focus in our study to control the outbreaks of an epidemic that affects a hypothetical population belonging to a specific region. Firstly, we analyze the epidemic model when the control strategy is based on the vaccination control only, and secondly, when the travel-blocking control is added, we finish with the introduction of the treatment control. The optimal control theory, based on Pontryagin’s maximum principle, is applied thrice in this paper, for the characterizations of the vaccination, travel-blocking, and treatment controls. The numerical results associated with the multipoint boundary value problems are obtained based on the forward-backward sweep method combined with progressive-regressive Runge–Kutta fourth-order schemes.
Highlights
Conventional mathematical models have focused only on the temporal spread of infection but ignored or neglected spatial dynamics, the spatial spread of the epidemic was observed and many infectious diseases were transmitted from one region to another
Among the examples in human history there is the black death which spread in Europe in the 13th century, measles and smallpox between the 16th and 17th centuries which spread in the New World [1, 2], HIV/AIDS in the year 1981, the West Nile virus that emerged in North America in the late 1990s [3], SARS that appeared in Asia in 2003 [4], and recently the new corona virus (COVID-19) that appeared at the end of 2019 [5]
We present numerical simulations associated with the aforementioned optimal control problem
Summary
Conventional mathematical models have focused only on the temporal spread of infection but ignored or neglected spatial dynamics, the spatial spread of the epidemic was observed and many infectious diseases were transmitted from one region to another. E epidemic could spread over large and remote areas It can reach continents, and among the examples in human history there is the black death (plague) which spread in Europe in the 13th century, measles and smallpox between the 16th and 17th centuries which spread in the New World [1, 2], HIV/AIDS in the year 1981, the West Nile virus that emerged in North America in the late 1990s [3], SARS that appeared in Asia in 2003 [4], and recently the new corona virus (COVID-19) that appeared at the end of 2019 [5]. Before this disease was diagnosed at the end of 2019, infected people were moving from city to city, and after diagnosing some infected cases and identifying the problem, it took some time for the political decision to prevent all movement, especially from the affected areas, and the regions were divided according to the number of infected cases it contains (from one affected continent to another, or from one affected country to another, or from one affected city to another . . .) [7]. us, in order to control spatial spread of the disease, it was necessary to follow all precautions and take all necessary measures to reduce displacement to and from the affected areas
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