Mathematical analysis and optimal control applied to the treatment of leukemia
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Mathematical analysis and optimal control applied to the treatment of leukemia Mst. Shanta Khatun1 · Md. Haider Ali Biswas1 Received: 27 October 2019 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract Leukemia is a worldwide public health problem. It is the result of unconstrained growth of immature white blood cells in the blood. In this paper, we propose a compartmental model of leukemia in terms of a system of ordinary nonlinear differential equations and apply optimal control technique in obtaining optimal strategies to minimize the number of infected cells in the blood. Initially, a mathematical model of leukemia is formulated. We verify the stability of the equilibria (disease-free and endemic equilibrium point) of the model based on the basic reproduction number. After words, in order to minimize the number of infected cells as well as the costs of the control, we develop the proper optimal control model by using Pontryagin’s maximum principle. Finally, numerical simulation of the proposed model and optimal control model are performed to show that the effectiveness of immune boosting drugs as well as immunotherapy to defend leukemia in the blood. Keywords Leukemia · Treatment · Mathematical model · Optimal control · Numerical analysis Mathematics Subject Classification Primary: 93A30 · 49K15 · 92D30 · Secondary: 34A34
1 Introduction Recently, cancer has become one of the top most causes of death of human being among the non-communicable diseases. It is the second leading cause of death in the world [37]. It takes our valuable life every year. It is said that “cancer attacks a body
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Mst. Shanta Khatun [email protected] Md. Haider Ali Biswas [email protected]
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Mathematics Discipline, Science Engineering and Technology School, Khulna University, Khulna 9208, Bangladesh
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Mst. S. Khatun, Md. H. A. Biswas
Fig. 1 Global statistics of number of people died from leukemia, 1990 to 2018 [37]
Fig. 2 American statistics of number of leukemia cases from 1990 to 2018 [3]. Showing data labels and position left of the data points
without knowing and leaves the body after causing death”. There is no one in the world who does not know about the overwhelming shock of cancer. There are many types of cancer that take shelter in the human body. The mysterious leukemia is one of them. It is the uncontrolled growth of immature white blood cells in the body. In 1990, about 2,41,630 people died from leukemia all over the world and since then every year the rate of leukemia related death is increasing (see Figs. 1, 2). Leukemia is mysterious because when blood from a leukemic patient is transfused to a healthy person it doesn’t attack the person. This is because of the strong immune system of the healthy person. However, it is more devastating as it aims our immune cells. Leukemia affects the blood and bone marrow. It has two possible phases: (1) acute (2) chronic. If the disease is acute, immature lymphoid cells grow rapidly in the blood. Acute leukemia frequently occurs in child
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