A Dynamic Model of Multiple Time-Delay Interactions between the Virus-Infected Cells and Body’s Immune System with Autoimmune Diseases

Abstract

The immune system is a complex interconnected network consisting of many parts including organs, tissues, cells, molecules and proteins that work together to protect the body from illness when germs enter the body. An autoimmune disease is a disease in which the body’s immune system attacks healthy cells. It is known that when the immune system is working properly, it can clearly recognize and kill the abnormal cells and virus-infected cells. But when it doesn't work properly, the human body will not be able to recognize the virus-infected cells and, therefore, it can attack the body’s healthy cells when there is no invader or does not stop an attack after the invader has been killed, resulting in autoimmune disease; This paper presents a mathematical modeling of the virus-infected development in the body’s immune system considering the multiple time-delay interactions between the immune cells and virus-infected cells with autoimmune disease. The proposed model aims to determine the dynamic progression of virus-infected cell growth in the immune system. The patterns of how the virus-infected cells spread and the development of the body’s immune cells with respect to time delays will be derived in the form of a system of delay partial differential equations. The model can be used to determine whether the virus-infected free state can be reached or not as time progresses. It also can be used to predict the number of the body’s immune cells at any given time. Several numerical examples are discussed to illustrate the proposed model. The model can provide a real understanding of the transmission dynamics and other significant factors of the virus-infected disease and the body’s immune system subject to the time delay, including approaches to reduce the growth rate of virus-infected cell and the autoimmune disease as well as to enhance the immune effector cells.

Mathematical Modeling the Time-Delay Interactions between Tumor Viruses and the Immune System with the Effects of Chemotherapy and Autoimmune Diseases

“…In this section, we discuss a time-delay model consisting of three partial differential equations including a growth rate of immune cells, growth rate of tumor cells, and growth amount of the concentration of chemotherapy drug based on a recent work by the author [67]. Although many assumptions below can be found in [29,67], for the sake of convenience, we will list them here as well.…”

“…Notation We use the following notation, partially from [67], throughout the paper: a = the intrinsic growth rate per unit time b = the elimination rate of the tumor cells by the healthy immune system (effector cells) per cells and unit time c = the death rate of the healthy immune system per unit time d = degree of recruitment of maximum immune-effector cells in relation with tumor cells per unit time e = capacity of the tumor cells per unit time f = the rate that the immune system attacks the body's own healthy (effector) cells, resulting in autoimmune disease per cells and unit time g = constant factor of growth rate per unit time h = the half saturation constant (cells) k = the half saturation for tumor cleanup (cells) m = the degree of inactivation of effector cells by tumor cells per cells and unit time p = parameter of tumor cells cleanup by immune-effector cells per unit time s = growth rate of immune-effector cells per unit time I(t) = the number of healthy immune-effector cells at time t V(t) = the number of tumor cells at time t Q(t) = the amount of the concentration of chemotherapy drug α = rate of decrease in concentration of chemotherapy drug β = the occurrence of a drug from outside the body β(t) = the amount of chemotherapy drug injected to a patient at a given time r = rate of the influence of the interaction between effector cells and the chemotherapy drug u = rate of the influence of the interaction between tumor cells and the chemotherapy drug.…”

“…They noticed that the number of undetected recoveries overpasses the number of detected recoveries for the proposed countries. Pham [67] recently discussed a virus-immune time-delay model of virus-infected cells and immune system but did not consider a chemotherapy drug treatment. Further research study of the time delay between the virus infection, tumor cell growth, and chemotherapy drug treatment is crucial in determining how long the current growth could continue before the slowdown owing to a newly-introduced intervention of chemotherapy treatment becoming visible.…”

“…Further research study of the time delay between the virus infection, tumor cell growth, and chemotherapy drug treatment is crucial in determining how long the current growth could continue before the slowdown owing to a newly-introduced intervention of chemotherapy treatment becoming visible. This paper aims to extend Pham's work [67] by considering the effects of time-delay chemotherapy drug treatment. In other words, this paper studies the mathematical modeling of the time-delay interactions between tumor cells and the immune system and the effects of chemotherapy and autoimmune diseases.…”

Modern Problems of Mathematical Physics and Their Applications

“…Pham in [7] has presented a mathematical modeling of the virus-infected development in the body's immune system considering the multiple time-delay interactions between the immune cells and virus-infected cells with autoimmune disease. In the proposed model, he tried to determine the dynamic progression of virus-infected cell growth in the immune system.…”