Department of Mathematics
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Item Optimal control for therapeutic drug treatment on a delayed model incorporating immune response(World Scientific, 2016) Dubey, Balram; Dubey, Uma S.Millions of people get infected every year by viral pathogens. Newly emergent diseases such as Ebola, Swine-flu, HIV/AIDS, etc. are spreading worldwide at an alarming rate. We introduced a delayed mathematical model with immune response and therapeutic drug treatment to understand the dynamics of pathogenimmune interaction. Here, we are considering the innate immune response and the two major component of the acquired immune response, namely, cytotoxic T lymphocytes (CTLs) and humoral immunity. This model also incorporates the absorption of pathogens i.e. loss of pathogens and its related mechanisms. Further, an optimal control model is formulated with two optimal controls i.e. maximization of uninfected cells count and minimization of cost of treatments. This is done by using the Pontryagins' Maximum Principle. Existence of non-negative equilibria is established and their stability behavior is studied using theory of ordinary differential equations. Further, numerical simulations are carried out to exemplify the qualitative results.Item A mathematical model for the effect of toxicant on the immune system(World Scientific, 2007) Dubey, Balram; Dubey, Uma S.In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of environmental toxicant on the immune response of the body. Criteria for local stability, instability and global stability are obtained. It is shown that the immune response of the body decreases as the concentration of environmental toxicant increases, and certain criteria are obtained under which it settles down at its equilibrium level. In the absence of toxicant, an oscillatory behavior of immune system and pathogenic growth is observed. However, in the presence of toxicant, oscillatory behavior is not observed. These studies show that the toxicant may have a grave effect on our body's defense mechanism.Item Modeling the interaction between avascular cancerous cells and acquired immune response(World Scientific, 2008) Dubey, Balram; Dubey, Uma S.This paper deals with the interaction between dispersed cancer cells and the major populations of the immune system, namely, the T helper cells, T Cytotoxic cells, B cells, and antibodies produced. The system is described by a set of five ordinary differential equations. Both local and global stability of the system has been investigated. It has been observed that under appropriate conditions this interaction is capable of controlling the growth of these cancer cells. The analytical findings are supported by numerical and computational analytical methods.Item Modeling effects of toxicant on uninfected cells, infected cells and immune response in the presence of virus(World Scientific, 2011) Dubey, Balram; Dubey, Uma S.In this paper, two mathematical models are proposed and analyzed. The first one deals with the interaction of uninfected cells, infected cells, viruses and immune response within humans. The second one deals with the effects of environmental toxicant on the first model. In each case, sufficient conditions for local stability and global stability of the equilibria are obtained, computer simulations are performed and the result is biologically interpreted. It has been seen that the environmental toxicant has detrimental effects on healthy cells, infected cells as well as on the immune response of the body.