Mathematical Modeling and Analysis of Leukemia: Effect of External Engineered T Cells Infusion

In this paper, a nonlinear model is proposed and analyzed to study the spread of Leukemia by considering the effect of genetically engineered patients T cells to attack cancer cells. The model is governed by four dependent variables namely; naive or susceptible blood cells, infected or dysfunctional blood cells, cancer cells and immune cells. The model is analyzed by using the stability theory of differential equations and numerical simulation. We have observed that the system is stable in the local and global sense if antigenicity rate or rate of stimulation of immune cells is greater than a threshold value dependent on the density of immune cells. Further, external infusion of T cells (immune cells) reduces the concentration of cancer cells and infected cells in the blood. It is observed that the infected cells decrease with the increase in antigenicity rate or stimulation rate of immune response due to abnormal cancer cells present in the blood. This indicates that immune cells kill cancer cells on being stimulated and as antigenicity rate increases rate of destruction of cancer cells also increase leading to decrease in the concentration of cancer cells in the body. This decrease in cancer cells further causes decrease in the concentration of infected or dysfunctional cells in the body

Modeling Spread of Polio with the Role of Vaccination

In this paper, we have proposed and analyzed a nonlinear mathematical model for the spread of Polio in a population with variable size structure including the role of vaccination. A threshold parameter, R , is found that completely determines the stability dynamics and outcome of the disease. It is found that if R 1, the disease free equilibrium is stable and the disease dies out. However, if R \u3e1, there exists a unique endemic equilibrium that is locally asymptotically stable. Conditions for the persistence of the disease are determined by means of Fonda’s theorem. Moreover, numerical simulation of the proposed model is also performed by using fourth order Runge - Kutta method. Numerically, it has been found that the system exhibits steady state bifurcation for some parameter values. It is concluded from our analysis that endemic level of infective population increases with the increase in rate of transmission of infection due to infective among susceptible class that further enhances because of transmission of infection due to latent hosts. A particular value of disease transmission coefficient r is found for which exposed and infective population dies out. It is found that periodic outbreak of the disease occurs when infection due to exposed and infective class occurs at the same rate. It is also observed from our analysis that although vaccination helps in eradicating polio by decreasing endemic equilibrium level yet careful administration of vaccination is desired because if vaccine is administered during incubation period, endemic equilibrium level increases and disease persists in the population