Chicken Pox (also called Varicella) is a disease caused by a virus known as Varicella Zoster Virus (VZV) also known as human herpes virus 3 (HHV -3). Varicella Zoster Virus (VZV) is a DNA virus of the Herpes group, transmitted by direct contact with infective individuals. In this work, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination strategy was solved, using Adomian Decomposition Method (ADM) and Fourth-Fifth Rungekutta Felhberg Method and Approximate solutions were realized. ADM, yields analytical solution in terms of rapidly convergent infinite power series with easily computed terms. This solution was realized by applying Adomian polynomials to the nonlinear terms in the system. Similarly, fourth-fifth-order Runge-Kutta Felberg method with degree four interpolant (RK45F) was used to compute a numerical solution that was used as a reference solution to compare with the semianalytical approximations. The main advantage of the ADM is that it yields an approximate series solution in close form with accelerated convergence. The effect of Varicella was considered in five compartments: The Susceptible, the Vaccinated, the Exposed, the Infective and the Recovered class. The Varicella Zoster virus model which is a nonlinear system can only be solved conveniently using powerful semi-analytic tool such as the ADM. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.
A compartmentalized deterministic mathematical model for the dynamics of HIV/AIDS under the use of male and female condoms has been formulated and studied qualitatively. Disease-free equilibria of the sub-models have been found to be locally and asymptotically stable. Stability results revealed threshold values for the proportions of susceptible and infected subpopulations that must use condom in order to achieve control, and possibly, eradication of HIV/AIDS in heterosexual populations. Condom use rate for the susceptible subpopulations has been found to be bounded above by the population's birth rate, while that of the infected subpopulations is bounded below by a given threshold.
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