Continuous multistep methods for volterra integro-differential equations of the second order

A new class of numerical methods for Volterra integro-differential equations of the second order is developed. The methods are based on interpolation and collocation of the shifted Legendre polynomial as basis function with Trapezoidal quadrature rules. The convergence analysis revealed that the methods are consistent and zero stable, hence their convergence. Numerical examples revealed that the methods compared favourably with existing standard methods.Keywords: Consistency, Zero stable, Continuous multistep methods, Volterra integro-differential equation, Convergent, Trapezoidal rul

Modelling Hepatitis B Virus Transmission Dynamics In The Presence of Vaccination and Treatment

Hepatitis B is a global threat as over a billion people have been infected and about 300 million people die annually across the world. In this paper, a mathematical model for the transmission dynamics of hepatitis B virus infection considering vaccination and treatment as control parameters in the host population is presented. First, the disease-free equilibrium state of the model was determined. The next generation method was used to determine the basic reproduction number,  as a threshold parameter, in terms of the given model parameters.  was analytically evaluated for its sensitivity to vaccination and treatment parameters. It was proved that the disease-free equilibrium state is locally asymptotically stable if the  is below unity, otherwise, it is unstable. Local stability of the endemic equilibrium state was established using the centre manifold theory. The analytical results of the  show that increasing the proportion of people who receive vaccines, either at birth or later in life, reduces it below unity. Similarly, increasing the proportion of carriers who receive treatment achieves the same purpose. The result of the local stability analysis of the disease-free equilibrium state shows that the disease can be eliminated if  is below unity. The result of the centre manifold theory on the endemic equilibrium state shows that the disease can persist as the value of  increases above one. The findings of this study strongly suggest a combination of effective vaccination and treatment as a control strategy is crucial to the success of HBV disease control

Simulation of A Mathematical Model Of Hepatitis B Virus Transmission Dynamics In The Presence Of Vaccination And Treatment

In this paper, a mathematical model for the transmission dynamics of hepatitis B virus (HBV) infection incorporating vaccination and treatment as control parameters is presented. The basic reproduction number, , as a threshold parameter, was constructed, in terms of the given model parameters, by the next generation method.   was numerically assessed for its sensitivity to vaccination and treatment parameters. A unique disease-free equilibrium state was determined, indicating possibility of control of HBV disease. The model was solved numerically using Runge-Kutta method of order four to evaluate the effects of vaccination and treatment parameters on the prevalence of the disease. The numerical results of the sensitivity analysis show that increasing either vaccination or treatment rate has the potential of reducing  below unity. The results of the numerical simulations of the model show that effective vaccination, treatment or a combination of both of them as a control strategy can eradicate HBV disease, with the combination being far better than either of them. Finally, these findings strongly suggest that high coverage of vaccination and treatment are crucial to the success of HBV disease control

Stability results of a mathematical model for the control of HIV/AIDS with the use of male and female condoms in heterosexual populations

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.KEYWORDS: Locally and asymptotically stable, disease-free equilibrium, HIVAIDS contro