This paper considers the Susceptible-Infected-Vaccinated-Recovered (SIRV) deterministic model with a non linear force of infection and treatment, where individual humans that are vaccinated losses their vaccination after some time and become vulnerable to infections. The basic reproduction number R 0 obtained from the model system is an epidemic threshold that determines if a disease will continue to ravage the human population or not. The model state equations considered in this paper possess two steady-state solutions such that if R 0 < 1, the infection-absent steady-state solutions are locally and globally asymptotically stable. Also, if R 0 > 1, a unique infection-persistent steady-state solutions are established, which is also locally and globally asymptotically stable. Thus, it leads to the persistence of infections in the human host population. Finally, numerical simulations were carried out to validate our theoretical results.
This paper presents the solution of wave equations on transmission lines where leakage to ground on the line is very small. As a result of the leakages to ground on the transmission lines which are negligible, the conductance and the inductance, which are responsible for leakages on the line, are set to zero in the model of the general wave equation of the transmission line. The Laplace transform method was now applied to transform the resulting partial differential equation into ordinary differential equation and the method of variation of parameters was used to get the particular solution to the problem.
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