On the Exact analytical solution and Van der Pol like equation of infectious diseases model with the time-dependent total population

Abstract

In this paper, we derive the exact analytical solution in the parametric form of the infectious diseases (SIR) model, taking into account the population migration and vaccines SIRVN. By applying derivatives and substitutions, we convert the SIRVN equation into nonlinear third-order differential equation, and get an approximate semi-analytical solution in the form of a parametric function. The long-time oscillatory behavior of SIRVN model studies reduces to Van der Pol like equation with nonlinear damping. An analytic solution is obtained by multi-scale analysis and the Laplace transform methods. The result shows the comparison between the exact solution and the Jakarta outbreak data correlate of about R2 = 0.99. We also found that the vaccine effectively reduces the outbreak’s peak, and the asymptotic stability implies that Jakarta will change from the pandemic to the endemic. Finally, the solutions of Van der Pol-like equation show that the existence of multiple outbreak waves can be explained by this model.