Analysis of a delayed malaria transmission model including vaccination with waning immunity and reinfection

To study the impact of infection delays in human and mosquito populations, vaccination with waning immunity and reinfection on the malaria transmission process, a malaria transmission model with these factors is developed and investigated. The local stability of disease-free and endemic equilibria have been discussed explicitly. By taking the delay as the bifurcation parameter, the existence of Hopf bifurcation is analyzed in four cases. Using normal form theory and center manifold theorem, direction and stability of Hopf bifurcation are discussed. Numerically, the bifurcation diagrams show that both delays can destabilize the endemic equilibrium and cause Hopf bifurcation and irregular oscillations, and that stability switches can occur mainly because of the delay in human. In addition, the malaria transmission case of Nigeria is studied. Numerical analysis reveals that ignoring the waning of immunity and reinfection may underestimate the infection risk and enlarge the critical value of Hopf bifurcation. Moreover, combined with sensitivity analysis, we can see that even though vaccination is not so effective in reducing the basic reproduction number, it is efficient for controlling the disease.