Analyzing the dynamics of an SIRS vaccination model with waning natural and vaccine-induced immunity

• Published in: Nonlinear analysis: real world applications 2011-10, Vol.12 (5), p.2692-2705
• Main Authors: Elbasha, E.H., Podder, C.N., Gumel, A.B.
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Backward bifurcation; Equilibria; Immunity; Nonlinearity; Protective; Reproduction; Reproduction number; Stability; Unity; Vaccination; Vaccines
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2011.03.015

This paper characterizes the qualitative dynamics of a vaccination model, with waning natural and vaccine-induced immunity, representing the transmission of an infectious agent in a population. The deterministic SIRS model with vaccination undergoes backward bifurcation when the associated reproduction number ( R v ) is less than unity. It is shown that the backward bifurcation phenomenon does not occur under three scenarios, namely when (i) the vaccine is fully protective, (ii) the vaccine is partially protective and the rate of loss of natural immunity does not exceed that of waning of vaccine-induced immunity, or (iii) the vaccine is partially protective but the duration of vaccine failure (measured in terms of the product of the residual susceptibility to infection in vaccinated individuals and the duration of vaccine protection) is below a certain critical value. The model has a unique endemic equilibrium when R v > 1 . For the case where the rate of loss of natural immunity does not exceed that of vaccine-derived immunity, it is shown, using a non-linear Lyapunov function of Goh–Volterra type, that the disease-free equilibrium of the model is globally asymptotically stable when R v < 1 . Furthermore, for this case, the unique endemic equilibrium of the model is globally asymptotically stable if the reproduction threshold ( R v ) exceeds unity. The model is extended to incorporate additional vaccine characteristics and to allow for two classes of vaccine-induced immunity (high and low), thereby representing the duration of vaccine-derived immunity using a gamma distribution. The disease-free equilibrium of the extended model is locally asymptotically stable if the associated reproduction number is less than unity. The extended model has a unique endemic equilibrium if the reproduction number exceeds unity.