Global dynamics of a periodically forced SI disease model of Lotka–Volterra type

Abstract

In this paper, we investigate the dynamics of an SI disease model of Lotka–Volterra type in the presence of a periodically fluctuating environment. We give a global analysis of the dynamical behavior of the model. Interestingly, our results show that the permanence guarantees the existence of a unique positive harmonic time-periodic solution which is globally attracting when the horizontal disease transmission has a weaker impact than the intraspecific competition. While for the case when the horizontal disease transmission has a stronger impact than the intraspecific competition, we numerically show that complex dynamics such as chaos can occur in a permanent system. Nonetheless, we provide sufficient conditions for the existence and uniqueness of the positive harmonic time-periodic solution for the latter case. The impact of the environment on the spread of disease is studied by using a bifurcation analysis. We show that in each of the qualitatively different cases of the associated autonomous SI model in a constant environment, an alternative possibility can appear in the periodic model.