The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model

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We study dynamical behavior and bifurcation structure of a multi-strain SIR epidemiological model with seasonal forcing in transmission rate. The conventional single strain SIR dynamics with seasonal forcing is known to show a cascade of bifurcations as the strength of seasonality increases. In this study with an extension to multiple strains of pathogens, we first investigate the bifurcation patterns of the two strain SIR model. In the two strain SIR model, a new parameter called cross-immunity between strains plays a key role in the dynamical behavior. As analogous to the single strain model, we found that the period doubling bifurcation occurs as the strength of seasonal forcing is increased. However, bifurcation patterns differ greatly both in their subharmonic periods and the relative phase of cycles between strains, depending mostly on the degree of the cross-immunity between strains. With strong and weak cross-immunities, the period doubling cascade proceeds gradually toward chaotic cycles. On the other hand, with intermediate cross-immunity, the loss of stability of annual cycle attractor is immediately followed by chaotic cycles. Asynchronous cycles of two strains are robust outcome when annual cycles lose stability, but the population can converge to perfectly synchronized biennial cycles if two strains are antigenically distant from each other. Second, we found that there are simultaneously stable multiple attractors in two strain SIR model. Biennial and chaotic attractors, e.g., occasionally coexist at the same degree of seasonal forcing, and the trajectories converge to one of them depending on the initial conditions. Such multiple attractors are widely seen between biennial and annual cycles, or among the different types of biennial cycles, at the same strength of the seasonality. Finally, we found that the population may switch from one attractor to another by introducing a small random noise in seasonally varying transmission rate.