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Behavior change significantly influences the transmission of diseases during outbreaks. To incorporate spontaneous preventive measures, we propose a model that integrates behavior change with disease transmission. The model represents behavior change through an imitation process, wherein players exclusively adopt the behavior associated with higher payoff. We find that relying solely on spontaneous behavior change is insufficient for eradicating the disease. The dynamics of behavior change are contingent on the basic reproduction number R a corresponding to the scenario where all players adopt non-pharmaceutical interventions (NPIs). When R a < 1 , partial adherence to NPIs remains consistently feasible. We can ensure that the disease stays at a low level or maintains minor fluctuations around a lower value by increasing sensitivity to perceived infection. In cases where oscillations occur, a further reduction in the maximum prevalence of infection over a cycle can be achieved by increasing the rate of behavior change. When R a > 1 , almost all players consistently adopt NPIs if they are highly sensitive to perceived infection. Further consideration of saturated recovery leads to saddle-node homoclinic and Bogdanov–Takens bifurcations, emphasizing the adverse impact of limited medical resources on controlling the scale of infection. Finally, we parameterize our model with COVID-19 data and Tokyo subway ridership, enabling us to illustrate the disease spread co-evolving with behavior change dynamics. We further demonstrate that an increase in sensitivity to perceived infection can accelerate the peak time and reduce the peak size of infection prevalence in the initial wave. [ABSTRACT FROM AUTHOR]

The phase reduction approach has manifested its efficacy in investigating synchronization behaviors in limit-cycle oscillators. However, spatial distributions of the phase value on the limit cycle may lead to illusions of synchronizations for oscillators close to bifurcations. In this paper, we compared the phase sensitivity function in the spatial domain and time domain for oscillators close to saddle-node homoclinic (SNH) bifurcation, also known as saddle-node bifurcation on an invariant circle. It was found that the phase sensitivity function in the spatial domain can show the phase accumulation feature on the limit cycle, which can be ignored in the phase sensitivity function in the time domain. As an example, the synchronization distributions of uncoupled SNH oscillators driven by common and independent noises were investigated, where the space-dependent coupling function was considered on common noise. These results shed some light on the phase dynamics of oscillators close to bifurcations. [ABSTRACT FROM AUTHOR]

In this paper, we systematical study the rich dynamics and complex bifurcations of a non-monotonic predator–prey system with a constant releasing rate for the predator. We prove that the system can have at most three positive equilibria, and can undergo a sequence of bifurcations, including transcritical, saddle-node, Hopf, degenerate Hopf, double limit cycle, saddle-node homoclinic bifurcation (or homoclinic loop with a saddle-node), cusp bifurcation of codimension 2, and Bogdanov–Takens bifurcation of codimension 2 and 3. And the system can generate very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexistent periodic orbits, homoclinic loops, etc. Moreover, our results show that the dynamical behaviors highly rely on the constant releasing rate of predators and the initial conditions. That is, there exists a critical value of the constant releasing rate of predators such that (i) when the constant releasing rate is greater than the critical value, the prey goes to extinction for all admissible initial populations of both species; (ii) when the constant releasing rate is less than the critical value, the prey can always coexist with the predator. Numerical simulations are presented to verify the main results. [ABSTRACT FROM AUTHOR]