An SIRS model is developed to account for precautionary measures and immunity loss. The existence and stability of equilibria are studied. It is shown that precautionary measures can induce Hopf bifurcations leading to the occurrence of cyclical behavior. The model is then extended to a two-patch scenario to explore how disease spread patterns are influenced by dispersal (travel). Theoretical analyses establish the stability of the disease-free equilibrium, the basic reproduction number, and thresholds related to travel rates. The persistence of the system, as well as the existence of boundary and endemic equilibria, are also discussed. Using Hopf bifurcation theory, we further examine the interaction between nonlinear incidence functions, travel rates, and precaution delay effects in shaping the stability of the endemic equilibrium. The findings reveal a strong connection between reduced infection rates due to precaution and the emergence of Hopf bifurcations, emphasizing the importance of timely and accurate disease information in curbing the spread of diseases. Additionally, the study highlights the significant impact of different infection force functions on equilibrium stability, underscoring the critical role of precautionary measures in disease transmission mechanisms. The results also show the diverse effects of travel rates on disease spread, suggesting that restricting travel may not always lead to favorable outcomes. This underscores the necessity for governments to consider multiple factors comprehensively in their efforts to prevent and control diseases.
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