We construct an SIR-type compartmental model for the spread of COVID-19 which incorporates as a parameter the susceptible individuals' cautiousness level, and utilises nonlinear incidence and recovery rates. We determine the model's basic reproduction number, study the stability of the existing equilibria analytically, and carry out a sensitivity analysis to confirm that the cautiousness level is one of the parameters upon which the basic reproduction number depends most sensitively. Fixing specific values for all other parameters, we study numerically the rich dynamics of the model as the cautiousness level varies over its domain, revealing backward transcritical, Hopf, and saddle-node bifurcations of equilibria, as well as homoclinic and fold bifurcations of limit cycles with the aid of AUTO. Finally, in view of some key events affecting the pandemic in Indonesia, we design a scenario in which the cautiousness level varies over time, and show that the model exhibits a hysteresis, whereby, a slight cautiousness decrease (e.g., from $0.17$ to $0.16$) could bring a disease-free equilibrium state to endemic, and this is subsequently reversible only by a drastic cautiousness increase (from $0.16$, not merely back to $0.17$, but to exceed $0.35$), thereby mathematically justifying the importance of a high cautiousness level for resolving the pandemic.
Resolving COVID-19: How Important Is the People's Cautiousness Level for Its Successfulness?