Dynamics of a SCIS model with treatment and a general force of infection

We developed an SCIS mathematical model with a general nonlinear force of infection in which asymptomatic carriers are able to transmit the disease. The model incorporates distinct force of infection functions for carriers and symptomatic infectious individuals, thereby allowing for nonlinear and nonmonotonic transmission effects. We proved global asymptotic stability of the disease-free equilibrium when \(R_0 \leq 1\) using a Lyapunov function, and of the endemic equilibrium when \(R_0 > 1\) by means of a Dulac function. Both results were established under the additional assumption that the incidence functions do not increase faster than the corresponding compartments, i.e., \(f(I)/I\) and \(g(C)/C\) are non-increasing functions, meaning the infection forces associated with infectious individuals and carriers grow at most linearly relative to \(I\) and \(C\), respectively. In addition, local stability conditions for both equilibria are established through linearization and eigenvalue analysis. We also derive conditions on the nonlinear force of infection and the parameters for the occurrence of a Hopf bifurcation. These results extend previous SCIS-type epidemic models by explicitly incorporating treatment, temporary immunity, and transmission driven by asymptomatic carriers within a broad class of infection mechanisms.