SIR epidemiological model with ratio-dependent incidence: influence of
preventive vaccination and treatment control strategies on disease
dynamics
Abstract
In this paper, we study an SIR epidemic model with ratio dependent
incident rate function. We explore the impact of vaccination and
treatment on the transmission dynamics of the disease. The treatment
control strategies depend on the availability of maximal treatment
capacity: treatment rate is constant when the number of infected
individuals is greater than the maximal capacity of treatment and
proportional to the number of infected individuals when the number of
infected individuals is less than the maximal capacity of treatment. The
existence and stability of the endemic equilibria are governed by the
basic reproduction number and treatment control strategies. By carrying
out rigorous mathematical analysis and numerical evaluations, it has
been shown that (1) the sufficiently large value of the preventive
vaccination rate can control the spread of disease, (2) a threshold
level of the psychological (or inhibitory) effects in the incidence rate
function is enough to decrease the infective population. Model system
also undergoes a transcritical and a saddle-node bifurcation with
respect to disease contact rate. In the presence of treatment
strategies, system have multiple endemic equilibria and undergoes a
backward bifurcation. The number of infected individuals decreases with
respect to maximal treatment capacity and disease dies out from the
system for large capacity of the treatment when constant treatment
strategy is applied. Further, it is also found that the spread of
disease can be suppressed by increasing treatment rate. Sensitivity
analysis shows that the transmission and treatment rates are most
sensitive parameters on the model system.