Backward bifurcation and permanence of a disease‐severity‐structured epidemic model with treatment

Abstract

When an unknown pathogen is encountered, developing medicines and vaccine to counter its effects becomes a potentially urgent task. In addition to these primary medical issues, there exists a secondary problem of medical collapse; it is caused by limited treatment capacity that includes limited number of medical supplies, doctors, nurses, beds, and other medical equipment, and must be seriously considered from a public health perspective. To discuss the effects of treatment capacity on disease transmission, we present a disease-severity-structured epidemic model with necessary treatment only for severely infective individuals. We demonstrate the occurrence of backward bifurcation, wherein a stable endemic equilibrium coexists with a stable disease-free equilibrium when the basic reproduction number R 0 $\mathcal {R}_0$ is less than 1, and if the treatment capacity is relatively small. This epidemiological implication states that when there is insufficient capacity for treatment, the requirement R 0 < 1 $\mathcal {R}_0 <1$ is not sufficient for effective disease control, and disease outbreak can reach a high endemic level even though R 0 < 1 $\mathcal {R}_0 <1$ . Permanence of our model (i.e., uniform strong persistence of the disease) is also discussed for the case R 0 > 1 $\mathcal {R}_0>1$ .