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

Abstract

AbstractWhen 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 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 is not sufficient for effective disease control, and disease outbreak can reach a high endemic level even though . Permanence of our model (i.e., uniform strong persistence of the disease) is also discussed for the case .