The spreading fronts of an infective environment in a man–environment–man epidemic model

• Published in: Applied mathematical modelling 2016-08, Vol.40 (15-16), p.7082-7101
• Main Authors: Ahn, Inkyung, Baek, Seunghyeon, Lin, Zhigui
• Format: Article
• Language: English
• Subjects: Bacteria; Epidemic model; Epidemics; Free boundaries; Free boundary; Mathematical analysis; Mathematical models; Reaction–diffusion systems; Reproduction; Spreading; Spreading and vanishing
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2016.02.038

•A free boundary problem is used to describe the expanding of bacteria in a man–environment–man epidemic model.•The basic reproduction number introduced here depends on the time.•Sufficient conditions for the spreading or vanishing of the bacteria are given.•It seems the first time to present numerical simulations for such free boundary problems. A reaction–diffusion model is investigated to understand infective environments in a man–environment–man epidemic model. The free boundary is introduced to describe the expanding front of an infective environment induced by fecally–orally transmitted disease. The basic reproduction number R0 for the non-spatial epidemic model is defined and the basic reproduction number R0F(t) for the free boundary problem is introduced, and the behavior of positive solutions to the reaction–diffusion system is discussed. Sufficient conditions for the bacteria to vanish or spread are given. We show that, if R0 ≤ 1, the bacteria always vanish, and if R0F(t0)≥1 for some t0 ≥ 0, the bacteria must spread, while if R0F(0)