A nonstandard Volterra difference equation for the SIS epidemiological model

作者：Lubuma Jean M S^*; Terefe Yibeltal A

来源：Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A-Matematicas, 2015, 109(2): 597-602.

DOI：10.1007/s13398-014-0203-5

• SIS model
• Volterra integral equation
• Nonstandard finite difference scheme
• Dynamics preserving scheme

By considering the contact rate as a function of infective individuals and by using a general distribution of the infective period, the SIS-model extends to a Volterra integral equation that exhibits complex behaviour such as the backward bifurcation phenomenon. We design a nonstandard finite difference (NSFD) scheme, which is reliable in replicating this complex dynamics. It is shown that the NSFD scheme has no spurious fixed-points compared to the equilibria of the continuous model. Furthermore, there exist two threshold parameters and , such that the disease-free fixed-point is globally asymptotically stable (GAS) for , the basic reproduction number, less than and unstable for , while it is locally asymptotically stable (LAS) and coexists with a LAS endemic fixed-point for . A unique GAS endemic fixed-point exists when and . Numerical experiments that support the theory are provided.

• 出版日期2015-9