A disease transmission model in a nonconstant population

Abstract

A general SIRS disease transmission model is formulated under assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. For a class of incidence functions it is shown that the model has no periodic solutions. By contrast, for a particular incidence function, a combination of analytical and numerical techniques are used to show that (for some parameters) periodic solutions can arise through homoclinic loops or saddle connections and disappear through Hopf bifurcations.

The Basic Reproduction Number of an Infectious Disease in a Stable Population: The Impact of Population Growth Rate on the Eradication Threshold

“…Following a pioneering study by May and Anderson [40], this issue has been discussed mainly based on a system governed by ordinary differential equations ( [41], [4], [5], [6], [7], [47]). Mathematical extensions have been made particularly with respect to the assumption of the loss of acquired immunity which was incorporated in the similar system ( [9], [19], [35]). In practice, only a few quantitative studies investigated practical implications of varying population size using the observed data, e.g., in ageing and decreasing population in Italy ( [36], [58]).…”

“…ψ denotes the normalized stable age distribution, (a) the survival rate with respect to the natural death rate, r 0 the intrinsic rate of natural increase determined by the real root of the Euler-Lotka equation: 9) and b 0 is the crude birth rate of the stable population given by…”

“…Several studies assuming homogeneous patterns of transmission have dealt with growing host population, the basic system of which was usually governed by ordinary differential equations ( [40], [41], [47], [50]). In addition, a rigorous series of mathematical studies was reported on the transmission dynamics in a growing population with loss of acquired immunity ( [35], [5], [6], [7], [9], [19]). However, except for a few studies ( [32], [33], [52], [36], [58]), limited number of epidemic models have taken into account both the age structure of host population and its growth.…”

Stability analysis of an SIR model with immunity and modified transmission function

“…[12,13]. A very general form of non-linear incidence rate was considered by Derrick and Driessche [14].One of the most fundamental quantities used by epidemiologists is certainty the basic reproduction number consider and analyzed by Anderson R M, May R M [2].In this paper the result is written in terms of basic reproduction number and stability of the equilibriums are investigated.…”

Modeling Adaptive Behavior in Influenza Transmission

“…In [2], Capasso and Serio proposed a contact rate that is a decreasing function of disease prevalence. More general infection forces were investigated by Liu, Hethcote and Levin [11], Liu, Levin and Iwasa [12], Derrick and van den Driessche [4], and by Wang [22]. Ruan and Wang [15] studied the saturated contact rate of an epidemic model where saddle-node bifurcation, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation can occur and there exist two limit cycles.…”