A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness

Abstract

The study of the dynamics of human infectious disease using deterministic models is typically carried out under the assumption that a critical mass of individuals is available and involved in the transmission process. However, in the study of animal disease dynamics where demographic considerations often play a significant role, this assumption must be weakened. Models of the dynamics of animal populations often naturally assume that the presence of a minimal number of individuals is essential to avoid extinction. In the ecological literature, this a priori requirement is commonly incorporated as an Allee effect. The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion in as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.

Dynamics Analysis of a Stochastic SIR Epidemic Model

“…That is, the solutions of model (3) are stochastically permanent. From Figures 1(a) and 1(b), one can see that with increasing the noise intensities, the solutions of model (3) will be oscillating strongly around the endemic point * of model (4).…”

“…Almost all mathematical models for the transmission of infectious diseases descend from the classical susceptibleinfective-removed (SIR) model of Kermack and McKendrick [1]. The dynamic behavior of different epidemic models and a lot of their extensions is well investigated by a number of scholars; see [2][3][4][5][6][7][8][9][10][11]. The basic and important research subjects for recent studies are the existence of the threshold values which distinguish whether the disease dies out, the stability of the disease-free and the endemic equilibria, permanence, and extinction [12].…”

Nonlinear dynamics of avian influenza epidemic models

“…Controlling the disease in the avian population is very difficult and the basic reproduction numbers do not provide effective control measures for the human population. Notice that β h (the transmission rate from infective avian to susceptible human) appears in the expressions (34) and (37) for the steady state values of I * * h , the number of infective human individuals. In fact, it should be understood that β h = c h p h , where c h is the contact rate between a susceptible human and an infective bird and p h is the probability of transmitting the virus per contact.…”

“…In wild populations of Serins (Serinus serinus), Senar and Conroy [48] reported that avian pox infections were very virulent and survival rates of infected birds were half that of uninfected ones. Recently, great attention has been paid to the theoretical modeling and analysis of the joint interplay of infectious disease and Allee effects (see Hilker et al [26,27], Thieme et al [51], Friedman and Yakubu [20,21], Kang and Castillo-Chavez [34,35], and the references cited therein). On one hand, it has been observed that recurrent infectious disease outbreaks tend to enhance the deleterious role of Allee effects within diseases capable of inducing reductions in host fitness (Kang and Castillo-Chavez [35]).…”