Dynamics of SI epidemic with a demographic Allee eect 

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CHAPTER 4 Dynamical behavior of an epidemiological model with a demographic Allee e ect

Introduction

While Chapter 3 is primarily concerned with providing ample opportunity for taking into account major contributors to Allee e ect by modeling both demo-graphic functions as quadratic polynomials, this Chapter focuses on bifurcation behavior of an extended version of model (3.7). We extend the SI model analyzed in Chapter 3 by adding a compartment of exposed individuals and considering frequency-dependent incidence instead of density-dependent transmission.
In recent years, a number of authors reported that models with Allee e ect in the host demographics exhibit complex dynamics such as periodic oscillation, multiple stable steady states, and a series of bifurcations. Such bifurcations in-clude sub- and super-critical bifurcations, Bogdanov-Takens bifurcation etc. (see for instance, [20, 23, 25]). In a similar note, the model developed by Hilker [16] seems to be the rst account of the presence of another type of bifurcation be-havior in an epidemiological model with a demographic Allee e ect similar to the backward bifurcation discussed in Section 2.5.3 of Chapter 2. The author explored the di erences between bifurcation behaviors in epidemiological models without Allee e ect that exhibit backward bifurcations and the epidemiological model with the Allee e ect as given in Table 2.1 of Chapter 2. It is highlighted in [16] that another saddle-node bifurcation is possible in the Allee e ect model, resulting in the re-emergence of two endemic equilibria since highly pathogenic parasites cause their own extinction but not that of their host. In addition, the author noted that the second saddle-node bifurcation might not be detected by any computer software such as MatCont [71], AUTO [72] or XPPAUT [73].
In conservation biology, one of the primary goals is to understand the ecological mechanisms that make some species more prone than others to population decline and extinction [74, 75]. Such information play relevant role for guiding manage-ment actions as it would allow biologists to predict the vulnerability of species to extinction even before they decline, thereby improving the species’ chances of survival [74]. There is an evidence that some species are more vulnerable to extinction than others [76]. More precisely, the Allee e ect is to be more likely to occur when individuals bene t from the presence of conspeci cs [1, 26]. Some species, however, su er heavy mortality at low population because they rely on mass numbers and a strategy of predator dilution for survival [77]. In light of this, we deduce that the mortality rate of species whose individuals bene t from the presence of conspeci cs decreases when small, whereas the mortality rate of those whose individuals do not bene t from the presence of conspeci cs increases in such a situation. As discussed in Chapter 3, when > 0 the mortality rate decreases, while it increases if 0 for small population.
It can be observed that the de nition of the Allee e ect as given in the intro-ductory part of Chapter 1, refers to low population levels. However, whether or not the mechanisms responsible for the Allee e ect at low density or small popu-lation size a ect the dynamics of a population at high density or large population size need to be investigated. The main purpose of this chapter is to investigate the combined impact of the Allee e ect and infectious disease at high population level and to determine which species are more vulnerable to extinction than others under such a situation.

Model formulation

Let N(t) be the host total population size at time t. This population is subdi-vided into three disjoint compartments of individuals that are susceptible (S(t)), exposed (have been infected but are not yet infectious) (E(t)) and infectious (I(t)), so that N(t) = S(t) + E(t) + I(t). The respective transfer rates are given on the ow diagram depicted in Figure 4.1.

Threshold quantities

There are two well known ways of a disease control for disease transmission models with varying population size (i.e. a population with increasing or decreasing total size) due to demographic e ects [45, 78]. The rst way requires that the proportion i(t) of infectives goes to zero, whereas the second requirement is that the absolute number I(t) of infectives approaches to zero. These notions of disease elimination were given and discussed in some detail in [78]. Thus, the conditions for the linear stability of disease free equilibria and for the existence and stability of endemic proportion equilibria are required. The pertinent threshold parameters are as follows according as the population is at its carrying capacity (p = 1), minimum survival level (p = u) or extinction state (p = 0), respectively.
It is important to note that the demographic functions b(p) and d(p) are equal at the carrying capacity state and Allee threshold state. Thus, the threshold parameters R₀ and R_u The proof follows from Lemma 4.3.1 and Theorem 4.3.4. Furthermore, the results of Theorem 4.3.5 assert that if R_u 1, then the disease cannot invade a population at the edge of extinction due to the Allee e ect. This leads to a bistable system that approaches either one of the extinction states E₀=E_s or the carrying capacity state E₂. On the other hand, if R_u > 1, depending on the initial condition the host population either goes extinct or settles at its carrying capacity. Hence host extinction is possible even if the initial size is above the Allee threshold. Therefore, the disease increases the basin of extinction beyond the Allee threshold.

Bifurcation Analysis

It is evident that, when an epidemiological model admits multiple non-trivial equi-libria, the model usually exhibits complex dynamical behavior such as backward bifurcation and forward hysteresis [34, 42, 79].
In order to investigate the existence of such phenomena in model (4.2) with (4.16), we rearrange equation (4.14) with (4.16) and obtain after algebraic ma-nipulations the following quartic polynomial for p.
(p) = Q₄p⁴ + Q₃p³ + Q₂p² + Q₁p + Q₀;
It can be deduced from Case (ii) of Theorem 4.4.1 and Figure 4.2 (C) that model (4.2) with (4.16) can have two non-trivial steady states when R₀ > 1. Keep-ing all the parameters xed other than , we take as the bifurcation parameter. It should be noted that as bifurcation parameters, and R₀ can be considered essentially equivalent. More precisely, R₀ can be regarded as a function of so that R₀ is varied by varying transmissibility . We denote by ^c the value of at which the two endemic states coincide as in Figure 4.2 (D) with corresponding critical reproduction number R^c₀ = R₀( ^c). However, in the rest of this section we will discuss the dynamical behavior of model (4.2) with (4.16) in terms of R₀
Using a numerical bifurcation software ‘MatCont’, we show in Figure 4.3 how the total population p and prevalence i change with varying the threshold pa-rameter R₀. If R₀ < 1, the disease cannot invade from arbitrarily introductions into the population at carrying capacity. If R₀ > 1, however, the disease-free equilibrium E₂ loses stability, resulting in the emergence of locally stable endemic equilibrium E₂ by transcritical bifurcation. This endemic equilibrium coexists with unstable endemic equilibrium E₁ that already arises when R₀ > R^u₀. The two endemic equilibria coalesce and disappear by a saddle-node (SN) bifurcation at R₀ = R^c₀. Hence, the population goes extinct.
Furthermore, the sub-threshold R^c₀ is a tipping point for an unanticipated population collapse. Therefore, the dynamics of the system is rendered monostable whenever R₀ > R^c₀ with a semi-trivial extinction state E_s being globally stable. If R₀ < R^c₀, the system is bistable with one of the attractors being an extinction state, either E₀ or E_s according as R_e < 1 or R_e > 1, respectively. The other attractor is either the carrying capacity state E₂ if R₀ < 1 or the endemic state E₂ when R₀ > 1.
One can observe from Figure 4.6 that the value of the tipping point for the abrupt population collapse R^c₀ increases with decreasing value of . As it was highlighted in Chapter 3, the mortality rate decreases when > 0 and slowly in-creases for 0 at low population. The biological implication of the parameter as discussed earlier in the introductory section of this chapter is that the mor-tality rate of species whose individuals bene t from the presence of conspeci cs decreases slowly when small. On the other hand, species whose individuals do not bene t from the presence of conspeci cs have an increasing mortality rate at low population level. The bifurcation results depicted in Figure 4.6 indicate that the abrupt population collapse from a level of large population size p₂ is faster when > 0 than for 0. This reveals that species whose individuals bene t from the presence of conspeci cs are more vulnerable to decline and extinction at high population level. The essential mechanism behind this scenario is the simultane-ous population size depression and the increase of the extinction threshold owing to disease virulence and the Allee e ect.
It is worth mentioning here that all the results of model (4.2) with (4.16) hold true for its special cases. These are the cases when the demographic function d(p) in (4.16) becomes linear and constant for = 0 and = = 0, respectively. For the rst case, if = 1=ku, then the demographic functions in (4.16) are similar to the dimensional forms of those given in equation (1.4) of Chapter 1.

Declaration 
Dedication 
Acknowledgements 
Abstract 
1 Introduction 
2 Mathematical background 
2.1 Autonomous Systems .
2.2 Hartman -Grobman Theorem
2.2.1 Local Stability of Equilibria
2.2.1.1 Linearization Theory
2.2.1.2 Next generation method
2.3 Global Stability of Equilibria
2.3.1 Lyapunov functions and LaSalle’s Invariance Principle
2.3.1.2 Limit Sets and Invariance Principle
2.4 Qualitative Analysis
2.5 Bifurcation Theory
2.5.1 Saddle-node bifurcation
2.5.3 Backward bifurcation
2.6 Epidemiological Preliminaries
2.6.1 Incidence functions
2.6.2 The basic reproduction number
3 Dynamics of SI epidemic with a demographic Allee eect 
3.1 Introduction
3.2 Model formulation
3.3 Basic properties
3.3.1 Model (3.10) as a dynamical system.
3.3.2 Threshold quantities .
3.3.3 Existence and stability of equilibria
3.3.3.1 Disease-free equilibria
3.3.3.2 Endemic equilibria
3.3.4 The eect of disease-induced mortality on the model
3.3.4.1 The model with low disease-induced mortality:
3.3.4.2 The model with high disease-induced mortality:
3.4 Special cases
3.5 Persistence and Extinction
3.6 Summary
4 Dynamical behavior of an epidemiological model with a demographic Allee eect 
4.1 Introduction .
4.2 Model formulation
4.3 Basic properties
4.4 Bifurcation Analysis
4.5 Summary
5 Backward bifurcation analysis of an epidemiological model with par-tial immunity 
5.1 Introduction
5.2 Model formulation
5.3 Basic properties
5.4 Backward bifurcation analysis
5.5 Two special cases
5.6 Impact of vaccine
5.7 Summary
6 Conclusion and Future Work 
Bibliography
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