Abstract:Backward bifurcation has significant implications for disease control. Deriving necessary and sufficient conditions for backward bifurcation is of paramount importance to understand the reasons for its occurrence and devise effective control strategies. In this paper, we review the methods that lead to necessary and sufficient conditions for backward bifurcation in infectious disease models. We review separately the methods that apply to ODEs and methods that apply to PDEs. We further propose a new method, app… Show more
“…Stability of equilibria to the model ( 2) is somewhat unusual, particularly with regard to interior equilibria. In general, when a backward bifurcation occurs in epidemic models, the endemic equilibrium with a smaller proportion of infected individuals (i) is unstable, while the other endemic equilibrium with a larger proportion of infected individuals is stable [23]. However, this is not the case here (Figure 2(c)).…”
Section: Analytical and Numerical Results Of Model (2)mentioning
We propose two models inspired by the COVID-19 pandemic: a coupled disease-human behaviour (or disease-game theoretic), and a coupled disease-human behaviour-economic model, both of which account for the impact of social-distancing on disease control and economic growth. The models exhibit rich dynamical behaviour including multistable equilibria, a backward bifurcation, and sustained bounded periodic oscillations. Analyses of the first model suggests that the disease can be eliminated if everybody practices full social-distancing, but the most likely outcome is some level of disease coupled with some level of social-distancing. The same outcome is observed with the second model when the economy is weaker than the social norms to follow health directives. However, if the economy is stronger, it can support some level of socialdistancing that can lead to disease elimination.
“…Stability of equilibria to the model ( 2) is somewhat unusual, particularly with regard to interior equilibria. In general, when a backward bifurcation occurs in epidemic models, the endemic equilibrium with a smaller proportion of infected individuals (i) is unstable, while the other endemic equilibrium with a larger proportion of infected individuals is stable [23]. However, this is not the case here (Figure 2(c)).…”
Section: Analytical and Numerical Results Of Model (2)mentioning
We propose two models inspired by the COVID-19 pandemic: a coupled disease-human behaviour (or disease-game theoretic), and a coupled disease-human behaviour-economic model, both of which account for the impact of social-distancing on disease control and economic growth. The models exhibit rich dynamical behaviour including multistable equilibria, a backward bifurcation, and sustained bounded periodic oscillations. Analyses of the first model suggests that the disease can be eliminated if everybody practices full social-distancing, but the most likely outcome is some level of disease coupled with some level of social-distancing. The same outcome is observed with the second model when the economy is weaker than the social norms to follow health directives. However, if the economy is stronger, it can support some level of socialdistancing that can lead to disease elimination.
“…Several studies have examined the potential causes that lead to the appearance of a backward bifurcation. Among the most common caused are the use of imperfect vaccines, exogenous reinfection (e.g TB), vaccine-derived immunity waning at a slower rate than natural immunity, treatment, behavioral changes, superinfection, disease-induced mortality in vector-borne models, differential susceptibility in risk-structured models, and others (see [ 41 , 66 ] and the references therein). Fig.…”
Section: Mixing Functions and Disease Spreadmentioning
Since the start of the still ongoing COVID-19 pandemic, there have been many modeling efforts to assess several issues of importance to public health. In this work, we review the theory behind some important mathematical models that have been used to answer questions raised by the development of the pandemic. We start revisiting the basic properties of simple Kermack-McKendrick type models. Then, we discuss extensions of such models and important epidemiological quantities applied to investigate the role of heterogeneity in disease transmission e.g. mixing functions and superspreading events, the impact of non-pharmaceutical interventions in the control of the pandemic, vaccine deployment, herd-immunity, viral evolution and the possibility of vaccine escape. From the perspective of mathematical epidemiology, we highlight the important properties, findings, and, of course, deficiencies, that all these models have.
“…Having methods for deriving necessary and sufficient conditions for backward bifurcation can delineate the reasons for its occurrence and help devise control strategies that combat its negative effects. A recent review article [10] collects methods for analysis of backward bifurcation in ODE models and age-structured partial differential equation (PDE) models. The reader may find some mechanisms that create backward bifurcations for endemic steady states in [7].…”
Section: Introductionmentioning
confidence: 99%
“…The importance of this example is the development of the tools to handle the non-linear, non-local boundary condition associated with age-since-infection structured models. Example 3 is an age-since-infection structured cholera model considered as a test example in [10], where we derive a necessary and sufficient condition for backward bifurcation using several different methods. Here we use the newly proposed Lyapunov-Schmidt method and we reach a similar condition for backward bifurcation.…”
Section: Introductionmentioning
confidence: 99%
“…Here we use the newly proposed Lyapunov-Schmidt method and we reach a similar condition for backward bifurcation. Compared to the methods in [10] the Lyapunov-Schmidt method developed here is somewhat easier to use but it is based on more abstract approach.…”
Backward bifurcation is an important property of infectious disease models. A centre manifold method has been developed by Castillo-Chavez and Song for detecting the presence of backward bifurcation and deriving a necessary and sufficient condition for its occurrence in Ordinary Differential Equations (ODE) models. In this paper, we extend this method to partial differential equation systems. First, we state a main theorem. Next we illustrate the application of the new method on a chronological age-structured Susceptible-Infected-Susceptible (SIS) model with density-dependent recovery rate, an age-since-infection structured HIV/AIDS model with standard incidence and an age-since-infection structured cholera model with vaccination.
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