Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.Comment: 21 pages, 5 figures, Final revision to appear in Bulletin of Mathematical Biolog

Modelling of Human Behaviour and Response to the Spread of Infectious Diseases

We incorporate two types of human behavioural changes into the epidemic models. First, a two-subpopulation imitation dynamic model is constructed via the replicator dynamical equations to study the self-initiated pre-cautionary health protective behaviour under the cost-benefit considerations and group pressure. Second, the impacts of additional characteristics of imperfect vaccine and the asymmetric property of smoothed best response on the vaccination behaviour are studied within the vaccination population game framework, and via the Gompertz function, respectively

Differential equation and complex network approaches for epidemic modelling

This study consists of three parts. The first part focuses on bifurcation analysis of epidemic models with sub-optimal immunity and saturated treatment/recovery rate as well as nonlinear incidence rate. The second part of the research focuses on estimating the domain of attraction for sub-optimal immunity epidemic models. In the third part of the research, we develop a bond percolation model for community clustered networks with an arbitrarily specified joint degree distribution

Robustness of behaviourally-induced oscillations in epidemic models under a low rate of imported cases

This paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ODE model defined on contact networks. The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a non-oscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters.Comment: 17 pages, 11 figure

Dynamic analysis of a fractional-order SIRS model with time delay

Mathematical modeling plays a vital role in the epidemiology of infectious diseases. Policy makers can provide the effective interventions by the relevant results of the epidemic models. In this paper, we build a fractional-order SIRS epidemic model with time delay and logistic growth, and we discuss the dynamical behavior of the model, such as the local stability of the equilibria and the existence of Hopf bifurcation around the endemic equilibrium. We present the numerical simulations to verify the theoretical analysis

Spontaneous changes of human behaviors and intervention strategies: human and animal diseases

Doctor of PhilosophyDepartment of Industrial & Manufacturing Systems EngineeringChih-Hang WuThe topic of infectious disease epidemics has recently attracted substantial attentions in research communities and it has been shown that the changes of human behaviors have significant impacts on the dynamics of disease transmission. However, the study and understanding of human reactions into spread of infectious disease are still in the very beginning phase and how human behaviors change during the spread of infectious disease has not been systematically investigated. Moreover, the study of human behaviors includes not only various enforced measures by public authorities such as school closure, quarantine, vaccination, etc, but also the spontaneous self-protective actions which are triggered by risk perception and fear of diseases. Hence, the goal of this research is to study the impacts of human behaviors to the epidemic from these two perspectives: spontaneous behavioral changes and public intervention strategies. For the sake of studying spontaneous changes of human behaviors, this research first time applied evolutionary spatial game into the study of human reactions to the spread of infectious disease. This method integrated contact structures and epidemics information into the individuals’ decision processes, by adding two different types of information into the payoff functions: the local information and global information. The new method would not only advance the field of game theory, but also the field of epidemiology. In addition, this method was also applied to a classic compartmental dynamic system which is a widely used model for studying the disease transmission. With extensive numerical studies, the results first proved the consistency of two models for the sake of validating the effectiveness of the spatial evolutionary game. Then the impacts of changes of human behaviors to the dynamics of disease transmission and how information impacts human behaviors were discussed temporally and spatially. In addition to the spontaneous behavioral changes, the corresponding intervention strategies by policy-makers played the key role in process of mitigating the spread of infectious disease. For the purpose of minimizing the total lost, including the social costs and number of infected individuals, the intervention strategies should be optimized. Sensitivity analysis, stability analysis, bifurcation analysis, and optimal control methods are possible tools to understand the effects of different combination of intervention strategies or even find an appropriate policy to mitigate the disease transmission. One zoonotic disease, named Zoonotic Visceral Leishmaniasis (ZVL), was studied by adopting different methods and assumptions. Particularly, a special case, backward bifurcation, was discussed for the transmission of ZVL. Last but not least, the methodology and modeling framework used in this dissertation can be expanded to other disease situations and intervention applications, and have a broad impact to the research area related to mathematical modeling, epidemiology, decision-making processes, and industrial engineering. The further studies can combine the changes of human behaviors and intervention strategies by policy-makers so as to seek an optimal information dissemination to minimize the social costs and the number of infected individuals. If successful, this research should aid policy-makers by improving communication between them and the public, by directing educational efforts, and by predicting public response to infectious diseases and new risk management strategies (regulations, vaccination, quarantine, etc.)

The Kuramoto model in complex networks

181 pages, 48 figures. In Press, Accepted Manuscript, Physics Reports 2015 Acknowledgments We are indebted with B. Sonnenschein, E. R. dos Santos, P. Schultz, C. Grabow, M. Ha and C. Choi for insightful and helpful discussions. T.P. acknowledges FAPESP (No. 2012/22160-7 and No. 2015/02486-3) and IRTG 1740. P.J. thanks founding from the China Scholarship Council (CSC). F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grants No. 2011/50761-2 and No. 2013/26416-9) for financial support. J.K. would like to acknowledge IRTG 1740 (DFG and FAPESP).Peer reviewedPreprin

Understanding Recurrent Disease: A Dynamical Systems Approach

Recurrent disease, characterized by repeated alternations between acute relapse and long re- mission, can be a feature of both common diseases, like ear infections, and serious chronic diseases, such as HIV infection or multiple sclerosis. Due to their poorly understood etiology and the resultant challenge for medical treatment and patient management, recurrent diseases attract much attention in clinical research and biomathematics. Previous studies of recurrence by biomathematicians mainly focus on in-host models and generate recurrent patterns by in- corporating forcing functions or stochastic elements. In this study, we investigate deterministic in-host models through the qualitative analysis of dynamical systems, to reveal the possible intrinsic mechanisms underlying disease recurrence. Recurrence in HIV infection is referred to as “viral blips”, that is, transient periods of high viral replication separated by long periods of quiescence. A 4-dimensional HIV antioxidant- therapy model exhibiting viral blips is studied using bifurcation theory. Four conditions for the existence of viral blips in a deterministic in-host model are proposed. Guided by the four con- ditions, the simplest 2-dimensional infection model which shows recurrence is obtained. One key point for recurrence is identified, that is an increasing and saturating infectivity function. Furthermore, Hopf and generalized Hopf bifurcations, Bogdanov-Takens bifurcation, and ho- moclinic bifurcation are proved to exist in this 2-dimensional model. Bogdanov-Takens bifur- cation and homoclinic bifurcation provide a new mechanism for generating recurrence. From the viewpoint of modelling, the increasing and saturating infectivity function gives rise to a convex incidence rate, which further induces backward bifurcation and Hopf bifurcation, and allows the infection model to exhibit rich dynamical behavior, such as bistability, recurrence, and regular oscillation. The relapse-remission cycle in autoimmune disease is investigated based on a regulatory T cell model. By introducing a newly discovered class of regulatory T cells, Hopf bifurcation oc- curs in the autoimmune model with negative backward bifurcation, and gives rise to a recurrent pattern. The main insight of this thesis is that recurrent disease can arise naturally from the de- terministic dynamics of populations. It will provide a starting point for further research in dynamical systems theory, and recurrence in other physical systems