Stochastic and deterministic SIS patch model

Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. Furthermore, we show the existence and the global stability of a unique endemic equilibrium provided that the migration rates of susceptible and infectious individuals are equal. Finally, we compare the equilibra with those of the homogeneous model, and with those of isolated patches

Impact of network assortativity on epidemic and vaccination behaviour

The resurgence of measles is largely attributed to the decline in vaccine adoption and the increase in mobility. Although the vaccine for measles is readily available and highly successful, its current adoption is not adequate to prevent epidemics. Vaccine adoption is directly affected by individual vaccination decisions, and has a complex interplay with the spatial spread of disease shaped by an underlying mobility (travelling) network. In this paper, we model the travelling connectivity as a scale-free network, and investigate dependencies between the network's assortativity and the resultant epidemic and vaccination dynamics. In doing so we extend an SIR-network model with game-theoretic components, capturing the imitation dynamics under a voluntary vaccination scheme. Our results show a correlation between the epidemic dynamics and the network's assortativity, highlighting that networks with high assortativity tend to suppress epidemics under certain conditions. In highly assortative networks, the suppression is sustained producing an early convergence to equilibrium. In highly disassortative networks, however, the suppression effect diminishes over time due to scattering of non-vaccinating nodes, and frequent switching between the predominantly vaccinating and non-vaccinating phases of the dynamics.Comment: 17 pages, 13 figure

Consequences of Short Term Mobility Across Heterogeneous Risk Environments: The 2014 West African Ebola Outbreak

abstract: In this dissertation the potential impact of some social, cultural and economic factors on Ebola Virus Disease (EVD) dynamics and control are studied. In Chapter two, the inability to detect and isolate a large fraction of EVD-infected individuals before symptoms onset is addressed. A mathematical model, calibrated with data from the 2014 West African outbreak, is used to show the dynamics of EVD control under various quarantine and isolation effectiveness regimes. It is shown that in order to make a difference it must reach a high proportion of the infected population. The effect of EVD-dead bodies has been incorporated in the quarantine effectiveness. In Chapter four, the potential impact of differential risk is assessed. A two-patch model without explicitly incorporate quarantine is used to assess the impact of mobility on communities at risk of EVD. It is shown that the overall EVD burden may lessen when mobility in this artificial high-low risk society is allowed. The cost that individuals in the low-risk patch must pay, as measured by secondary cases is highlighted. In Chapter five a model explicitly incorporating patch-specific quarantine levels is used to show that quarantine a large enough proportion of the population under effective isolation leads to a measurable reduction of secondary cases in the presence of mobility. It is shown that sharing limited resources can improve the effectiveness of EVD effective control in the two-patch high-low risk system. Identifying the conditions under which the low-risk community would be willing to accept the increases in EVD risk, needed to reduce the total number of secondary cases in a community composed of two patches with highly differentiated risks has not been addressed. In summary, this dissertation looks at EVD dynamics within an idealized highly polarized world where resources are primarily in the hands of a low-risk community – a community of lower density, higher levels of education and reasonable health services – that shares a “border” with a high-risk community that lacks minimal resources to survive an EVD outbreak.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201

Understanding the Impact of Social Factors on the Transmission Dynamics of Infectious Diseases Across Highly Heterogeneous Risk Environments.

abstract: This dissertation explores the impact of environmental dependent risk on disease dynamics within a Lagrangian modeling perspective; where the identity (defined by place of residency) of individuals is preserved throughout the epidemic process. In Chapter Three, the impact of individuals who refuse to be vaccinated is explored. MMR vaccination and birth rate data from the State of California are used to determine the impact of the anti-vaccine movement on the dynamics of growth of the anti-vaccine sub-population. Dissertation results suggest that under realistic California social dynamics scenarios, it is not possible to revert the influence of anti-vaccine contagion. In Chapter Four, the dynamics of Zika virus are explored in two highly distinct idealized environments defined by a parameter that models highly distinctive levels of risk, the result of vector and host density and vector control measures. The underlying assumption is that these two communities are intimately connected due to economics with the impact of various patterns of mobility being incorporated via the use of residency times. In short, a highly heterogeneous community is defined by its risk of acquiring a Zika infection within one of two "spaces," one lacking access to health services or effective vector control policies (lack of resources or ignored due to high levels of crime, or poverty, or both). Low risk regions are defined as those with access to solid health facilities and where vector control measures are implemented routinely. It was found that the better connected these communities are, the existence of communities where mobility between risk regions is not hampered, lower the overall, two patch Zika prevalence. Chapter Five focuses on the dynamics of tuberculosis (TB), a communicable disease, also on an idealized high-low risk set up. The impact of mobility within these two highly distinct TB-risk environments on the dynamics and control of this disease is systematically explored. It is found that collaboration and mobility, under some circumstances, can reduce the overall TB burden.Dissertation/ThesisDoctoral Dissertation Applied Mathematics for the Life and Social Sciences 201

Mathematical Investigation of the Spatial Spread of an Infectious Disease in a Heterogeneous Environment

Outbreaks of infectious diseases can devastate a population. Researchers thus study the spread of an infection in a habitat to learn methods of control. In mathematical epidemiology, disease transmission is often assumed to adhere to the law of mass action, yet there are numerous other incidence terms existing in the literature. With recent global outbreaks and epidemics, spatial heterogeneity has been at the forefront of these epidemiological models. We formulate and analyze a model for humans in a homogeneous population with a nonlinear incidence function and demographics of birth and death. We allow for the combination of host immunity after recovery from infection or host susceptibility once the infection has run its course in the individual. We compute the basic reproduction number, R0, for the system and determine the global stability of the equilibrium states. If R0 \u3c = 1, the population tends towards a disease-free state. If R0 \u3e 1, an endemic equilibrium exists, and the disease is persistent in the population. This work provides the framework needed for a spatially heterogeneous model. The model is then expanded to include a set of cities (or patches), each of which is structured from the homogeneous model. Movement is introduced, allowing travel between the cities at different rates. We assume there always exists a potentially non-direct route between two cities, and the movement need not be symmetric between two patches. Further, each city has its own nonlinear incidence function, demographics, and recovery rates, allowing for realistic interpretations of country-wide network structures. New global stability results are established for the disease-free equilibrium and endemic equilibrium, the latter utilizing a graph theoretic approach and Lyapunov functions. Asymptotic profiles are determined for both the disease-free equilibrium and basic reproduction number as the diffusion of human individuals is faster than the disease dynamics. A numerical investigation is performed on a star network, emulating a rural-urban society with a center city and surrounding suburbs. Numerical simulations give rise to similar and contrasting behavior for symmetric movement to the proposed asymmetric movement. Conjectures are made for the monotonicty of the basic reproduction number in terms of the diffusion of susceptible and infectious individuals. The limiting behavior of the system as the diffusion of susceptibles halts is shown to experience varying behavior based on the location of hot spots and biased movement

Mathematical Modelling of Ecological Systems in Patchy Environments

In this thesis, we incorporate spatial structure into different ecological/epidemiological systems by applying the patch model. Firstly, we consider two specific costs of dispersal: (i) the period of time spent for migration; (ii) deaths during the dispersal process. Together with the delayed logistic growth, we propose a two-patch model in terms of delay differential equation with two constant time delays. The costs of dispersal, by themselves, only affect the population sizes at equilibrium and may even drive the populations to extinction. With oscillations induced by the delay in logistic growth, numerical examples are provided to illustrate the impact of loss by dispersal. Secondly, we study a predator-prey system in a two-patch environment with indirect effect (fear) considered. When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. The benefit of an anti-predation response is also included. We investigate the effect of anti-predation response on population dynamics by analyzing the model with a fixed response level and study the anti-predation strategies from an evolutionary perspective by applying adaptive dynamics. Thirdly, we explore the short-term or transient dynamics of some SIR infectious disease models over a patchy environment. Employing the measurements of reactivity of equilibrium and amplification rates previously used in ecology to study the response of an ecological system to perturbations to an equilibrium, we analyze the impact of the dispersals/travels between patches and other disease-related parameters on short term dynamics of these spatially structured disease models. This contrasts with most existing works on modelling the dynamics of infectious disease which are only interested in long-term disease dynamics in terms of the basic reproduction number