Identifying the steady states of a population is a key issue in theoretical ecology, that includes the study of spatially heterogeneous populations. There are several examples of real ecosystems in patchy environments where the habitats are heterogeneous in their local density dependence. We investigate a multi-patch model of a single species with spatial dispersal, where the growth of the local population is logistic in some localities (negative density dependence) while other patches exhibit a strong Allee effect (positive density dependence). When the local dynamics is logistic in each patch and the habitats are interconnected by dispersal then the total population has only the extinction steady state and a componentwise positive equilibrium, corresponding to persistence in each patch. We show that animal populations in patchy environments can have a large number of steady states if local density dependence varies over the locations. It is demonstrated that, depending on the network topology of migration routes between the patches, the interaction of spatial dispersal and local density dependence can create a variety of coexisting stable positive equilibria. We give a detailed description of the multiple ways dispersal can rescue local populations from extinction.
Abstract. National boundaries have never prevented infectious diseases from reaching distant territories; however, the speed at which an infectious agent can spread around the world via the global airline transportation network has significantly increased during recent decades. We introduce an SEAIRbased, antigravity model to investigate the spread of an infectious disease in two regions which are connected by transportation. As a submodel, an age-structured system is constructed to incorporate the possibility of disease transmission during travel, where age is the time elapsed since the start of the travel. The model is equivalent to a large system of differential equations with dynamically defined delayed feedback. After describing fundamental but biologically relevant properties of the system, we detail the calculation of the basic reproduction number and obtain disease transmission dynamics results in terms of R0. We parametrize our model for influenza and use real demographic and air travel data for the numerical simulations. To understand the role of the different characteristics of the regions in the propagation of the disease, three distinct origin-destination pairs are considered. The model is also fitted to the first wave of the influenza A(H1N1) 2009 pandemic in Mexico and Canada. Our results highlight the importance of including travel time and disease dynamics during travel in the model: the invasion of disease-free regions is highly expedited by elevated transmission potential during transportation.
We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptibleinfected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations.
We describe a new approach for investigating the control strategies of compartmental disease transmission models. The method rests on the construction of various alternative next-generation matrices, and makes use of the type reproduction number and the target reproduction number. A general metapopulation SIRS (susceptible-infected-recovered-susceptible) model is given to illustrate the application of the method. Such model is useful to study a wide variety of diseases where the population is distributed over geographically separated regions. Considering various control measures such as vaccination, social distancing, and travel restrictions, the procedure allows us to precisely describe in terms of the model parameters, how control methods should be implemented in the SIRS model to ensure disease elimination. In particular, we characterize cases where changing only the travel rates between the regions is sufficient to prevent an outbreak. ARTICLE HISTORY
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