We develop a multi-group epidemic framework via virtual dispersal where the risk of infection is a function of the residence time and local environmental risk. This novel approach eliminates the need to define and measure contact rates that are used in the traditional multi-group epidemic models with heterogeneous mixing. We apply this approach to a general n-patch SIS model whose basic reproduction number R0 is computed as a function of a patch residence-times matrix ℙ. Our analysis implies that the resulting n-patch SIS model has robust dynamics when patches are strongly connected: there is a unique globally stable endemic equilibrium when R0 > 1 while the disease free equilibrium is globally stable when R0 ≤ 1. Our further analysis indicates that the dispersal behavior described by the residence-times matrix ℙ has profound effects on the disease dynamics at the single patch level with consequences that proper dispersal behavior along with the local environmental risk can either promote or eliminate the endemic in particular patches. Our work highlights the impact of residence times matrix if the patches are not strongly connected. Our framework can be generalized in other endemic and disease outbreak models. As an illustration, we apply our framework to a two-patch SIR single outbreak epidemic model where the process of disease invasion is connected to the final epidemic size relationship. We also explore the impact of disease prevalence driven decision using a phenomenological modeling approach in order to contrast the role of constant versus state dependent ℙ on disease dynamics.

The dynamics, control, and evolution of communicable and vectorborne diseases are intimately connected to the joint dynamics of epidemiological, behavioral, and mobility processes that operate across multiple spatial, temporal, and organizational scales. The identification of a theoretical explanatory framework that accounts for the pattern regularity exhibited by a large number of host-parasite systems, including those sustained by host-vector epidemiological dynamics, is but one of the challenges facing the coevolving fields of computational, evolutionary, and theoretical epidemiology. Host-parasite epidemiological patterns, including epidemic outbreaks and endemic recurrent dynamics, are characteristic to well-identified regions of the world; the result of processes and constraints such as strain competition, host and vector mobility, and population structure operating over multiple scales in response to recurrent disturbances (like El Niño) and climatological and environmental perturbations over thousands of years. It is therefore important to identify and quantify the processes responsible for observed epidemiological macroscopic patterns: the result of individual interactions in changing social and ecological landscapes. In this perspective, we touch on some of the issues calling for the identification of an encompassing theoretical explanatory framework by identifying some of the limitations of existing theory, in the context of particular epidemiological systems. Fostering the reenergizing of research that aims at disentangling the role of epidemiological and socioeconomic forces on disease dynamics, better understood as complex adaptive systems, is a key aim of this perspective.infectious disease | risk | complex adaptive systems | mobility | behavior

International audienceWe consider SIS, SIR and MSIR models with standard mass action and varying population, with $n$ different pathogen strains of an infectious disease. We also consider the same models with vertical transmission. We prove that under generic conditions a competitive exclusion principle holds. To each strain a basic reproduction ratio can be associated. It corresponds to the case where only this strain exists. The basic reproduction ratio of the complete system is the maximum of each individual basic reproduction ratio. Actually we also define an equivalent threshold for each strain. The winner of the competition is the strain with the maximum threshold. It turns out that this strain is the most virulent, i.e., this is the strain for which the endemic equilibrium gives the minimum population for the susceptible host population. This can be interpreted as a pessimization principle.On considère les modèles SIS, SIR et MSIR avec la loi de l'action de masse standard et une population non constante, avec n différentes souches de pathogènes. Nous considérons aussi les même modèles avec transmission verticale. On prouve que sous une condition générique, le principe de compétition exclusive est vérifié. Pour chaque souche, un nombre de reproduction de base est associé. Il correspond au cas où seule cette souche existe. Le nombre de reproduction de base du système complet est le maximum de tous les nombres de reproduction de base pris individuellement. Nous définissons aussi un seuil équivalent pour chaque souche. La souche qui gagne la compétition est celle qui maximise le nombre de reproduction de base. C'est aussi la souche la plus virulente, i.e., c'est la souche pour laquelle l'équilibre endémique donne le minimum des individus susceptibles dans la population hôte. C'est le principe de pessimisation

A multi-patch and multi-group modeling framework describing the dynamics of a class of diseases driven by the interactions between vectors and hosts structured by groups is formulated. Hosts' dispersal is modeled in terms of patch-residence times with the nonlinear dynamics taking into account the effective patch-host size. The residence times basic reproduction number R 0 is computed and shown to depend on the relative environmental risk of infection. The model is robust, that is, the disease free equilibrium is globally asymptotically stable (GAS) if R 0 ≤ 1 and a unique interior endemic equilibrium is shown to exist that is GAS whenever R 0 > 1 whenever the configuration of host-vector interactions is irreducible. The effects of patchiness and groupness, a measure of host-vector heterogeneous structure, on the basic reproduction number R 0 , are explored. Numerical simulations are carried out to highlight the effects of residence times on disease prevalence.Mathematics Subject Classification: 92D30, 34D23, 34A34, 34C12 .

We develop a multi-patch and multi-group model that captures the dynamics of an infectious disease when the host is structured into an arbitrary number of groups and interacts into an arbitrary number of patches where the infection takes place. In this framework, we model host mobility that depends on its epidemiological status, by a Lagrangian approach. This framework is applied to a general SEIRS model and the basic reproduction number [Formula: see text] is derived. The effects of heterogeneity in groups, patches and mobility patterns on [Formula: see text] and disease prevalence are explored. Our results show that for a fixed number of groups, the basic reproduction number increases with respect to the number of patches and the host mobility patterns. Moreover, when the mobility matrix of susceptible individuals is of rank one, the basic reproduction number is explicitly determined and was found to be independent of the latter if the matrix is also stochastic. The cases where mobility matrices are of rank one capture important modeling scenarios. Additionally, we study the global analysis of equilibria for some special cases. Numerical simulations are carried out to showcase the ramifications of mobility pattern matrices on disease prevalence and basic reproduction number.

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In November 2015, El Salvador reported their first case of Zika virus (ZIKV) infection, an event followed by an explosive outbreak that generated over 6000 suspected cases in a period of two months. National agencies began implementing control measures that included vector control and recommending an increased use of repellents. Further, in response to the alarming and growing number of microcephaly cases in Brazil, the importance of avoiding pregnancies for two years was stressed. In this paper, we explore the role of mobility within communities characterized by extreme poverty, crime and violence. Specifically, the role of short term mobility between two idealized interconnected highly distinct communities is explored in the context of ZIKV outbreaks. We make use of a Lagrangian modeling approach within a two-patch setting in order to highlight the possible effects that short-term mobility, within highly distinct environments, may have on the dynamics of ZIKV outbreak when the overall goal is to reduce the number of cases not just in the most affluent areas but everywhere. Outcomes depend on existing mobility patterns, levels of disease risk, and the ability of federal or state public health services to invest in resource limited areas, particularly in those where violence is systemic. The results of simulations in highly polarized and simplified scenarios are used to assess the role of mobility. It quickly became evident that matching observed patterns of ZIKV outbreaks could not be captured without incorporating increasing levels of heterogeneity. The number of distinct patches and variations on patch connectivity structure required to match ZIKV patterns could not be met within the highly aggregated model that is used in the simulations.

We formulate a two-patch mathematical model for Ebola Virus Disease dynamics in order to evaluate the effectiveness of cordons sanitaires, mandatory movement restrictions between communities while exploring their role on disease dynamics and final epidemic size. Simulations show that severe restrictions in movement between high and low risk areas of closely linked communities may have a deleterious impact on the overall levels of infection in the total population.