Analysis of the dynamics of a class of models for vector-borne diseases with host circulation (original) (raw)
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Large number of endemic equilibria for disease transmission models in patchy environment
Mathematical biosciences, 2014
We show that disease transmission models in a spatially heterogeneous environment can have a large number of coexisting endemic equilibria. A general compartmental model is considered to describe the spread of an infectious disease in a population distributed over several patches. For disconnected regions, many boundary equilibria may exist with mixed disease free and endemic components, but these steady states usually disappear in the presence of spatial dispersal. However, if backward bifurcations can occur in the regions, some partially endemic equilibria of the disconnected system move into the interior of the nonnegative cone and persist with the introduction of mobility between the patches. We provide a mathematical procedure that precisely describes in terms of the local reproduction numbers and the connectivity network of the patches, whether a steady state of the disconnected system is preserved or ceases to exist for low volumes of travel. Our results are illustrated on a ...
Mathematical Biosciences, 2002
A precise definition of the basic reproduction number, R o , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R o < 1, then the disease free equilibrium is locally asymptotically stable; whereas if R o > 1, then it is unstable. Thus, R o is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super-and sub-threshold endemic equilibria for R o near one. This criterion, together with the definition of R o , is illustrated by treatment, multigroup, staged progression, multistrain and vectorhost models and can be applied to more complex models. The results are significant for disease control.
Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate
Chaos, Solitons & Fractals, 2009
In this paper, the SEIR epidemic model with vertical transmission and the saturating contact rate is studied. It is proved that the global dynamics are completely determined by the basic reproduction number R 0 (p, q), where p and q are fractions of infected newborns from the exposed and infectious classes, respectively. If R 0 (p, q) 6 1, the disease-free equilibrium is globally asymptotically stable and the disease always dies out. If R 0 (p, q) > 1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists.
GLOBAL STABILITY OF AN EPIDEMIC MODEL IN A PATCHY ENVIRONMENT
We investigate an SIR compartmental epidemic model in a patchy environment where individuals in each compartment can travel among n patches. We derive the basic reproduction number R 0 and prove that, if R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable. In the case of R 0 > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable.
Transmission dynamics for vector-borne diseases in a patchy environment
Journal of Mathematical Biology, 2013
In this paper, a mathematical model is derived to describe the transmission and spread of vector-borne diseases over a patchy environment. The model incorporates into the classic Ross-MacDonald model two factors: disease latencies in both hosts and vectors, and dispersal of hosts between patches. The basic reproduction number R 0 is identified by the theory of the next generation operator for structured disease models. The dynamics of the model is investigated in terms of R 0. It is shown that the disease free equilibrium is asymptotically stable if R 0 < 1, and it is unstable if R 0 > 1; in the latter case, the disease is endemic in the sense that the variables for the infected compartments are uniformly persistent. For the case of two patches, more explicit formulas for R 0 are derived by which, impacts of the dispersal rates on disease dynamics are also explored. Some numerical computations for R 0 in terms of dispersal rates are performed which show visually that the impacts could be very complicated: in certain range of the parameters, R 0 is increasing with respect to a dispersal rate while in some other range, it can be decreasing with respect to the same dispersal rate. The results can be useful to health organizations at various levels for setting guidelines or making policies for travels, as far as malaria epidemics is concerned.
Modelling the Spread of Infectious Diseases in Complex Metapopulations
Mathematical Modelling of Natural Phenomena, 2010
Two main approaches have been considered for modelling the dynamics of the SIS model on complex metapopulations, i.e, networks of populations connected by migratory flows whose configurations are described in terms of the connectivity distribution of nodes (patches) and the conditional probabilities of connections among classes of nodes sharing the same degree. In the first approach migration and transmission/recovery process alternate sequentially, and, in the second one, both processes occur simultaneously. Here we follow the second approach and give a necessary and sufficient condition for the instability of the disease-free equilibrium in generic networks under the assumption of limited (or frequency-dependent) transmission. Moreover, for uncorrelated networks and under the assumption of non-limited (or density-dependent) transmission, we give a bounding interval for the dominant eigenvalue of the Jacobian matrix of the model equations around the disease-free equilibrium. Finally, for this latter case, we study numerically the prevalence of the infection across the metapopulation as a function of the patch connectivity.
Stability Analysis of a Deterministic Epidemic Model in Metapopulation Setting
Advances in Pure Mathematics, 2018
We present in this article an epidemic model with saturated in metapopulation setting. We develop the mathematical modelling of HIV transmission among adults in Metapopulation setting. We discussed the positivity of the system. We calculated the reproduction number, If 0 1 j R ≤ for 1,2,3,4 j = , then each infectious individual in Sub-Population j infects on average less than one other person and the disease is likely to die out. Otherwise, if 0 1 j R > for 1,2,3,4 j = , then each infectious individual in Sub-Population j infects on average more than one other person; the infection could therefore establish itself in the population and become endemic. An epidemic model, where the presence or absence of an epidemic wave is characterized by the value of 0 j R both ideas of the inner equilibrium point of stability properties are discussed.
Epidemic spread and bifurcation effects in two-dimensional network models with viral dynamics
Physical Review E, 2001
We extend a previous network model of viral dynamics to include host populations distributed in two space dimensions. The basic dynamical equations for the individual viral and immune effector densities within a host are bilinear with a natural threshold condition. In the general model, transmission between individuals is governed by three factors: a saturating function g(•) describing emission as a function of originating host virion level; a four-dimensional array B that determines transmission from each individual to every other individual; and a nonlinear function F, which describes the absorption of virions by a host for a given net arrival rate. A summary of the properties of the viral-effector dynamical system in a single host is given. In the numerical network studies, individuals are placed at the mesh points of a uniform rectangular grid and are connected with an m 2 ϫn 2 four-dimensional array with terms that decay exponentially with distance between hosts; g is linear and F has a simple step threshold. In a population of Nϭmn individuals, N 0 are chosen randomly to be initially infected with the virus. We examine the dependence of maximal population viral load on the population dynamical parameters and find threshold effects that can be related to a transcritical bifurcation in the system of equations for individual virus and host effector populations. The effects of varying demographic parameters are also examined. Changes in ␣, which is related to mobility, and contact rate  also show threshold effects. We also vary the density of ͑randomly chosen͒ initially infected individuals. The distribution of final size of the epidemic depends strongly on N 0 but is invariably bimodal with mass concentrated mainly near either or both ends of the interval ͓1,N͔. Thus large outbreaks may occur, with small probability, even with only very few initially infected hosts. The effects of immunization of various fractions of the population on the final size of the epidemic are also explored. The distribution of the final percentage infected is estimated by simulation. The mean of this quantity is obtained as a function of immunization rate and shows an almost linear decline for immunization rates up to about 0.2. When the immunization rate is increased past 0.2, the extra benefit accrues more slowly. We include a discussion of some approximations that illuminate threshold effects in demographic parameters and indicate how a mean-field approximation and more detailed studies of various geometries and rates of immunization could be a useful direction for future analysis.
Mathematical Biosciences, 2013
A wide range of infectious diseases are both vertically and horizontally transmitted. Such diseases are spatially transmitted via multiple species in heterogeneous environments, typically described by complex meta-population models. The reproduction number, R 0 , is a critical metric predicting whether the disease can invade the meta-population system. This paper presents the reproduction number for a generic disease vertically and horizontally transmitted among multiple species in heterogeneous networks, where nodes are locations, and links reflect outgoing or incoming movement flows. The metapopulation model for vertically and horizontally transmitted diseases is gradually formulated from two species, two-node network models. We derived an explicit expression of R 0 , which is the spectral radius of a matrix reduced in size with respect to the original next generation matrix. The reproduction number is shown to be a function of vertical and horizontal transmission parameters, and the lower bound is the reproduction number for horizontal transmission. As an application, the reproduction number and its bounds for the Rift Valley fever zoonosis, where livestock, mosquitoes, and humans are the involved species are derived. By computing the reproduction number for different scenarios through numerical simulations, we found the reproduction number is affected by livestock movement rates only when parameters are heterogeneous across nodes. To summarize, our study contributes the reproduction number for vertically and horizontally transmitted diseases in heterogeneous networks. This explicit expression is easily adaptable to specific infectious diseases, affording insights into disease evolution.