The network level reproduction number for infectious diseases with both vertical and horizontal transmission (original) (raw)
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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.
Mathematical Modeling of Infectious Disease Transmission Dynamics in a Metapopulation
IOSR Journals , 2019
Epidemic modeling is an important theoretical approach for investigating the transmission dynamics of infectious diseases. It formulates mathematical models to describe the mechanisms of disease transmissions and dynamics of infectious agents and then informs the health control practitioners the likely impact of the control methods. In this paper we investigate the spread of an infectious disease in a human population structured into n-patches. The population is initially fully susceptible until an infectious individual is introduced in one of the patches. The interaction between patches is dominated by movement of individuals between patches and also the migration of individuals and therefore any infection occurring in one patch will have a force of infection on the susceptible individuals on the other patches. We build a mathematical model for a metapopulation consisting of í µí± patches. The patches are connected by movement of individuals. For í µí± = 2, we obtained the basic reproduction number and obtained the condition under which the disease free equilibrium will be asymptotically stable. We further described in terms of the model parameters how control methods could be applied to ensure that the epidemic does not occur and validated the results by the use of the numerical simulation. We showed that the global basic reproduction number cannot exceed one unless the local basic reproduction number is greater than one in at least one of the sub-populations. We further showed that the control of the epidemic in this case can be achieved by applying a control method that decreases the transmission parameters in patches where the local basic reproduction number is greater than one.
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.
Journal of Biological Dynamics, 2011
A deterministic model with spatial consideration for a class of human disease-transmitting vectors is presented and analysed. The model takes the form of a nonlinear system of delayed ordinary differential equations in a compartmental framework. Using the model, existence conditions of a non-trivial steadystate vector population are obtained when more than one breeding site and human habitat site are available. Model analysis confirms the existence of a non-trivial steady state, uniquely determined by a threshold parameter, R j 0 , whose value depends on the distribution and distance of breeding site j to human habitats. Results are based on the existence of a globally and asymptotically stable non-trivial steady-state human population. The explicit form of the Hopf bifurcation, initially reported by Ngwa [On the population dynamics of the malaria vector, Bull. Math. Biol. 68 (2006), pp. 2161-2189, is also obtained and used to show that the vector population oscillates with time. The modelling exercise points to the possibility of spatial-temporal patterns and oscillatory behaviour without external seasonal forcing.
A network-based meta-population approach to model Rift Valley fever epidemics
Journal of theoretical biology, 2012
Rift Valley fever virus (RVFV) has been expanding its geographical distribution with important implications for both human and animal health. The emergence of Rift Valley fever (RVF) in the Middle East, and its continuing presence in many areas of Africa, has negatively impacted both medical and veterinary infrastructures and human morbidity, mortality, and economic endpoints. Furthermore, worldwide attention should be directed towards the broader infection dynamics of RVFV, because suitable host, vector and environmental conditions for additional epidemics likely exist on other continents; including Asia, Europe and the Americas. We propose a new compartmentalized model of RVF and the related ordinary differential equations to assess disease spread in both time and space; with the latter driven as a function of contact networks. Humans and livestock hosts and two species of vector mosquitoes are included in the model. The model is based on weighted contact networks, where nodes of ...
Journal of Applied Mathematics and Computing
Human mobility has been significantly influencing public health since time immemorial. A susceptible-infected-deceased epidemic reaction diffusion network model using asymptotic transmission rate is proposed to portray the spatial spread of the epidemic among two cities due to population dispersion. Qualitative behaviour including global attractor and persistence property are obtained. We also study asymptotic behaviour of the whole network with the help of asymptotic behaviour at individual cities. The epidemic model shows up two equilibria, (i) the disease-free, and (ii) unique endemic equilibria. An expression that can be used to calculate the basic reproduction number for heterogeneous environment, for the entire network is obtained. We use graph theory to analyze the global stability of our diffusive two-city model. We also performed bifurcation analysis and discovered that endemic equilibrium changes stability via Hopf bifurcations. A significant reduction in the number of infectives were observed when proper migration rate is maintained between the cities. Numerical results are provided to illuminate and clarify theoretical findings. Simulation experiments for two-dimensional spatial models show that infectious populations will increase if contact heterogeneity is increased, but it will decline if infective populations perform more local random movement. We observe that infection risk may be understated if the parameters used to estimate the basic reproduction number remains unchanged through space or time.
Spatial epidemic network models with viral dynamics
Physical Review E, 1998
A mathematical model is presented for the spread of viral diseases within human or other populations in which both the dynamics of viral growth within individuals and the interactions between individuals are taken into account. We thus bridge the classical macroscopic approach to the growth and population dynamics of disease at the microscopic level. Each member, i, of the population of n individuals is represented by a vector function of time whose components are antibody numbers a i (t), and the virion level v i (t). These quantities evolve according to 2n differential equations, which are coupled via a transmission matrix B with elements  i j , i, jϭ1,...,n, such that  i j v i is the expected rate of transmission of infectious particles from individual i to individual j. We study nearest-neighbor interaction and transmission which declines exponentially with distance between the individuals. Results are shown to be related to those of classical macroscopic ͑SIR͒ models. We find threshold effects in the occurrence of epidemics as the parameters of the viral and antibody dynamics change. The distribution of the final size of an epidemic is estimated, for various initial patterns of infection, at various values of the parameter which describes the mobility of the population. We also determine the final size in the cases of extreme clustering and dispersion of infected individuals.
Network-level reproduction number and extinction threshold for vector-borne diseases
Mathematical Biosciences and Engineering, 2015
The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction contribute crucial knowledge on disease control, elimination, and mitigation of infectious diseases. Relationships between the basic reproduction numbers of two network-based ordinary differential equation vector-host models, and extinction thresholds of corresponding continuous-time Markov chain models are derived under some assumptions. Numerical simulation results for malaria and Rift Valley fever transmission on heterogeneous networks are in agreement with analytical results without any assumptions, reinforcing the relationships may always exist and proposing a mathematical problem of proving their existences in general. Moreover, numerical simulations show that the reproduction number is not monotonically increasing or decreasing with the extinction threshold. Key parameters in predicting uncertainty of extinction thresholds are identified using Latin Hypercube Sampling/Partial Rank Correlation Coefficient. Consistent trends of extinction probability observed through numerical simulations provide novel insights into mitigation strategies to increase the disease extinction probability. Research findings may improve understandings of thresholds for disease persistence in order to control vector-borne diseases.
Large-Scale Spatial-Transmission Models of Infectious Disease
Science, 2007
During transmission of seasonal endemic diseases such as measles and influenza, spatial waves of infection have been observed between large distant populations. Also, during the initial stages of an outbreak of a new or reemerging pathogen, disease incidence tends to occur in spatial clusters, which makes containment possible if you can predict the subsequent spread of disease. Spatial models are being used with increasing frequency to help characterize these large-scale patterns and to evaluate the impact of interventions. Here, I review several recent studies on four diseases that show the benefits of different methodologies: measles (patch models), foot-and-mouth disease (distance-transmission models), pandemic influenza (multigroup models), and smallpox (network models). This review highlights the importance of the household in spatial studies of human diseases, such as smallpox and influenza. It also demonstrates the need to develop a simple model of household demographics, so ...
Multiscale, resurgent epidemics in a hierarchical metapopulation model
Proceedings of The National Academy of Sciences, 2005
Although population structure has long been recognized as relevant to the spread of infectious disease, traditional mathematical models have understated the role of nonhomogenous mixing in populations with geographical and social structure. Recently, a wide variety of spatial and network models have been proposed that incorporate various aspects of interaction structure among individuals. However, these more complex models necessarily suffer from limited tractability, rendering general conclusions difficult to draw. In seeking a compromise between parsimony and realism, we introduce a class of metapopulation models in which we assume homogeneous mixing holds within local contexts, and that these contexts are embedded in a nested hierarchy of successively larger domains. We model the movement of individuals between contexts via simple transport parameters and allow diseases to spread stochastically. Our model exhibits some important stylized features of real epidemics, including extreme size variation and temporal heterogeneity, that are difficult to characterize with traditional measures. In particular, our results suggest that when epidemics do occur the basic reproduction number R 0 may bear little relation to their final size. Informed by our model's behavior, we suggest measures for characterizing epidemic thresholds and discuss implications for the control of epidemics. math model ͉ population structure