Reinfection induced disease in a spatial SIRI model

Spatial models are widely used in epidemiology to investigate persistence and extinction of disease as well as their spatial patterns. One of the most important issues in studying epidemic models is the role of infection on the persistence and extinction of the disease. In this paper, we investigate a spatial susceptible–infected–recovered–infected model using cellular automata. We show that, in the regime where disease disappears in the susceptible–infected–recovered–susceptible model, spiral and target waves will emerge in the two-dimensional space due to the reinfection. The obtained results may point out that reinfection has great influence on the epidemic spreading, which enriches the findings of spatiotemporal dynamics in epidemic models.     

Stochastic eco-epidemiological model of dengue disease transmission by Aedes aegypti mosquito

We present a stochastic dynamical model for the transmission of dengue that takes into account seasonal and spatial dynamics of the vector Aedes aegypti. It describes disease dynamics triggered by the arrival of infected people in a city. We show that the probability of an epidemic outbreak depends on seasonal variation in temperature and on the availability of breeding sites. We also show that the arrival date of an infected human in a susceptible population dramatically affects the distribution of the final size of epidemics and that early outbreaks have a low probability. However, early outbreaks are likely to produce large epidemics because they have a longer time to evolve before the winter extinction of vectors. Our model could be used to estimate the risk and final size of epidemic outbreaks in regions with seasonal climatic variations.     

Stochastic epidemics in dynamic populations: quasi-stationarity and extinction

Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours. Received: 14 February 2000 / Revised version: 5 June 2000 / Published online: 24 November 2000     

The effects of disease dispersal and host clustering on the epidemic threshold in plants

For an epidemic to occur in a closed population, the transmission rate must be above a threshold level. In plant populations, the threshold depends not only on host density, but on the distribution of hosts in space. This paper presents an alternative analysis of a previously presented stochastic model for an epidemic in continuous space (Bolker, 1999, Bull. Math. Biol. 61, 849–874). A variety of moment closures are investigated to determine the dependence of the epidemic threshold on host spatial distribution and pathogen dispersal. Local correlations that arise during the early phase of the outbreak determine whether a true global epidemic will occur.     

The effect of incubation time distribution on the extinction characteristics of a rabies epizootic

The continuous model of Anderson et al. (1981), Nature 289, 765–771, is successful in describing certain characteristics of rabies epizootics, in particular, the secondary recurrences which follow the initial outbreak; however, it also predicts the occurrence of exponentially small minima in the infected population, which would realistically imply extinction of the virus. Here we show that inclusion of a more realistic distribution of incubation times in the model can explain why extinction will not occur, and we give explicit parametric estimates for the minimum infected fox density which will occur in the model, in terms of the incubation time distribution.     

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton–Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.     

A Comparison of Continuous and Discrete-time West Nile Virus Models

The first recorded North American epidemic of West Nile virus was detected in New York state in 1999, and since then the virus has spread and become established in much of North America. Mathematical models for this vector-transmitted disease with cross-infection between mosquitoes and birds have recently been formulated with the aim of predicting disease dynamics and evaluating possible control methods. We consider discrete and continuous time versions of the West Nile virus models proposed by Wonham et al. [Proc. R. Soc. Lond. B 271:501–507, 2004] and by Thomas and Urena [Math. Comput. Modell. 34:771–781, 2001], and evaluate the basic reproduction number as the spectral radius of the next-generation matrix in each case. The assumptions on mosquito-feeding efficiency are crucial for the basic reproduction number calculation. Differing assumptions lead to the conclusion from one model [Wonham, M.J. et al., [Proc. R. Soc. Lond. B] 271:501–507, 2004] that a reduction in bird density would exacerbate the epidemic, while the other model [Thomas, D.M., Urena, B., Math. Comput. Modell. 34:771–781, 2001] predicts the opposite: a reduction in bird density would help control the epidemic.     

Using Combined Diagnostic Test Results to Hindcast Trends of Infection from Cross-Sectional Data

Infectious disease surveillance is key to limiting the consequences from infectious pathogens and maintaining animal and public health. Following the detection of a disease outbreak, a response in proportion to the severity of the outbreak is required. It is thus critical to obtain accurate information concerning the origin of the outbreak and its forward trajectory. However, there is often a lack of situational awareness that may lead to over- or under-reaction. There is a widening range of tests available for detecting pathogens, with typically different temporal characteristics, e.g. in terms of when peak test response occurs relative to time of exposure. We have developed a statistical framework that combines response level data from multiple diagnostic tests and is able to ‘hindcast’ (infer the historical trend of) an infectious disease epidemic. Assuming diagnostic test data from a cross-sectional sample of individuals infected with a pathogen during an outbreak, we use a Bayesian Markov Chain Monte Carlo (MCMC) approach to estimate time of exposure, and the overall epidemic trend in the population prior to the time of sampling. We evaluate the performance of this statistical framework on simulated data from epidemic trend curves and show that we can recover the parameter values of those trends. We also apply the framework to epidemic trend curves taken from two historical outbreaks: a bluetongue outbreak in cattle, and a whooping cough outbreak in humans. Together, these results show that hindcasting can estimate the time since infection for individuals and provide accurate estimates of epidemic trends, and can be used to distinguish whether an outbreak is increasing or past its peak. We conclude that if temporal characteristics of diagnostics are known, it is possible to recover epidemic trends of both human and animal pathogens from cross-sectional data collected at a single point in time.     

Modeling the Impact of Ebola and Bushmeat Hunting on Western Lowland Gorillas

The 2003 outbreak of Ebola in the Republic of Congo killed 114 people and up to 800 western lowland gorillas. This outbreak and all outbreaks between 2001–2003 began with human handling of infected animal carcasses. Ebola has since spread, putting the entire gorilla population at risk. An epidemiological model is presented to describe the combined effects of Ebola and hunting on persistence of gorillas. The number of infected gorillas also provides a means of assessing the risk of transmission to humans. Under current harvest practices and the estimated annual outbreak rate, the gorilla population is predicted to undergo a 97% decline within 100 years. Controlling bushmeat hunting may not be enough to prevent extinction if frequent outbreaks occur.     

Household epidemic models with varying infection response

This paper is concerned with SIR (susceptible → infected → removed) household epidemic models in which the infection response may be either mild or severe, with the type of response also affecting the infectiousness of an individual. Two different models are analysed. In the first model, the infection status of an individual is predetermined, perhaps due to partial immunity, and in the second, the infection status of an individual depends on the infection status of its infector and on whether the individual was infected by a within- or between-household contact. The first scenario may be modelled using a multitype household epidemic model, and the second scenario by a model we denote by the infector-dependent-severity household epidemic model. Large population results of the two models are derived, with the focus being on the distribution of the total numbers of mild and severe cases in a typical household, of any given size, in the event that the epidemic becomes established. The aim of the paper is to investigate whether it is possible to determine which of the two underlying explanations is causing the varying response when given final size household outbreak data containing mild and severe cases. We conduct numerical studies which show that, given data on sufficiently many households, it is generally possible to discriminate between the two models by comparing the Kullback–Leibler divergence for the two fitted models to these data.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

Extinction thresholds in deterministic and stochastic epidemic models

The basic reproduction number, ?(0), one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if ?(0)>1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if ?(0)>1 and i infectious individuals are introduced into a susceptible population, then the probability of a major outbreak is approximately 1-(1/?(0))( i ). With multiple infectious groups from which the disease could emerge, this result no longer holds. Stochastic thresholds for multiple groups depend on the number of individuals within each group, i ( j ), j=1, …, n, and on the probability of disease extinction for each group, q ( j ). It follows from multitype branching processes that the probability of a major outbreak is approximately [Formula: see text]. In this investigation, we summarize some of the deterministic and stochastic threshold theory, illustrate how to calculate the stochastic thresholds, and derive some new relationships between the deterministic and stochastic thresholds.     

Persistence and resurgence of sleeping sickness caused by Trypanosoma brucei gambiense in historic foci. Biomathematical approach of an epidemiologic enigma

Since the end of the 19th century, historic endemic foci of Trypanosoma brucei gambiense sleeping sickness have proven very persistent. A five-compartment mathematical model with open vector populations was developed in order to study the dynamics of this disease in Central Africa. Of particular interest is the rate at which the disease spreads or goes to extinction at the beginning of an epidemic outbreak. A measure of this rate is the initial halving/doubling time T(o) of the numbers infected; T(o) is a doubling time when the basic reproduction number Ro > 1 and a halving time when Ro < 1. For realistic parameter values, T(o) can be quite large (i.e. several years or even decades) which corresponds to a persistent low-level endemic brought about by an Ro either just above 1 (slow spread) or just below 1 (slow extinction). A resurgence of historical foci can then be caused by a small shift in parameter values that brings Ro well above 1 and decreases T(o). In addition, when Ro is less than 1 (in the absence of vector migrations), simulations show that a very small percentage of infected immigrant flies can bring about high prevalence rates in the human population. The model is validated with field data from historical Congolese, Central and West African foci of the past.     

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multiple transmission pathways exist for many waterborne diseases, including cholera, Giardia, Cryptosporidium, and Campylobacter. Theoretical work exploring the effects of multiple transmission pathways on disease dynamics is incomplete. Here, we consider a simple ODE model that extends the classical SIR framework by adding a compartment (W) that tracks pathogen concentration in the water. Infected individuals shed pathogen into the water compartment, and new infections arise both through exposure to contaminated water, as well as by the classical SIR person–person transmission pathway. We compute the basic reproductive number (ℛ₀), epidemic growth rate, and final outbreak size for the resulting “SIWR” model, and examine how these fundamental quantities depend upon the transmission parameters for the different pathways. We prove that the endemic disease equilibrium for the SIWR model is globally stable. We identify the pathogen decay rate in the water compartment as a key parameter determining when the distinction between the different transmission routes in the SIWR model is important. When the decay rate is slow, using an SIR model rather than the SIWR model can lead to under-estimates of the basic reproductive number and over-estimates of the infectious period.     

The Potential Impact of Disease on the Migratory Structure of a Partially Migratory Passerine Population

Since its introduction into eastern North America in the 1940s, the eastern population of house finches (Carpodacus mexicanus) has become partially migratory, unlike its nonmigratory source population in southern California (Able and Belthoff in Proc. R. Soc. Lond. 265 (1410), 2063–2071, 1998; Belthoff and Gauthreaux in Condor 93, 374–382, 1991). The infectious disease mycoplasmal conjunctivitis (pathogen Mycoplasma gallisepticum or “MG”), which has been monitored in the house finch population since its appearance around 1993 (Dhondt et al. in J. Wild. Dis. 34 (2), 265–280, 1998), may induce higher mortality rates among populations in more northerly latitudes relative to more southerly populations. Here, we investigate the potential impact of this differential disease mortality on the migratory structure of the eastern house finch population using an epidemic modeling approach. Analytical and computational results suggest the ongoing MG epidemic in the eastern house finch could lead to increases in the percentage of and the total number of migrating individuals in a population despite overall population declines, assuming relatively high winter mortality rates in the north eastern part of their range. These results also suggest that empirical evidence of such a change in migratory structure would be most noticeable in northerly inland populations that showed significant declines following the initial outbreak of MG in the east.     

Deterministic extinction effect of parasites on host populations

 Experimental studies have shown that parasites can reduce host density and even drive host population to extinction. Conventional mathematical models for parasite-host interactions, while can address the host density reduction scenario, fail to explain such deterministic extinction phenomena. In order to understand the parasite induced host extinction, Ebert et al. (2000) formulated a plausible but ad hoc epidemiological microparasite model and its stochastic variation. The deterministic model, resembles a simple SI type model, predicts the existence of a globally attractive positive steady state. Their simulation of the stochastic model indicates that extinction of host is a likely outcome in some parameter regions. A careful examination of their ad hoc model suggests an alternative and plausible model assumption. With this modification, we show that the revised parasite-host model can exhibit the observed parasite induced host extinction. This finding strengthens and complements that of Ebert et al. (2000), since all continuous models are likely break down when all population densities are small. This extinction dynamics resembles that of ratio-dependent predator-prey models. We report here a complete global study of the revised parasite-host model. Biological implications and limitations of our findings are also presented. Received: 30 October 2001 / Revised version: 11 February 2002 / Published online: 17 October 2002 Work is partially supported by NSF grant DMS-0077790 Mathematics Subject Classification (2000): 34C25, 34C35, 92D25. Keywords or phrases: Microparasite model – Ratio-dependent predator-prey model – Host extinction – Global stability – Biological control     

Variational data assimilation with epidemic models

Mathematical modelling is playing an increasing role in developing an understanding of the dynamics of communicable disease and assisting the construction and implementation of intervention strategies. The threat of novel emergent pathogens in human and animal hosts implies the requirement for methods that can robustly estimate epidemiological parameters and provide forecasts. Here, a technique called variational data assimilation is introduced as a means of optimally melding dynamic epidemic models with epidemiological observations and data to provide forecasts and parameter estimates. Using data from a simulated epidemic process the method is used to estimate the start time of an epidemic, to provide a forecast of future epidemic behaviour and estimate the basic reproductive ratio. A feature of the method is that it uses a basic continuous-time SIR model, which is often the first point of departure for epidemiological modelling during the early stages of an outbreak. The method is illustrated by application to data gathered during an outbreak of influenza in a school environment.     

Optimal control of epidemics with limited resources

We extend the existing work on the time-optimal control of the basic SIR epidemic model with mass action contact rate. Previous results have focused on minimizing an objective function that is a linear combination of the cost associated with using control and either the outbreak size or the infectious burden. We instead, provide analytic solutions for the control that minimizes the outbreak size (or infectious burden) under the assumption that there are limited control resources. We provide optimal control policies for an isolation only model, a vaccination only model and a combined isolation–vaccination model (or mixed model). The optimal policies described here contain many interesting features especially when compared to previous analyses. For example, under certain circumstances the optimal isolation only policy is not unique. Furthermore the optimal mixed policy is not simply a combination of the optimal isolation only policy and the optimal vaccination only policy. The results presented here also highlight a number of areas that warrant further study and emphasize that time-optimal control of the basic SIR model is still not fully understood.