Simple Epidemic Models Research Articles

Applications of occupancy urn models to epidemiology

This paper shows how occupancy urn models can be used to derive useful results in epidemiology. First we show how simple epidemic models can be re-interpreted in terms of occupancy problems. We use this reformulation to derive an expression for the expected epidemic size, that is, the total number of infected at the end of an outbreak. We also use this approach to derive point and interval estimates of the Basic Reproduction Ratio, R0 . We show that this construction does not require that the underlying SIR model be a homogeneous Poisson process, leading to a geometric distribution for the number of contacts before removal, but instead it supports a general distribution. The urn model construction is easy to handle and represents a rich field for further exploitation.

The Allee Effect and Infectious Diseases: Extinction, Multistability, and the (Dis‐)Appearance of Oscillations

Infectious diseases that affect their host on a long timescale can regulate the host population dynamics. Here we show that a strong Allee effect can lead to complex dynamics in simple epidemic models. Generally, the Allee effect renders a population bistable, but we also identify conditions for tri- or monostability. Moreover, the disease can destabilize endemic equilibria and induce sustained oscillations. These disappear again for high transmissibilities, with eventually vanishing host population. Disease-induced extinction is thus possible for density-dependent transmission and without any alternative reservoirs. The overall complexity suggests that the system is very sensitive to perturbations and control methods, even in parameter regions with a basic reproductive ratio far beyond R(0) = 1. This may have profound implications for biological conservation as well as pest management. We identify important threshold quantities and attribute the dynamical behavior to the joint interplay of a strong Allee effect and infection.

Using simple models to review the application and implications of different approaches used to simulate transmission of pathogens among aquatic animals

Disease is an important issue affecting aquatic animal populations. Aquatic pathogens may be transmitted in ways that could result in qualitatively different impacts to those of terrestrial diseases. I analyse simple SIR epidemic models with different functions to describe transmission. Four forms of transmission are applied: density-dependent, density-independent, non-linear density-dependent and constant infection pressure; the first two are similar to terrestrial systems, the second two are based on specifically aquatic modes of transmission. Observed diseases and existing models are reviewed in terms of these simple forms. The significance of mode of transmission to host populations, to strategies to prevent or control diseases, and to wild–farm interactions are analysed. Different diseases are simulated by different transmission models, for example furunculosis depends on host density, while spread of phocine distemper virus is density-independent, and sea lice infestation pressure may result from open transmission processes that are not dependent on local infested hosts. Appropriate transmission model may also depend on the scale of interest (inter- or intra-population). These different models result in very different responses to intervention strategies, for example culling may be effective for controlling density-dependent disease but may be counter-productive when pathogens depend on open recruitment. It is therefore important for management that appropriate models (whether existing or novel) be selected and this paper aims to provide a basic framework for cataloguing and management of aquatic diseases.

Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models

The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcation. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a Ricker recruitment function in an SIS model and obtained a three component discrete Hopf (Neimark-Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.

Resonance of the epidemic threshold in a periodic environment

Resonance between some natural period of an endemic disease and a seasonal periodic contact rate has been the subject of intensive study. This paper does not focus on resonance for endemic diseases but on resonance for emerging diseases. Periodicity can have an important impact on the initial growth rate and therefore on the epidemic threshold. Resonance occurs when the Euler-Lotka equation has a complex root with an imaginary part (i.e., a natural frequency) close to the angular frequency of the contact rate and a real part not too far from the Malthusian parameter. This is a kind of continuous-time analogue of work by Tuljapurkar on discrete-time population models, which in turn was motivated by the work by Coale on continuous-time demographic models with a periodic birth. We illustrate this resonance phenomenon on several simple epidemic models with contacts varying periodically on a weekly basis, and explain some surprising differences, e.g., between a periodic SEIR model with an exponentially distributed latency and the same model but with a fixed latency.

Spatial epidemics: critical behavior in one dimension

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are considered, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson–Watanabe processes with killing, respectively.

On representing network heterogeneities in the incidence rate of simple epidemic models

Mean-field ecological models ignore space and other forms of contact structure. At the opposite extreme, high-dimensional models that are both individual-based and stochastic incorporate the distributed nature of ecological interactions. In between, moment approximations have been proposed that represent the effect of correlations on the dynamics of mean quantities. As an alternative closer to the typical temporal models used in ecology, we present here results on “modified mean-field equations” for infectious disease dynamics, in which only mean quantities are followed and the effect of heterogeneous mixing is incorporated implicitly. We specifically investigate the previously proposed empirical parameterization of heterogeneous mixing in which the bilinear incidence rate SI is replaced by a nonlinear term kS p I q , for the case of stochastic SIRS dynamics on different contact networks, from a regular lattice to a random structure via small-world configurations. We show that, for two distinct dynamical cases involving a stable equilibrium and a noisy endemic steady state, the modified mean-field model approximates successfully the steady state dynamics as well as the respective short and long transients of decaying cycles. This result demonstrates that early on in the transients an approximate power-law relationship is established between global (mean) quantities and the covariance structure in the network. The approach fails in the more complex case of persistent cycles observed within the narrow range of small-world configurations.

Some simple epidemic models

The SARS epidemic of 2002-3 led to the study of epidemic models including management measures and other generalizations of the original 1927 epidemic model of Kermack and McKendrick. We consider some natural extensions of the Kermack-McKendrick model and show that they share the main properties of the original model.

Allometry and simple epidemic models for microparasites.

Simple mathematical models for microparasites offer a useful way to examine the population dynamics of different viral and bacterial pathogens. One constraint in applying these models in free-living host populations is the paucity of data with which to estimate transmission rates. Here we recast a standard epidemiological model by setting the birth and death rates of the host population and its density as simple allometric functions of host body weight. We then use standard threshold theorems for the model in order to estimate the minimum rate of transmission for the parasite to establish itself in a mammalian host population. Transmission rates that produce different comparable values of the parasites' basic reproductive number, RO, are themselves allometric functions of host body size. We have extended the model to show that hosts having different body sizes suffer epidemic outbreaks whose frequency scales with body size. The expected epidemic periods for pathogens in different mammalian populations correspond to cycles observed in free-living populations.

A Study of Stochastic Epidemic Model with Genetic Susceptibility to Infection

Abstract A simple stochastic epidemic model involving genetic susceptibility to infection is proposed which incldues as a particular case, some simple stochastic epidemic models. A recursive procedure is given leading to a solution for the state probabilities at any point in time. A numerical example is made to explain the simplicity of the theory.

A long term epidemiological study of bovine viral diarrhoea infections in a large herd of dairy cattle

Epidemiological aspects of bovine viral diarrhoea virus (BVDV) infections were studied longitudinally in a large dairy herd for three years. At the start of the study, practically all the cows more than four years old had BVDV antibody titres, whereas the younger stock were almost all seronegative. The spread of the virus was monitored in a part of the population that contained only transiently viraemic cattle and in another part that contained persistently viraemic calves. Among the lactating cows the virus circulated for two-and-a-half years, although they had no direct contact with persistently viraemic cattle during this period. The highest transmission rate occurred when a large number of susceptible heifers was added to the population of cows that contained transiently viraemic cattle. The circulation of BVDV among the lactating cows ceased while 27 seronegative cows were still present. Both findings are in accordance with predictions from simple epidemic models. The susceptibility of the cows that remained seronegative was confirmed experimentally. In contrast with the limited circulation of BVDV caused by transiently viraemic cattle, virtually all susceptible cattle that came into contact with a persistently viraemic calf became seropositive within three months. Transplacental BVDV infections were not detected in the calves born to cows that had antibodies against the virus due to an infection that had occurred at least four years earlier. Transplacental transmission of BVDV did not occur in most of the pregnant cows that were infected before approximately the 60th day of gestation, but when cows became infected later in gestation the virus virtually always invaded the fetus.(ABSTRACT TRUNCATED AT 250 WORDS)

Simplifying simple epidemic models.

Interest has recently revived in the use of simple models for epidemic diseases. In particular, Anderson et al. have introduced an improved simple differential equation model for diseases such as fox rabies which regulate the population density of their host. Here I describe how such apparently simple models can be dissected into their basic components. This dissection facilitates a structural sensitivity analysis in which we explore the dependence of features of a model's behaviour on the assumptions regarding each component. I report that particular features can be related to particular components, for example, oscillations depend mainly on population growth and the generation gap of the disease, while estimates of the effect of potential control strategies such as vaccination or culling can be related directly to assumptions concerning the infection term and the way it changes with population density. Some features, for example, the level of prevalence of the disease and the period of any oscillations, turn out to be robust, depending mainly on basic ecological parameters. Others, including the crucial estimates regarding control, prove very sensitive to the details of the model. This is unfortunate, as the detailed form of the components is to a large extent chosen, not for ecological reasons, but to keep models simple, for example, the infectious period of a disease is often assumed to have an exponential distribution because this implies a constant death or recovery rate for infectious individuals which is mathematically very convenient for a continuous-time model. Because our dissection is in terms of components with straightforward ecological interpretations, any required improvements in modelling can be related to observational evidence which either exists or for which experiments can be suggested.

Optimization over Time. Dynamic Programming and Stochastic Control. Volume 1

"Optimization over Time. Dynamic Programming and Stochastic Control. Volume 1." Journal of the Operational Research Society, 34(3), p. 255

Simple Epidemic Models with Positive Latent Period

This paper deals with two types of simple epidemic models, namely, deterministic and stochastic wherein the latent period is assumed to be positive. In the deterministic epidemic model, the distributions of susceptibles, inactive infectives, active infectives and that of epidemic curve which gives the rate at which new infections take place have been obtained. The expression for the expected time of the entire epidemic has been derived. Also the partial differential equation for the moment generating function of the proportion of susceptibles in the population is established. In the end, we have studied a stochastic approach of the system.

On interarrival times in simple stochastic epidemic models

The probability distributions of the size and the duration of simple stochastic epidemic models are well known. However, in most instances, the solutions are too complicated to be of practical use. In this note, interarrival times of the infectives are utilized to study asymptotic distributions of the duration of the epidemic for a class of simple epidemic models. A brief summary of the results on simple epidemic models in terms of interarrival times is included.

On interarrival times in simple stochastic epidemic models

The probability distributions of the size and the duration of simple stochastic epidemic models are well known. However, in most instances, the solutions are too complicated to be of practical use. In this note, interarrival times of the infectives are utilized to study asymptotic distributions of the duration of the epidemic for a class of simple epidemic models. A brief summary of the results on simple epidemic models in terms of interarrival times is included.

Epidemics in competition.

When two or more epidemic agents are simultaneously present in a population, they may interact to increase or decrease each other's effectiveness. One form of interaction is "competition" where each agent confers immunity to the others. Such competition occurs, for example, between different strains of myxomatosis in rabbit populations. We consider some consequences of introducing competition into mathematical epidemic models. Both deterministic and stochastic simple epidemic models are examined. In either case the conclusions are similar: the faster spreading epidemic has a considerable advantage.

The convergence in distribution of some simple epidemics

A group of n susceptible individuals exposed to a simple epidemic is modeled by a stochastic process X n ( t), t∈[0, ∞), representing the number of infective individuals at time t. In this paper our analysis is restricted to two classes of simple epidemic models. It is shown, for the first class of simple epidemics, that X n ( t) converges in distribution as n→∞ to a negative binomial random variable, and that EX n ( t), Var{ X n ( t)} converge as n→∞ for all t∈(0, ∞). For the second class of simple epidemics it is shown that X n ( t), with suitable standardization, converges in distribution as n→∞ to a normal variable for all tϵ(0, ∞).

A new look at the simple epidemic process

Classical epidemic models have invariably proved to be mathematically intractable. By considering the distribution of the number of infectives in a simple epidemic process as a convolution of exponential waiting times, the solution to the classical model is obtained easily giving more insight into the underlying structure. The idea can be extended to other simple epidemic models.

A new look at the simple epidemic process

Classical epidemic models have invariably proved to be mathematically intractable. By considering the distribution of the number of infectives in a simple epidemic process as a convolution of exponential waiting times, the solution to the classical model is obtained easily giving more insight into the underlying structure. The idea can be extended to other simple epidemic models.