SIS epidemic attractors in periodic environments

The demographic dynamics are known to drive the disease dynamics in constant environments. In periodic environments, we prove that the demographic dynamics do not always drive the disease dynamics. We exhibit a chaotic attractor in an SIS epidemic model, where the demograhic dynamics are asymptotically cyclic. Periodically forced SIS epidemic models are known to exhibit multiple attractors. We prove that the basins of attraction of these coexisting attractors have infinitely many components.     

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.     

Transients and attractors in epidemics

Historical records of childhood disease incidence reveal complex dynamics. For measles, a simple model has indicated that epidemic patterns represent attractors of a nonlinear dynamic system and that transitions between different attractors are driven by slow changes in birth rates and vaccination levels. The same analysis can explain the main features of chickenpox dynamics, but fails for rubella and whooping cough. We show that an additional (perturbative) analysis of the model, together with knowledge of the population size in question, can account for all the observed incidence patterns by predicting how stochastically sustained transient dynamics should be manifested in these systems.     

Dynamics and topology of idiotypic networks

Jerne's idiotypic network was previously modelled using simple proliferation dynamics and a homogeneous tree as a connection structure. The present paper studies analytically and numerically the genericity of the previous results when the network connection structure is randomized, e.g. with loops and varying connection intensities. The main feature of the dynamics is the existence of different localized attractors that can be interpreted in terms of vaccination and tolerance. This feature is preserved when loops are added to the network, with a few exceptions concerning some regular lattices. Localized attractors might be destroyed by the introduction of a continuous distribution of connection intensities. We conclude by discussing possible modifications of the elementary model that preserve localization of the attractors and functionality of the network.     

Real-time investigation of measles epidemics with estimate of vaccine efficacy

As part of measles elimination effort, evaluation of the vaccination program and real-time assessment of the epidemic dynamics constitute two important tasks to improve and strengthen the control. The present study aimed to develop an epidemiological modeling method which can be applied to estimating the vaccine efficacy at an individual level while conducting the timely investigation of the epidemic. The multivariate renewal process model was employed to describe the temporal evolution of infection by vaccination history, jointly estimating the time-dependent reproduction number and the vaccine efficacy. Analyzing the enhanced surveillance data of measles in Aichi prefecture, Japan from 2007-08, the vaccine efficacy was estimated at 96.7% (95% confidence interval: 95.8, 97.4). Using an age structured model, the vaccine efficacy among those aged from 5-19 years was shown to be smaller than that among those from 0-4 years. The age-dependent vaccine efficacy estimate informs the age-groups to be targeted for revaccination. Because the estimation method can rest on readily available epidemiological data, the proposed model has a potential to be integrated with routine surveillance.     

Pulse vaccination strategy in the SIR epidemic model

Theoretical results show that the measles ‘pulse’ vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. Using the SIR epidemic model we showed that under a planned pulse vaccination regime the system converges to a stable solution with the number of infectious individuals equal to zero. We showed that pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and on the period of the pulses are adhered to. Our theoretical results are confirmed by numerical simulations. The introduction of seasonal variation into the basic SIR model leads to periodic and chaotic dynamics of epidemics. We showed that under seasonal variation, in spite of the complex dynamics of the system, pulse vaccination still leads to epidemic eradication. We derived the conditions for epidemic eradication under various constraints and showed their dependence on the parameters of the epidemic. We compared effectiveness and cost of constant, pulse and mixed vaccination policies.     

Predicting the impact of a nonsterilizing vaccine against human immunodeficiency virus

Studies of human immunodeficiency virus (HIV) vaccines in animal models suggest that it is difficult to induce complete protection from infection (sterilizing immunity) but that it is possible to reduce the viral load and to slow or prevent disease progression following infection. We have developed an age-structured epidemiological model of the effects of a disease-modifying HIV vaccine that incorporates the intrahost dynamics of infection, a transmission rate and host mortality that depend on the viral load, the possible evolution and transmission of vaccine escape mutant viruses, a finite duration of vaccine protection, and possible changes in sexual behavior. Using this model, we investigated the long-term outcome of a disease-modifying vaccine and utilized uncertainty analysis to quantify the effects of our lack of precise knowledge of various parameters. Our results suggest that the extent of viral load reduction in vaccinated infected individuals (compared to unvaccinated individuals) is the key predictor of vaccine efficacy. Reductions in viral load of about 1 log(10) copies ml(-1) would be sufficient to significantly reduce HIV-associated mortality in the first 20 years after the introduction of vaccination. Changes in sexual risk behavior also had a strong impact on the epidemic outcome. The impact of vaccination is dependent on the population in which it is used, with disease-modifying vaccines predicted to have the most impact in areas of low prevalence and rapid epidemic growth. Surprisingly, the extent to which vaccination alters disease progression, the rate of generation of escape mutants, and the transmission of escape mutants are predicted to have only a weak impact on the epidemic outcome over the first 25 years after the introduction of a vaccine.     

A model for the control of malaria using genetically modified vectors

Recent works have considered the problem of using transgenic mosquitoes to control a malaria epidemic. These insects have been genetically engineered to reduce their capacity to infect humans with malaria parasites. We analyze a model of the mosquito population dynamics when genetically modified individuals are introduced into a wild type population so that the effect of their introduction can be assessed. The model describes the dynamics of gene selection under sexual reproduction in a closed vector population. Our results show that the fitness of the resulting heterozygous population is the key parameter for the success of the invasion, independently of the fitness of homozygous vectors. The vector population dynamics model is then combined with an epidemiological model to study the feasibility of controlling a malaria epidemic. Basic reproductive numbers are calculated for both models, and conditions are obtained for preventing reappearance of the epidemic. Simulations on this model show that it may be possible to reduce or even eradicate the epidemic only if the heterozygous population is better adapted than the wild type. They also show that this can be achieved without completely eliminating the wild type mosquitoes.     

Health newscasts for increasing influenza vaccination coverage: an inductive reasoning game approach

Both pandemic and seasonal influenza are receiving more attention from mass media than ever before. Topics such as epidemic severity and vaccination are changing the way in which we perceive the utility of disease prevention. Voluntary influenza vaccination has been recently modeled using inductive reasoning games. It has thus been found that severe epidemics may occur because individuals do not vaccinate and, instead, attempt to benefit from the immunity of their peers. Such epidemics could be prevented by voluntary vaccination if incentives were offered. However, a key assumption has been that individuals make vaccination decisions based on whether there was an epidemic each influenza season; no other epidemiological information is available to them. In this work, we relax this assumption and investigate the consequences of making more informed vaccination decisions while no incentives are offered. We obtain three major results. First, individuals will not cooperate enough to constantly prevent influenza epidemics through voluntary vaccination no matter how much they learned about influenza epidemiology. Second, broadcasting epidemiological information richer than whether an epidemic occurred may stabilize the vaccination coverage and suppress severe influenza epidemics. Third, the stable vaccination coverage follows the trend of the perceived benefit of vaccination. However, increasing the amount of epidemiological information released to the public may either increase or decrease the perceived benefit of vaccination. We discuss three scenarios where individuals know, in addition to whether there was an epidemic, (i) the incidence, (ii) the vaccination coverage and (iii) both the incidence and the vaccination coverage, every influenza season. We show that broadcasting both the incidence and the vaccination coverage could yield either better or worse vaccination coverage than broadcasting each piece of information on its own.     

Spontaneous behavioural changes in response to epidemics

We study how spontaneous reduction in the number of contacts could develop, as a defensive response, during an epidemic and affect the course of infection events. A model is proposed which couples an SIR model with selection of behaviours driven by imitation dynamics. Therefore, infection transmission and population behaviour become dynamical variables that influence each other. In particular, time scales of behavioural changes and epidemic transmission can be different. We provide a full qualitative characterization of the solutions when the dynamics of behavioural changes is either much faster or much slower than that of epidemic transmission. The model accounts for multiple outbreaks occurring within the same epidemic episode. Moreover, the model can explain “asymmetric waves”, i.e., infection waves whose rising and decaying phases differ in slope. Finally, we prove that introduction of behavioural dynamics results in the reduction of the final attack rate.     

Antibody dynamics in childhood diseases: waning and boosting of immunity and the impact of vaccination

The introduction of vaccination against acute diseases such as measles induced a dramatic decline in the prevalence of the disease, and a more gradual rise in the proportion of the population whose immunity is derived solely from vaccination. These two factors combine to constitute an important shift in the dynamics of immunity, especially in highly vaccinated populations. We develop a general model to describe both loss of immunity in the absence of disease, and boosting of immunity corresponding to subclinical infection in individuals whose immunity has waned. We consider the interaction between infection and immunity and identify the key parameters that determine the eradication threshold. We explore the dynamics in the years following the introduction of vaccination using a stochastic version of the model, and consider the effect of different assumptions concerning the nature of immunity. A comparison of the model results with recently published data suggests that heterogeneity in host immune response is an important feature of the antibody dynamics.     

Influenza drift and epidemic size: the race between generating and escaping immunity

Influenza in humans is characterised by strongly annual dynamics and antigenic evolution leading to partial escape from prior host immunity. The variability of new epidemic strains depends on the amount of virus currently circulating. In this paper, the amount of antigenic variation produced each year is dependent on the epidemic size. Our model reduces to a one-dimensional map and a full mathematical analysis is presented. This simple system suggests some basic principles which may be more generally applicable. In particular, for diseases with antigenic drift, vaccination may be doubly beneficial. Not only does it protect the population through classical herd immunity, but the overall case reduction reduces the chance of new variants being produced; hence, subsequent epidemics may be milder as a result of this positive feedback. Also, a disease with a high innate rate of antigenic variation will always be able to invade a susceptible population, whereas a disease with less potential for variation may require several introduction events to become endemic.     

Dynamic equilibria in an epidemic model with voluntary vaccinations

The dynamics of an epidemic model with voluntary vaccinations are studied. Individual vaccination decisions are modelled using an economic/game-theoretic approach: agents in the model decide whether to vaccinate or not by weighing the cost and benefit of vaccination and choose the action that maximizes their net benefit. It is shown that, when vaccine efficacy is low, there are parameter values for which multiple steady-state equilibria and periodic equilibria coexist. When multiplicity of steady states is obtained, which one the population reaches in some cases depends entirely on agents' expectations concerning the future course of an epidemic and not on the initial conditions of the model. (?)Comments and suggestions from anonymous referees of the journal are gratefully acknowledged. This paper is dedicated to the loving memory of Lucy Hauser.     

Modeling toxoplasmosis spread in cat populations under vaccination

In this paper we present an epidemiological model to study the transmission dynamics of toxoplasmosis in a cat population under a continuous vaccination schedule. We explore the dynamics of toxoplasmosis at the population level using a mathematical model that includes the effect of oocyst, since the probability of acquisition of Toxoplasma Gondii infection depends on the environmental load of the parasite. This model considers indirectly the infection of prey through the oocyst shedding by cats. We prove that the basic reproduction number R₀ is a threshold value that completely determines the global dynamics and the outcome of the disease. Numerical computer simulations are presented to investigate different scenarios. These simulations show the effectiveness of a constant vaccination program.     

Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality

Stochastic differential equations that model an SIS epidemic with multiple pathogen strains are derived from a system of ordinary differential equations. The stochastic model assumes there is demographic variability. The dynamics of the deterministic model are summarized. Then the dynamics of the stochastic model are compared to the deterministic model. In the deterministic model, there can be either disease extinction, competitive exclusion, where only one strain persists, or coexistence, where more than one strain persists. In the stochastic model, all strains are eventually eliminated because the disease-free state is an absorbing state. However, if the population size and the initial number of infected individuals are sufficiently large, it may take a long time until all strains are eliminated. Numerical simulations of the stochastic model show that coexistence cases predicted by the deterministic model are an unlikely occurrence in the stochastic model even for short time periods. In the stochastic model, either disease extinction or competitive exclusion occur. The initial number of infected individuals, the basic reproduction numbers, and other epidemiological parameters are important determinants of the dominant strain in the stochastic epidemic model.     

Evidence for and against the existence of alternate attractors on coral reefs

Synthesis Coral reefs are widely thought to exhibit multiple attractors which have profound implications for people that depend on them. If reefs become ‘stuck’ within a self‐reinforcing state dominated by seaweed, it becomes disproportionately difficult and expensive for managers to shift the system back towards its natural, productive, coral state. The existence of multiple attractors is controversial. We assess various forms of evidence and conclude that there remains no incontrovertible proof of multiple attractors on reefs. However, the most compelling evidence, which combines ecological models and field data, is far more consistent with multiple attractors than the competing hypothesis of only a single, coral attractor. Managers should exercise caution and assume that degraded reefs can become stuck there. Testing for the existence of alternate attractors in ecosystems that possess slow dynamics and frequent pulse perturbation is exceptionally challenging. Coral reefs typify such conditions and the existence of alternate attractors is controversial. We analyse different forms of evidence and assess whether they support or challenge the existence of multiple attractors on Caribbean reefs, many of which have shown profound phase shifts in community structure from coral to algal dominance. Field studies alone provide no insight into multiple attractors because the non‐equilibrial nature of reef dynamics prevents equilibria from being observed. Statistical models risk failing to sample the parameter space in which multiple attractors occur, and have failed to account for the confounding effects of heterogeneous environments, anthropogenic drivers (e.g. fishing), and major disturbances (e.g. hurricanes). Simple and complex models all find multiple attractors over some – though not all – regions of a system driver (fishing). Tests of model predictions with field data closely match theory of alternate attractors but a forward‐leaning monotonic curve with only a single attractor can also be fitted to these data. Deeper consideration of the assumptions of this monotonic relationship reveal significant ecological problems which disappear under a model of multiple attractors. To date, there is no evidence against the existence of multiple attractors on Caribbean reefs and while there remains no definitive proof, the balance of evidence and ecological reasoning favours their existence. Theory predicts that Caribbean reefs do not exhibit alternate attractors in their natural state but that disease‐induced loss of two key functional groups has generated bistability. Whether alternate attractors becomes a persistent element of reef dynamics or a brief moment in their geological history will depend, in part, on the ability of functional groups to recover and the impacts of climate change and ocean acidification on coral growth and mortality.     

Can influenza epidemics be prevented by voluntary vaccination?

Previous modeling studies have identified the vaccination coverage level necessary for preventing influenza epidemics, but have not shown whether this critical coverage can be reached. Here we use computational modeling to determine, for the first time, whether the critical coverage for influenza can be achieved by voluntary vaccination. We construct a novel individual-level model of human cognition and behavior; individuals are characterized by two biological attributes (memory and adaptability) that they use when making vaccination decisions. We couple this model with a population-level model of influenza that includes vaccination dynamics. The coupled models allow individual-level decisions to influence influenza epidemiology and, conversely, influenza epidemiology to influence individual-level decisions. By including the effects of adaptive decision-making within an epidemic model, we can reproduce two essential characteristics of influenza epidemiology: annual variation in epidemic severity and sporadic occurrence of severe epidemics. We suggest that individual-level adaptive decision-making may be an important (previously overlooked) causal factor in driving influenza epidemiology. We find that severe epidemics cannot be prevented unless vaccination programs offer incentives. Frequency of severe epidemics could be reduced if programs provide, as an incentive to be vaccinated, several years of free vaccines to individuals who pay for one year of vaccination. Magnitude of epidemic amelioration will be determined by the number of years of free vaccination, an individuals' adaptability in decision-making, and their memory. This type of incentive program could control epidemics if individuals are very adaptable and have long-term memories. However, incentive-based programs that provide free vaccination for families could increase the frequency of severe epidemics. We conclude that incentive-based vaccination programs are necessary to control influenza, but some may be detrimental. Surprisingly, we find that individuals' memories and flexibility in adaptive decision-making can be extremely important factors in determining the success of influenza vaccination programs. Finally, we discuss the implication of our results for controlling pandemics.     

A discrete-time model with vaccination for a measles epidemic.

A discrete-time, age-independent SIR-type epidemic model is formulated and analyzed. The effects of vaccination are also included in the model. Three mathematically important properties are verified for the model: solutions are nonnegative, the population size is time-invariant, and the epidemic concludes with all individuals either remaining susceptible or becoming immune (a property typical of SIR models). The model is applied to a measles epidemic on a university campus. The simulated results are in good agreement with the actual data if it is assumed that the population mixes nonhomogeneously. The results of the simulations indicate that a rate of immunity greater than 98% may be required to prevent an epidemic in a university population. The model has applications to other contagious diseases of SIR type. Furthermore, the simulated results of the model can easily be compared to data, and the effects of a vaccination program can be examined.     

Inferring the causes of the three waves of the 1918 influenza pandemic in England and Wales

Past influenza pandemics appear to be characterized by multiple waves of incidence, but the mechanisms that account for this phenomenon remain unclear. We propose a simple epidemic model, which incorporates three factors that might contribute to the generation of multiple waves: (i) schools opening and closing, (ii) temperature changes during the outbreak, and (iii) changes in human behaviour in response to the outbreak. We fit this model to the reported influenza mortality during the 1918 pandemic in 334 UK administrative units and estimate the epidemiological parameters. We then use information criteria to evaluate how well these three factors explain the observed patterns of mortality. Our results indicate that all three factors are important but that behavioural responses had the largest effect. The parameter values that produce the best fit are biologically reasonable and yield epidemiological dynamics that match the observed data well.