Epidemiological effects of seasonal oscillations in birth rates

Seasonal oscillations in birth rates are ubiquitous in human populations. These oscillations might play an important role in infectious disease dynamics because they induce seasonal variation in the number of susceptible individuals that enter populations. We incorporate seasonality of birth rate into the standard, deterministic susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) epidemic models and identify parameter regions in which birth seasonality can be expected to have observable epidemiological effects. The SIR and SEIR models yield similar results if the infectious period in the SIR model is compared with the "infected period" (the sum of the latent and infectious periods) in the SEIR model. For extremely transmissible pathogens, large amplitude birth seasonality can induce resonant oscillations in disease incidence, bifurcations to stable multi-year epidemic cycles, and hysteresis. Typical childhood infectious diseases are not sufficiently transmissible for their asymptotic dynamics to be likely to exhibit such behaviour. However, we show that fold and period-doubling bifurcations generically occur within regions of parameter space where transients are phase-locked onto cycles resembling the limit cycles beyond the bifurcations, and that these phase-locking regions extend to arbitrarily small amplitude of seasonality of birth rates. Consequently, significant epidemiological effects of birth seasonality may occur in practice in the form of transient dynamics that are sustained by demographic stochasticity.     

The effect of seasonal host birth rates on disease persistence

In this paper, we add seasonality to the birth rate of an SIR model with density dependence in the death rate. We find that disease persistence can be explained by considering the average value of the seasonal term. If the basic reproductive ratio R(0)>1 with this average value then the disease will persist and if R(0)<1 with this average value then the disease will die out. However, if the underlying non-seasonal model displays oscillations towards the equilibrium then the dynamics of the seasonal model can become more complex. In this case, the seasonality can interact with the underlying oscillations, resonate and the population can display a range of complex behaviours including chaos. We discuss these results in terms of two examples, Cowpox in bank voles and Rabbit Haemorrhagic disease in rabbits.     

Impact of Birth Seasonality on Dynamics of Acute Immunizing Infections in Sub-Saharan Africa

We analyze the impact of birth seasonality (seasonal oscillations in the birth rate) on the dynamics of acute, immunizing childhood infectious diseases. Previous research has explored the effect of human birth seasonality on infectious disease dynamics using parameters appropriate for the developed world. We build on this work by including in our analysis an extended range of baseline birth rates and amplitudes, which correspond to developing world settings. Additionally, our analysis accounts for seasonal forcing both in births and contact rates. We focus in particular on the dynamics of measles. In the absence of seasonal transmission rates or stochastic forcing, for typical measles epidemiological parameters, birth seasonality induces either annual or biennial epidemics. Changes in the magnitude of the birth fluctuations (birth amplitude) can induce significant changes in the size of the epidemic peaks, but have little impact on timing of disease epidemics within the year. In contrast, changes to the birth seasonality phase (location of the peak in birth amplitude within the year) significantly influence the timing of the epidemics. In the presence of seasonality in contact rates, at relatively low birth rates (20 per 1000), birth amplitude has little impact on the dynamics but does have an impact on the magnitude and timing of the epidemics. However, as the mean birth rate increases, both birth amplitude and phase play an important role in driving the dynamics of the epidemic. There are stronger effects at higher birth rates.     

The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches

Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth–death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle’s study of a susceptible–infected–recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations.     

Seasonal dynamics and thresholds governing recurrent epidemics

Driven by seasonality, many common recurrent infectious diseases are characterized by strong annual, biennial and sometimes irregular oscillations in the absence of vaccination programs. Using the seasonally forced SIR epidemic model, we are able to provide new insights into the dynamics of recurrent diseases and, in some cases, specific predictions about individual outbreaks. The analysis reveals a new threshold effect that gives clear conditions for the triggering of future disease outbreaks or their absence. The threshold depends critically on the susceptibility S ₀ of the population after an outbreak. We show that in the presence of seasonality, forecasts based on the susceptibility S ₀ are more reliable than those based on the classical reproductive number R ₀ from the conventional theory.      

Effects of Immigration on the Dynamics of Simple Population Models

Many simple population models exhibit the period doubling route to chaos as a single parameter, commonly the growth rate, is increased. Here we examine the effect of an immigration process on such models and explain why in the case of one-dimensional ("single-humped") maps, immigration often tends to suppress chaos and stabilise equilibrium behaviour or cyclical oscillations of long period. The conditions for which an increase of immigration "simplifies" population dynamics are examined.     

Discrete and continuous state population models in a noisy world

Simple ecological models operate mostly with population densities using continuous variables. However, in reality densities could not change continuously, since the population itself consists of integer numbers of individuals. At first sight this discrepancy appears to be irrelevant, nevertheless, it can cause large deviations between the actual statistical behaviour of biological populations and that predicted by the corresponding models. We investigate the conditions under which simple models, operating with continuous numbers of individuals can be used to approximate the dynamics of populations consisting of integer numbers of individuals. Based on our definition for the (statistical) distance between the two models we show that the continuous approach is acceptable as long as sufficiently high biological noise is present, or, the dynamical behaviour is regular (non-chaotic). The concepts are illustrated with the Ricker model and tested on the Tribolium castaneum data series. Further, we demonstrate with the help of T. castaneum's model that if time series are not much larger than the possible population states (as in this practical case) the noisy discrete and continuous models can behave temporarily differently, almost independently of the noise level. In this case the noisy, discrete model is more accurate [OR has to be applied].     

Backward bifurcation and oscillations in a nested immuno-eco-epidemiological model

This paper introduces a novel partial differential equation immuno-eco-epidemiological model of competition in which one species is affected by a disease while another can compete with it directly and by lowering the first species' immune response to the infection, a mode of competition termed stress-induced competition. When the disease is chronic, and the within-host dynamics are rapid, we reduce the partial differential equation model (PDE) to a three-dimensional ordinary differential equation (ODE) model. The ODE model exhibits backward bifurcation and sustained oscillations caused by the stress-induced competition. Furthermore, the ODE model, although not a special case of the PDE model, is useful for detecting backward bifurcation and oscillations in the PDE model. Backward bifurcation related to stress-induced competition allows the second species to persist for values of its invasion number below one. Furthermore, stress-induced competition leads to destabilization of the coexistence equilibrium and sustained oscillations in the PDE model. We suggest that complex systems such as this one may be studied by appropriately designed simple ODE models.     

Discrete time piecewise affine models of genetic regulatory networks

We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. These models reduce to boolean networks in one limiting case of a parameter, and their asymptotic dynamics approaches that of a differential equation in another limiting case of this parameter. For intermediate values, the model present an original phenomenology which is argued to be due to delay effects. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle – with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.     

Instabilities in multiserotype disease models with antibody-dependent enhancement

This paper investigates the complex dynamics induced by antibody-dependent enhancement (ADE) in multiserotype disease models. ADE is the increase in viral growth rate in the presence of immunity due to a previous infection of a different serotype. The increased viral growth rate is thought to increase the infectivity of the secondary infectious class. In our models, ADE induces the onset of oscillations without external forcing. The oscillations in the infectious classes represent outbreaks of the disease. In this paper, we derive approximations of the ADE parameter needed to induce oscillations and analyze the associated bifurcations that separate the types of oscillations. We then investigate the stability of these dynamics by adding stochastic perturbations to the model. We also present a preliminary analysis of the effect of a single serotype vaccination in the model.     

Seasonality and wildlife disease: how seasonal birth, aggregation and variation in immunity affect the dynamics of Mycoplasma gallisepticum in house finches

We examine the role of host seasonal breeding, host seasonal social aggregation and partial immunity in affecting wildlife disease dynamics, focusing on the dynamics of house finch conjunctivitis (Mycoplasma gallisepticum (MG) in Carpodacus mexicanus). This case study of an unmanaged emerging infectious disease provides useful insight into the important role of seasonal factors in driving ongoing disease dynamics. Seasonal breeding can force recurrent epidemics through the input of fresh susceptibles, which will clearly affect a wide variety of wildlife disease dynamics. Seasonal patterns of social aggregation and foraging behaviour could change transmission dynamics. We use latitudinal variation in the timing of breeding, and social systems to model seasonal dynamics of house finch conjunctivitis across eastern North America. We quantify the patterns of seasonal breeding, and social aggregation across a latitudinal gradient in the eastern range of the house finch, supplemented with known field and laboratory information on immunity to MG in finches. We then examine the interactions of these factors in a theoretical model of disease dynamics. We find that both forms of seasonality could explain the dynamics of the house finch-MG system, and that these factors could have important effects on the dynamics of wildlife diseases generally. In particular, while either alone is sufficient to create recurrent cycles of prevalence in a population with an endemic disease, both are required to produce the specific semi-annual pattern of disease prevalence seen in the house finch conjunctivitis system.     

The Interplay of Public Intervention and Private Choices in Determining the Outcome of Vaccination Programmes

After a long period of stagnation, traditionally explained by the voluntary nature of the programme, a considerable increase in routine measles vaccine uptake has been recently observed in Italy after a set of public interventions aiming to promote MMR immunization, whilst retaining its voluntary aspect. To account for this take-off in coverage we propose a simple SIR transmission model with vaccination choice, where, unlike similar works, vaccinating behaviour spreads not only through the diffusion of “private” information spontaneously circulating among parents of children to be vaccinated, which we call imitation, but also through public information communicated by the public health authorities. We show that public intervention has a stabilising role which is able to reduce the strength of imitation-induced oscillations, to allow disease elimination, and to even make the disease-free equilibrium where everyone is vaccinated globally attractive. The available Italian data are used to evaluate the main behavioural parameters, showing that the proposed model seems to provide a much more plausible behavioural explanation of the observed take-off of uptake of vaccine against measles than models based on pure imitation alone.     

Inference for nonlinear epidemiological models using genealogies and time series

Phylodynamics - the field aiming to quantitatively integrate the ecological and evolutionary dynamics of rapidly evolving populations like those of RNA viruses - increasingly relies upon coalescent approaches to infer past population dynamics from reconstructed genealogies. As sequence data have become more abundant, these approaches are beginning to be used on populations undergoing rapid and rather complex dynamics. In such cases, the simple demographic models that current phylodynamic methods employ can be limiting. First, these models are not ideal for yielding biological insight into the processes that drive the dynamics of the populations of interest. Second, these models differ in form from mechanistic and often stochastic population dynamic models that are currently widely used when fitting models to time series data. As such, their use does not allow for both genealogical data and time series data to be considered in tandem when conducting inference. Here, we present a flexible statistical framework for phylodynamic inference that goes beyond these current limitations. The framework we present employs a recently developed method known as particle MCMC to fit stochastic, nonlinear mechanistic models for complex population dynamics to gene genealogies and time series data in a Bayesian framework. We demonstrate our approach using a nonlinear Susceptible-Infected-Recovered (SIR) model for the transmission dynamics of an infectious disease and show through simulations that it provides accurate estimates of past disease dynamics and key epidemiological parameters from genealogies with or without accompanying time series data.     

Allometric scaling and seasonality in the epidemics of wildlife diseases

We present a susceptibles-exposed-infectives (SEI) model to analyze the effects of seasonality on epidemics, mainly of rabies, in a wide range of wildlife species. Model parameters are cast as simple allometric functions of host body size. Via nonlinear analysis, we investigate the dynamical behavior of the disease for different levels of seasonality in the transmission rate and for different values of the pathogen basic reproduction number (R(0)) over a broad range of body sizes. While the unforced SEI model exhibits long-term epizootic cycles only for large values of R(0), the seasonal model exhibits multiyear periodicity for small values of R(0). The oscillation period predicted by the seasonal model is consistent with those observed in the field for different host species. These conclusions are not affected by alternative assumptions for the shape of seasonality or for the parameters that exhibit seasonal variations. However, the introduction of host immunity (which occurs for rabies in some species and is typical of many other wildlife diseases) significantly modifies the epidemic dynamics; in this case, multiyear cycling requires a large level of seasonal forcing. Our analysis suggests that the explicit inclusion of periodic forcing in models of wildlife disease may be crucial to correctly describe the epidemics of wildlife that live in strongly seasonal environments.     

Studying the recovery procedure for the time-dependent transmission rate(s) in epidemic models

Determining the time-dependent transmission function that exactly reproduces disease incidence data can yield useful information about disease outbreaks, including a range potential values for the recovery rate of the disease and could offer a method to test the “school year” hypothesis (seasonality) for disease transmission. Recently two procedures have been developed to recover the time-dependent transmission function, β(t), for classical disease models given the disease incidence data. We first review the β(t) recovery procedures and give the resulting formulas, using both methods, for the susceptible-infected-recovered (SIR) and susceptible-exposed-infected-recovered (SEIR) models. We present a modification of one procedure, which is then shown to be identical to the other. Second, we explore several technical issues that appear when implementing the procedure for the SIR model; these are important when generating the time-dependent transmission function for real-world disease data. Third, we extend the recovery method to heterogeneous populations modeled with a certain SIR-type model with multiple time-dependent transmission functions. Finally, we apply the β(t) recovery procedure to data from the 2002–2003 influenza season and for the six seasons from 2002–2003 through 2007–2008, for both one population class and for two age classes. We discuss the consequences of the technical conditions of the procedure applied to the influenza data. We show that the method is robust in the heterogeneous cases, producing comparable results under two different hypotheses. We perform a frequency analysis, which shows a dominant 1-year period for the multi-year influenza transmission function(s).     

Computationally exact methods for stochastic periodic dynamics: Spatiotemporal dispersal and temporally forced transmission

The dynamics of many diseases and populations possess distinct recurring phases. For example, many species breed only during a subset of the year and the infection dynamics of many pathogens have transmission rates that vary with season. Here I investigate computational methods for studying transient and long-term behaviour of stochastic models which have periodic phases—several different potential techniques for studying long-term behaviour will be contrasted. I illustrate the results with two studies: The first is of a spatially realistic metapopulation model of malleefowl (Leipoa ocellata), a species which disperses only during a quarter of the year; this model is used to highlight the advantages and disadvantages of the particular methods presented. The second study is of a model for disease dynamics which incorporates seasonality in both the rate of within-population transmission and also in the rate of transmission effected via aerosol importation. This model has applications to studying disease invasion and persistence in captive-breeding populations. We demonstrate, via comparison to appropriately matched models with constant transmission rates and also no aerosol transmission, that seasonality and aerosol importation may alter control choices, with possibly an increase in the threshold population size for local control surveillance, transfer of importance to limiting aerosol transmission, and the use of temporally targetted surveillance. The methodology presented is the gold-standard for dealing with many phased processes in ecology and epidemiology, but its application is limited to systems of small size.     

Exploring the relationship between incidence and the average age of infection during seasonal epidemics

The inverse relationship between the incidence and the average age of first infection for immunizing agents has become a basic tenet in the theory underlying the mathematical modeling of infectious diseases. However, this relationship assumes that the infection has reached an endemic equilibrium. In reality, most infectious diseases exhibit seasonal and/or long-term oscillations in incidence. We use a seasonally forced age-structured SIR model to explore the relationship between the number of cases and the average age of first infection over a single epidemic cycle. Contrary to the relationship for the equilibrium dynamics, we find that the average age of first infection is greatest at or near the peak of the epidemic when mixing is homogeneous. We explore the sensitivity of our findings to assumptions about the natural history of infection, population mixing behavior, the mechanism of seasonality, and of the timing of case reporting in relation to the infectious period. We conclude that seasonal variation in the average age of first infection tends to be greatest for acute infections, and the relationship between the number of cases and the average age of first infection can vary depending on the nature of population mixing and the natural history of infection.     

An SIR epidemic model with partial temporary immunity modeled with delay

The SIR epidemic model for disease dynamics considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately resusceptible. We study the case of temporary immunity in an SIR-based model with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay differential equations. We find conditions for which the endemic steady state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.      

Human birth seasonality: latitudinal gradient and interplay with childhood disease dynamics

More than a century of ecological studies have demonstrated the importance of demography in shaping spatial and temporal variation in population dynamics. Surprisingly, the impact of seasonal recruitment on infectious disease systems has received much less attention. Here, we present data encompassing 78 years of monthly natality in the USA, and reveal pronounced seasonality in birth rates, with geographical and temporal variation in both the peak birth timing and amplitude. The timing of annual birth pulses followed a latitudinal gradient, with northern states exhibiting spring/summer peaks and southern states exhibiting autumn peaks, a pattern we also observed throughout the Northern Hemisphere. Additionally, the amplitude of United States birth seasonality was more than twofold greater in southern states versus those in the north. Next, we examined the dynamical impact of birth seasonality on childhood disease incidence, using a mechanistic model of measles. Birth seasonality was found to have the potential to alter the magnitude and periodicity of epidemics, with the effect dependent on both birth peak timing and amplitude. In a simulation study, we fitted an susceptible-exposed-infected-recovered model to simulated data, and demonstrated that ignoring birth seasonality can bias the estimation of critical epidemiological parameters. Finally, we carried out statistical inference using historical measles incidence data from New York City. Our analyses did not identify the predicted systematic biases in parameter estimates. This may be owing to the well-known frequency-locking between measles epidemics and seasonal transmission rates, or may arise from substantial uncertainty in multiple model parameters and estimation stochasticity.     

Seasonality and the dynamics of infectious diseases

Seasonal variations in temperature, rainfall and resource availability are ubiquitous and can exert strong pressures on population dynamics. Infectious diseases provide some of the best-studied examples of the role of seasonality in shaping population fluctuations. In this paper, we review examples from human and wildlife disease systems to illustrate the challenges inherent in understanding the mechanisms and impacts of seasonal environmental drivers. Empirical evidence points to several biologically distinct mechanisms by which seasonality can impact host–pathogen interactions, including seasonal changes in host social behaviour and contact rates, variation in encounters with infective stages in the environment, annual pulses of host births and deaths and changes in host immune defences. Mathematical models and field observations show that the strength and mechanisms of seasonality can alter the spread and persistence of infectious diseases, and that population-level responses can range from simple annual cycles to more complex multiyear fluctuations. From an applied perspective, understanding the timing and causes of seasonality offers important insights into how parasite–host systems operate, how and when parasite control measures should be applied, and how disease risks will respond to anthropogenic climate change and altered patterns of seasonality. Finally, by focusing on well-studied examples of infectious diseases, we hope to highlight general insights that are relevant to other ecological interactions.