A simple influenza model with complicated dynamics

We propose and analyse a model for the dynamics of a single strain of an influenza-like infection. The model incorporates waning acquired immunity to infection and punctuated antigenic drift of the virus, employing a set of differential equations within a season and a discrete map between seasons. We show that the between-season map displays a variety of qualitatively different dynamics: fixed points, periodic solutions, or more complicated behaviour suggestive of chaos. For some example parameters we demonstrate the existence of two distinct basins of attraction, that is the initial conditions determine the long term dynamics. Our results suggest that there is no reason to expect influenza dynamics to be regular, or to expect past epidemics to give a clear indication of future seasons’ behaviour.

     

Population dynamics of botanical epidemics involving primary and secondary infection

In this paper we study the dynamical properties of models for botanical epidemics, especially for soil-borne fungal infection. The models develop several new concepts, involving dual sources of infection, host and inoculum dynamics. Epidemics are modelled with respect to the infection status of whole plants and plant organs (the G model) or to lesion density and size (the SW model). The infection can originate in two sources, either from the initial inoculum (primary infection) or by a direct transmission between plant tissue (secondary infection). The first term corresponds to the transmission through the free-living stages of macroparasites or an external source of infection in certain medical models, whereas the second term is equivalent to direct transmission between the hosts in microparasitic infections. The models allow for dynamics of host growth and inoculum decay. We show that the two models for root and lesion dynamics can be derived as special cases of a single generic model. Analytical and numerical methods are used to analyse the behaviour of the models for static, unlimited (exponential) and asymptotically limited host growth with and without secondary infection, and with and without decay of initial inoculum. The models are shown to exhibit a range of epidemic behaviour within single seasons that extends from simple monotonic increase with saturation of the host population, through temporary plateaux as the system switches from primary to secondary infection, to effective elimination of the pathogen by the host outgrowing the fungal infection. For certain conditions, the equilibrium values are shown to depend on initial conditions. These results have important consequences for the control of plant disease. They can be applied beyond soil-borne plant pathogens to mycorrhizal fungi and aerial pathogens while the principles of primary and secondary infection with host and inoculum dynamics may be used to link classical models for both microparasitic and macroparasitic infections.     

A game dynamic model for delayer strategies in vaccinating behaviour for pediatric infectious diseases

Several studies have found that some parents delay the age at which their children receive pediatric vaccines due to perception of higher vaccine risk at the recommended age of vaccination. This has been particularly apparently during the Measles-Mumps-Rubella scare in the United Kingdom. Under a voluntary vaccination policy, vaccine coverage in certain age groups is a potentially complex interplay between vaccinating behaviour, disease dynamics, and age-specific risk factors. Here, we construct an age-structured game dynamic model, where individuals decide whether to vaccinate according to imitation dynamics depending on age-dependent disease prevalence and perceived risk of vaccination. Individuals may be timely vaccinators, delayers, or non-vaccinators. The model exhibits multiple equilibria and a broad range of possible dynamics. For certain parameter regimes, the proportion of timely vaccinators and delayers oscillate in an anti-phase fashion in response to oscillations in infection prevalence. Under an exogenous change to the perceived risk of vaccination as might occur during a vaccine scare, the model can also capture an increase in delayer strategists similar in magnitude to that observed during the Measles-Mumps-Rubella vaccine scare in the United Kingdom. Our model also shows that number of delayers steadily increases with increasing severity of the scare, whereas it saturates to specific value with increases in duration of the scare. Finally, by comparing the model dynamics with and without the option of a delayer strategy, we show that adding a third delayer strategy can have a stabilizing effect on model dynamics. In an era where individual choice—rather than accessibility—is becoming an increasingly important determinant of vaccine uptake, more infectious disease models may need to use game theory or related techniques to determine vaccine uptake.     

Scaling properties and symmetrical patterns in the epidemiology of rotavirus infection

The rich epidemiological database of the incidence of rotavirus, as a cause of severe diarrhoea in young children, coupled with knowledge of the natural history of the infection, can make this virus a paradigm for studies of epidemic dynamics. The cyclic recurrence of childhood rotavirus epidemics in unvaccinated populations provides one of the best documented phenomena in population dynamics. This paper makes use of epidemiological data on rotavirus infection in young children admitted to hospital in Melbourne, Australia from 1977 to 2000. Several mathematical methods were used to characterize the overall dynamics of rotavirus infections as a whole and individually as serotypes G1, G2, G3, G4 and G9. These mathematical methods are as follows: seasonal autoregressive integrated moving-average (SARIMA) models, power spectral density (PSD), higher-order spectral analysis (HOSA) (bispectrum estimation and quadratic phase coupling (QPC)), detrended fluctuation analysis (DFA), wavelet analysis (WA) and a surrogate data analysis technique. Each of these techniques revealed different dynamic aspects of rotavirus epidemiology. In particular, we confirm the existence of an annual, biannual and a quinquennial period but additionally we found other embedded cycles (e.g. ca. 3 years). There seems to be an overall unique geometric and dynamic structure of the data despite the apparent changes in the dynamics of the last years. The inherent dynamics seems to be conserved regardless of the emergence of new serotypes, the re-emergence of old serotypes or the transient disappearance of a particular serotype. More importantly, the dynamics of all serotypes is multiple synchronized so that they behave as a single entity at the epidemic level. Overall, the whole dynamics follow a scale-free power-law fractal scaling behaviour. We found that there are three different scaling regions in the time-series, suggesting that processes influencing the epidemic dynamics of rotavirus over less than 12 months differ from those that operate between 1 and ca. 3 years, as well as those between 3 and ca. 5 years. To discard the possibility that the observed patterns could be due to artefacts, we applied a surrogate data analysis technique which enabled us to discern if only random components or linear features of the incidence of rotavirus contribute to its dynamics. The global dynamics of the epidemic is portrayed by wavelet-based incidence analysis. The resulting wavelet transform of the incidence of rotavirus crisply reveals a repeating pattern over time that looks similar on many scales (a property called self-similarity). Both the self-similar behaviour and the absence of a single characteristic scale of the power-law fractal-like scaling of the incidence of rotavirus infection imply that there is not a universal inherently more virulent serotype to which severe gastroenteritis can uniquely be ascribed.     

A nonlinear treatment of the protocell model by a boundary layer approximation

The “protocell” is a mathematical model of a self-maintaining unity based on the dynamics of simple reaction-diffusion processes and a self-controlled dynamics of the surface. In this paper its spatio-temporal behaviour far from the stationary structure is investigated by means of a boundary layer approximation. It is shown in detail how a simplified and mathematically feasible equation can be derived from the original parabolic problem. It turns out that the known instability which is initiated in the linear region around the stationary structure is continued further in the direction to a division by nonlinear dynamics.     

Host genetics and pathogen species modulate infection-induced changes in social aggregation behaviour

Identifying how infection modifies host behaviours that determine social contact networks is important for understanding heterogeneity in infectious disease dynamics. Here, we investigate whether group social behaviour is modified during bacterial infection in fruit flies (Drosophila melanogaster) according to pathogen species, infectious dose, host genetic background and sex. In one experiment, we find that systemic infection with four different bacterial species results in a reduction in the mean pairwise distance within infected female flies, and that the extent of this change depends on pathogen species. However, susceptible flies did not show any evidence of avoidance in the presence of infected flies. In a separate experiment, we observed genetic- and sex-based variation in social aggregation within infected, same-sex groups, with infected female flies aggregating more closely than infected males. In general, our results confirm that bacterial infection induces changes in fruit fly behaviour across a range of pathogen species, but also highlight that these effects vary between fly genetic backgrounds and can be sex-specific. We discuss possible explanations for sex differences in social aggregation and their consequences for individual variation in pathogen transmission.     

The dynamical implications of human behaviour on a social-ecological harvesting model

The dynamic aspects of human harvesting behaviour are often overlooked in resource management, such that models often neglect the complexities of dynamic human effort. Some researchers have recognized this, and a recent push has been made to understand how human behaviour and ecological systems interact through dynamic social-ecological systems. Here, we use a recent example of a social-ecological dynamical systems model to investigate the relationship between harvesting behaviour and the dynamics and stability of a harvested resource, and search for general rules in how relatively simple human behaviours can either stabilize or destabilize resource dynamics and yield. Our results suggest that weak to moderate behavioural and effort responses tend to stabilize dynamics by decreasing return times to equilibria or reducing the magnitude of cycles; however, relatively strong human impacts can readily lead to human-driven cycles, chaos, long transients and alternate states. Importantly, we further show that human-driven cycles are characteristically different from typical resource-driven cycles and, therefore, may be differentiated in real ecosystems. Given the potentially dramatic implications of harvesting on resource dynamics, it becomes critical to better understand how human behaviour determines harvesting effort through dynamic social-ecological systems.     

The effect of seasonal host birth rates on population dynamics: the importance of resonance

Many of the simple mathematical models currently in use often fail to capture important biological factors. Here we extend current models of insect-pathogen interactions to include seasonality in the birth rate. In particular, we consider the SIR model with self-regulation when applied to specific cases--rabbit haemorrhagic disease and fox rabies. In this paper, we briefly summarize the results of the model with a constant time-independent birth rate, a, which we then replace with the time dependent birth rate a(t), to investigate how this effects the dynamics of the host population. We can split parameter space into an area in which the model without seasonality has no oscillations, in which case a simple averaging rule predicts the behaviour. Alternatively, in the area where oscillations to the equilibrium do occur in the non-seasonal model, disease persistence is more complicated and we get more complex dynamical behaviour in this case. We apply resonance techniques to discover the structure of the subharmonic modes of the SIR model with self-regulation. We then look at whether many biological systems are likely to display these "resonant" dynamics and find that we would expect them to be widespread.     

Bubonic plague: a metapopulation model of a zoonosis.

Bubonic plague (Yersinia pestis) is generally thought of as a historical disease; however, it is still responsible for around 1000-3000 deaths each year worldwide. This paper expands the analysis of a model for bubonic plague that encompasses the disease dynamics in rat, flea and human populations. Some key variables of the deterministic model, including the force of infection to humans, are shown to be robust to changes in the basic parameters, although variation in the flea searching efficiency, and the movement rates of rats and fleas will be considered throughout the paper. The stochastic behaviour of the corresponding metapopulation model is discussed, with attention focused on the dynamics of rats and the force of infection at the local spatial scale. Short-lived local epidemics in rats govern the invasion of the disease and produce an irregular pattern of human cases similar to those observed. However, the endemic behaviour in a few rat subpopulations allows the disease to persist for many years. This spatial stochastic model is also used to identify the criteria for the spread to human populations in terms of the rat density. Finally, the full stochastic model is reduced to the form of a probabilistic cellular automaton, which allows the analysis of a large number of replicated epidemics in large populations. This simplified model enables us to analyse the spatial properties of rat epidemics and the effects of movement rates, and also to test whether the emergent metapopulation behaviour is a property of the local dynamics rather than the precise details of the model.     

A discrete stage-structured two-species competition model with sexual and clonal reproduction

In this paper, we analyse a discrete stage-structured model which is a generalization of the two-species competition model studied in [2]. Motivated by plant populations, each species is assumed to reproduce both sexually and clonally. We show that this model has a dynamical behaviour that is similar to that of the classical continuous two-dimensional Lotka-Volterra model under weak nonlinearities of the Beverton-Holt type. By allowing the species to have different competition efficiencies, we show that it is possible to obtain different dynamics including coexistence, bistability and competitive exclusion, in contrast with the model studied in [2], which exhibits only competitive exclusion behaviour.     

A generalized cholera model and epidemic-endemic analysis

The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R (0)=1 for the combined human-environment epidemiological model under biologically reasonable conditions.     

Influence of backward bifurcation in a model of hepatitis B and C viruses

In the 1990s, liver transplantation for hepatitis B and C virus (HBV and HCV) related-liver diseases was a very controversial issue since recurrent infection of the graft was inevitable. Significant progress has been made in the prophylaxis and treatment of recurrent hepatitis B/C (or HBV/HCV infection) after liver transplantation. In this paper, we propose a mathematical model of ordinary differential equations describing the dynamics of the HBV/HCV and its interaction with both liver and blood cells. A single model is used to describe infection of either virus since the dynamics in-host (infected of the liver) are similar. Analyzing the model, we observe that the system shows either a transcritical or a backward bifurcation. Explicit conditions on the model parameters are given for the backward bifurcation to be present. Consequently, we investigate possible factors that are responsible for HBV/HCV infection and assess control strategies to reduce HBV/HCV reinfection and improve graft survival after liver transplantation.     

Approximating the long-term behaviour of a model for parasitic infection

In a companion paper two stochastic models, useful for the initial behaviour of a parasitic infection, were introduced. Now we analyse the long term behaviour. First a law of large numbers is proved which allows us to analyse the deterministic analogues of the stochastic models. The behaviour of the deterministic models is analogous to the stochastic models in that again three basic reproduction ratios are necessary to fully describe the information needed to separate growth from extinction. The existence of stationary solutions is shown in the deterministic models, which can be used as a justification for simulation of quasi-equilibria in the stochastic models. Host-mortality is included in all models. The proofs involve martingale and coupling methods.     

Linking a spatially-explicit model of acacias to GIS and remotely-sensed data

Spatially-explicit and landscape-related simulation models are increasingly used in ecology, but are often criticized because their parameterization has high data requirements. A frequently suggested approach to overcome this difficulty is the linkage of spatially-explicit or landscape-related models with GIS (geographic information system) and remote-sensing technology. GIS can provide data on relevant landscape features, such as topography, and satellite images can be used to identify spatial vegetation distribution. In this paper, we use these techniques for simple, cost-inexpensive (in both time and money) parameterization based on readily-available GIS and remotely-sensed data. We use a previously developed, spatially-explicit model of the population dynamics of anAcacia species in the Negev desert of Israel (SAM, spatialAcacia model) to investigate if model initialization (measurement of current tree distribution) can be obtained from readily-available satellite images using a radiometric vegetation index (NDVI, normalized difference vegetation index). Furthermore, we investigate the applicability and the advantages of using an explicit consideration of landscape features in the model based on topographic data from a GIS. Using a DEM (digital elevation model), we compare the wadi topography to the current tree distribution observed in the field.     

Seasonality and Outbreaks in West Nile Virus Infection

In this paper we analyze the impact of seasonal variations on the dynamics of West Nile Virus infection. We are interested in the generation of new epidemic peaks starting from an endemic state. In many cases, the oscillations generated by seasonality in the dynamics of the infection are too small to be observable. The interplay of this seasonality with the epidemic oscillations can generate new outbreaks starting from the endemic state through a mechanism of parametric resonance. Using experimental data we present specific cases where this phenomenon is numerically observed.     

A deterministic size-structured population model for the worm Naidis elinguis

In this paper we model the population dynamics of the worm Nais elinguis, which reproduces by division into two unequal parts. By using renewal theory we derive the asymptotic behaviour of a Naidis elinguis population. In particular we prove a certain relation between the fraction of the population that was born small (respectively the fraction that was born large) and the inter-division times. Received 20 January 1999 / Revised version: 1 August 1999?Published online: 10 April 2001     

Understanding the dynamics of Salmonella infections in dairy herds: a modelling approach

There is evidence of variation in the infection dynamics of different Salmonella serotypes in cattle--ranging from transient epidemics to long term persistence and recurrence. We seek to identify possible causes of these differences. In this study we present mathematical models which describe both managed population dynamics and epidemiology and use these to investigate the effects of demographic and epidemiological factors on epidemic behaviour and threshold for invasion. In particular, when the system is perturbed by higher culling or pathogen-induced mortality we incorporate mechanisms to constrain the lactating herd size to remain constant in the absence of pathogen or to lie within a fairly small interval in the presence of pathogen. A combination of numerical and analytical techniques is used to analyse the models. We find that the epidemiologically dominating management group can change from the dry/lactating cycle to the weaned group with increasing culling rate. Pseudovertical transmission is found to have little effect on the invasion criteria, while indirect transmission has significant influence. Pathogen-induced mortality, recovery, immune response and pathogen removal are found to be factors inducing damped oscillations; variation in these factors between Salmonella serotypes may be responsible for some of the observed differences in within herd dynamics. Specifically, higher pathogen-induced mortality, shorter infectious period, more persistent immune response and more rapid removal of faeces result in a lower number of infectives and smaller epidemics but a greater tendency for damped oscillations.     

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.