Controlling chaos in ecology: From deterministic to individual-based models

The possibility of chaos control in biological systems has been stimulated by recent advances in the study of heart and brain tissue dynamics. More recently, some authors have conjectured that such a method might be applied to population dynamics and even play a nontrivial evolutionary role in ecology. In this paper we explore this idea by means of both mathematical and individual-based simulation models. Because of the intrinsic noise linked to individual behavior, controlling a noisy system becomes more difficult but, as shown here, it is a feasible task allowed to be experimentally tested.     

Contact-based transmission models in terrestrial gastropod populations infected with parasitic mites

Parasite transmission fundamentally affects the epidemiology of host-parasite systems, and is considered to be a key element in the epidemiological modelling of infectious diseases. Recent research has stressed the importance of detailed disease-specific variables involved in the transmission process. Riccardoella limacum is a hematophagous mite living in the mantle cavity of terrestrial gastropods. In this study, we experimentally examined whether the transmission success of R. limacum is affected by the contact frequency, parasite load and/or behaviour of the land snail Arianta arbustorum, a common host of R. limacum. In the experiment the transmission success was mainly affected by physical contacts among snails and slightly influenced by parasite intensity of the infected snail. Using these results we developed two different transmission models based on contact frequencies and transmission probability among host snails. As parameters for the models we used life-history data from three natural A. arbustorum populations with different population densities. Data on contact frequencies of video-recorded snail groups were used to fit the density response of the contact function, assuming either a linear relationship (model 1) or a second-degree polynomial relationship based on the ideal gas model of animal encounter (model 2). We calculated transmission coefficients (β), basic reproductive ratios (R₀) and host threshold population densities for parasite persistence in the three A. arbustorum populations. We found higher transmission coefficients (β) and larger R₀-values in model 1 than in model 2. Furthermore, the host population with the highest density showed larger R₀-values (16.47-22.59) compared to populations with intermediate (2.71-7.45) or low population density (0.75-4.10). Host threshold population density for parasite persistence ranged from 0.35 to 2.72 snails per m². Our results show that the integration of the disease-relevant biology of the organisms concerned may improve models of host-parasite dynamics.     

Effectiveness of realistic vaccination strategies for contact networks of various degree distributions

A "contact network" that models infection transmission comprises nodes (or individuals) that are linked when they are in contact and can potentially transmit an infection. Through analysis and simulation, we studied the influence of the distribution of the number of contacts per node, defined as degree, on infection spreading and its control by vaccination. Three random contact networks of various degree distributions were examined. In a scale-free network, the frequency of high-degree nodes decreases as the power of the degree (the case of the third power is studied here); the decrease is exponential in an exponential network, whereas all nodes have the same degree in a constant network. Aiming for containment at a very early stage of an epidemic, we measured the sustainability of a specific network under a vaccination strategy by employing the critical transmissibility larger than which the epidemic would occur. We examined three vaccination strategies: mass, ring, and acquaintance. Irrespective of the networks, mass preventive vaccination increased the critical transmissibility inversely proportional to the unvaccinated rate of the population. Ring post-outbreak vaccination increased the critical transmissibility inversely proportional to the unvaccinated rate, which is the rate confined to the targeted ring comprising the neighbors of an infected node; however, the total number of vaccinated nodes could mostly be fewer than 100 nodes at the critical transmissibility. In combination, mass and ring vaccinations decreased the pathogen's "effective" transmissibility each by the factor of the unvaccinated rate. The amount of vaccination used in acquaintance preventive vaccination was lesser than the mass vaccination, particularly under a highly heterogeneous degree distribution; however, it was not as less as that used in ring vaccination. Consequently, our results yielded a quantitative assessment of the amount of vaccination necessary for infection containment, which is universally applicable to contact networks of various degree distributions.     

Parameterization of Keeling's network generation algorithm

Simulation is increasingly being used to examine epidemic behaviour and assess potential management options. The utility of the simulations rely on the ability to replicate those aspects of the social structure that are relevant to epidemic transmission. One approach is to generate networks with desired social properties.Recent research by Keeling and his colleagues has generated simulated networks with a range of properties, and examined the impact of these properties on epidemic processes occurring over the network. However, published work has included only limited analysis of the algorithm itself and the way in which the network properties are related to the algorithm parameters.This paper identifies some relationships between the algorithm parameters and selected network properties (mean degree, degree variation, clustering coefficient and assortativity). Our approach enables users of the algorithm to efficiently generate a network with given properties, thereby allowing realistic social networks to be used as the basis of epidemic simulations. Alternatively, the algorithm could be used to generate social networks with a range of property values, enabling analysis of the impact of these properties on epidemic behaviour.     

Optimal control of drug therapy: melding pharmacokinetics with viral dynamics

Pharmacokinetics were melded with a viral dynamical model to design an optimal drug administration regimen such that the basic reproductive number for the virus was minimized. One-compartmental models with two kinds of drug delivery routes, intravenous and extravascular with multiple dosages, and two drug elimination rates, first order and Michaelis-Menten rates, were considered. We defined explicitly the basic reproductive number for the viral dynamical model melded with pharmacokinetics. When the average plasma drug concentration was constant, intravenous administration of the drug with small dosages applied frequently minimized the basic reproductive number. For extravascular administration, the basic reproductive number initially decreases to a trough point and then increases as the drug dosage increases. When a therapeutic window is considered, numerical studies indicate that the wider the window, the smaller the basic reproductive number. Once the width of the therapeutic window is fixed, the basic reproductive number monotonously declines as the minimum therapeutic level increases. The findings suggest that the existence of drug dosage and drug administration interval that minimize the basic reproductive number could help design the optimal drug administration regimen.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Partnership dynamics and strain competition

Models of epidemic spread that include partnership dynamics within the host population have demonstrated that finite length partnerships can limit the spread of pathogens. Here the influence of partnerships on strain competition is investigated. A simple epidemic and partnership formation model is used to demonstrate that, in contrast to standard epidemiological models, the constraint introduced by partnerships can influence the success of pathogen strains. When partnership turnover is slow, strains must have a long infectious period in order to persist, a requirement of much less importance when partnership turnover is rapid. By introducing a trade-off between transmission rate and infectious period it is shown that populations with different behaviours can favour different strains. Implications for control measures based on behavioural modifications are discussed, with such measures perhaps leading to the emergence of new strains.     

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.     

Contact rate calculation for a basic epidemic model

Mass-action epidemic models are the foundation of the majority of studies of disease dynamics in human and animal populations. Here, a kinetic model of mobile susceptible and infective individuals in a two-dimensional domain is introduced, and an examination of the contact process results in a mass-action-like term for the generation of new infectives. The conditions under which density dependent and frequency dependent transmission terms emerge are clarified. Moreover, this model suggests that epidemics in large mobile spatially distributed populations can be well described by homogeneously mixing mass-action models. The analysis generates an analytic formula for the contact rate (β) and the basic reproductive ratio (R₀) of an infectious pathogen, which contains a mixture of demographic and epidemiological parameters. The analytic results are compared with a simulation and are shown to give good agreement. The simulation permits the exploration of more realistic movement strategies and their consequent effect on epidemic dynamics.     

Expansion or extinction: deterministic and stochastic two-patch models with Allee effects

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment. Our main focus is on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and a stochastic counterpart. The deterministic model has either two, three or four attractors. The existence of a regime with exactly three attractors only appears when patches have distinct Allee thresholds. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either a global expansion or a global extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the global extinction and the basin of attraction of the global expansion. Above this threshold, for both the deterministic and the stochastic models, the patches tend to synchronize as the intensity of the dispersal increases. This results in either a global expansion or a global extinction. For the deterministic model, there are only two attractors, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee thresholds as the global population size in the deterministic and the stochastic models evolves as dictated by their single-patch counterparts. For all values of the dispersal parameter, Allee effects promote global extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.     

Improving estimates of the basic reproductive ratio: using both the mean and the dispersal of transition times

In both within-host and epidemiological models of pathogen dynamics, the basic reproductive ratio, R(0), is a powerful tool for gauging the risk associated with an emerging pathogen, or for estimating the magnitude of required control measures. Techniques for estimating R(0), either from incidence data or in-host clinical measures, often rely on estimates of mean transition times, that is, the mean time before recovery, death or quarantine occurs. In many cases, however, either data or intuition may provide additional information about the dispersal of these transition times about the mean, even if the precise form of the underlying probability distribution remains unknown. For example, we may know that recovery typically occurs within a few days of the mean recovery time. In this paper we elucidate common situations in which R(0) is sensitive to the dispersal of transition times about their respective means. We then provide simple correction factors that may be applied to improve estimates of R(0) when not only the mean but also the standard deviation of transition times out of the infectious state can be estimated.     

The deterministic limit of infectious disease models with dynamic partners

This paper analyzes the large population dynamics of an infectious disease model with contacts that occur during partnerships. The model allows for concurrent partnerships following a very broad class of dynamic laws. Previous work, with a stochastic version of the model, computed the reproductive number, the initial growth rate, and the final size. In the present paper, the deterministic system that is the limit for large populations is constructed. The construction is unusual in requiring two different scaling factors. Next, the approximation used by Watts and May for a related model is compared with the exact solution. This approximation is most accurate at the beginning of the epidemic and when partnerships are short. Lastly, the model is generalized to allow dependencies among partnerships. This generalization permits proportional mixing with an arbitrary distribution on the number of partners.     

Sexual mixing models: a comparison of analogue deterministic and stochastic models.

Models for sexual partner choice are discussed for the case of highly variable sexual activity in the population. It is demonstrated that the variances in the number of infected persons may be extremely large. For the random mixing model, higher order cumulants are also evaluated. On the basis of these results the applicability of deterministic models and models for expectations only are questioned. A general model is proposed for handling nonrandom, or correlated, mixing. The problem of inconsistency is overcome by considering the couples having sex as the natural unit in the model. In the case of s discrete homogeneous groups it is shown that only (s2) parameters defining the interaction between the groups can be chosen freely. Finally, the effect of correlation in partner choice is demonstrated by a bivariate lognormal model for partner choice.     

On the formulation and analysis of general deterministic structured population models I. Linear Theory

—We define a linear physiologically structured population model by two rules, one for reproduction and one for “movement” and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R ₀ and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step. Received 26 July 1996; received in revised form 3 September 1997     

Kinetics of cell-mediated cytotoxicity: Stochastic and deterministic multistage models

One of the body's major defenses against viral diseases and tumors is the killing of abnormal cells by host defense cells, such as T lymphocytes. The mechanism by which killing is accomplished is unknown. Here we develop both stochastic and deterministic models for the kinetics of killing in aggregates which contain a single lymphocyte and multiple target cells (LT_n conjugates), as might be seen early in an immune response, and in aggregates containing multiple lymphocytes and a single target cell (L_nT conjugates), which is characteristic of the late phase of a successful immune response. Comparing our models with data, we rule out the possibility of certain classes of lytic mechanisms and draw attention to the characteristics of likely mechanisms. Our stochastic model can be viewed as a specialized application of queueing theory to cell biology. For certain choices of arrival-time and service-time distributions, we find an exact correspondence between our stochastic and deterministic models.     

Stochastic analogues of deterministic single-species population models

Although single-species deterministic difference equations have long been used in modeling the dynamics of animal populations, little attention has been paid to how stochasticity should be incorporated into these models. By deriving stochastic analogues to difference equations from first principles, we show that the form of these models depends on whether noise in the population process is demographic or environmental. When noise is demographic, we argue that variance around the expectation is proportional to the expectation. When noise is environmental the variance depends in a non-trivial way on how variation enters into model parameters, but we argue that if the environment affects the population multiplicatively then variance is proportional to the square of the expectation. We compare various stochastic analogues of the Ricker map model by fitting them, using maximum likelihood estimation, to data generated from an individual-based model and the weevil data of Utida. Our demographic models are significantly better than our environmental models at fitting noise generated by population processes where noise is mainly demographic. However, the traditionally chosen stochastic analogues to deterministic models--additive normally distributed noise and multiplicative lognormally distributed noise--generally fit all data sets well. Thus, the form of the variance does play a role in the fitting of models to ecological time series, but may not be important in practice as first supposed.     

A new SEIR epidemic model with applications to the theory of eradication and control of diseases, and to the calculation of R0

We present a novel SEIR (susceptible-exposure-infective-recovered) model that is suitable for modeling the eradication of diseases by mass vaccination or control of diseases by case isolation combined with contact tracing, incorporating the vaccine efficacy or the control efficacy into the model. Moreover, relying on this novel SEIR model and some probabilistic arguments, we have found four formulas that are suitable for estimating the basic reproductive numbers R(0) in terms of the ratio of the mean infectious period to the mean latent period of a disease. The ranges of R(0) for most known diseases, that are calculated by our formulas, coincide very well with the values of R(0) estimated by the usual method of fitting the models to observed data.