Parametric dependence in model epidemics. II: Non-contact rate-related parameters

In a previous paper, we discussed the bifurcation structure of SEIR equations subject to seasonality. There, the focus was on parameters that affect transmission: the mean contact rate, β₀, and the magnitude of seasonality, ? _B . Using numerical continuation and brute force simulation, we characterized a global pattern of parametric dependence in terms of subharmonic resonances and period-doublings of the annual cycle. In the present paper, we extend this analysis and consider the effects of varying non-contact-related parameters: periods of latency, infection and immunity, and rates of mortality and reproduction, which, following the usual practice, are assumed to be equal. The emergence of several new forms of dynamical complexity notwithstanding, the pattern previously reported is preserved. More precisely, the principal effect of varying non-contact related parameters is to displace bifurcation curves in the β₀?? _B parameter plane and to expand or contract the regions of resonance and period-doubling they delimit. Implications of this observation with respect to modeling real-world epidemics are considered.     

Bifurcation analysis of periodic SEIR and SIR epidemic models

The bifurcations of the periodic solutions of SEIR and SIR epidemic models with sinusoidally varying contact rate are investigated. The analysis is carried out with respect to two parameters: the mean value and the degree of seasonality of the contact rate. The corresponding portraits in the two-parameter space are obtained by means of a numerical continuation method. Codimension two bifurcations (degenerate flips and cusps) are detected, and multiple stable modes of behavior are identified in various regions of the parameter space. Finally, it is shown how the parametric portrait of the SEIR model tends to that of the SIR model when the latent period tends to zero.     

On the dynamics of internal leaf carbon dioxide uptake

A model is proposed to investigate the hypothesis that the observed time course of whole leaf photosynthetic responses to changes in incident light energy are caused by diffusional limitations. The model leaf consists of a continuously distributed mesophyll with diffusion of CO₂ in the leaf interior governed by a reaction-diffusion equation. Biochemical activation is assumed to occur on a fast time scale. Both the cases of a homogeneous and a nonhomogeneous leaf interior are investigated. Using parameter estimates available in the literature, the model predicts CO₂ uptake equilibration times which are much smaller than those observed. The results strongly suggest that diffusional limitations do not significantly affect photosynthetic dynamics in variable light environments.     

A PDE Model for Imatinib-Treated Chronic Myelogenous Leukemia

We derive a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia (CML). This model is a continuous extension of the agent-based CML model of Roeder et al. (Nat. Med. 12(10), 1181–1184, 2006) and of its recent formulation as a system of difference equations (Kim et al. in Bull. Math. Biol. 70(3), 728–744, 2008). The new model is formulated as a system of partial differential equations that describe various stages of differentiation and maturation of normal hematopoietic cells and of leukemic cells. An imatinib treatment is also incorporated into the model. The simulations of the new PDE model are shown to qualitatively agree with the results that were obtained with the discrete-time (difference equation and agent-based) models. At the same time, for a quantitative agreement, it is necessary to adjust the values of certain parameters, such as the rates of imatinib-induced inhibition and degradation.     

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.     

On the uniqueness of the positive solution of an integral equation which appears in epidemiological models

 In this paper, we show that the positive solution of a non-linear integral equation which appears in classical SIR epidemiological models is unique. The demonstration of this fact is necessary to justify the correctness of any approximate or numerical solution. The SIR epidemiological model is used only for simplicity. In fact, the methods used can be easily extended to prove the existence and uniqueness of the more involved integral equations that appear when more biological realities are considered. Thus the inclusion of a latent class (SLIR models) and models incorporating variability in the infectiousness with duration of the infection and spatial distribution lead to integral equations to which the results derived in this paper apply immediately. Received: 7 May 1999     

Oscillatory reaction-diffusion equations on rings

We study the behavior of traveling waves in - systems on both homogeneous and inhomogeneous rings. The stability regions in parameter space of - waves were previously known [15, 19]; the results are extended here. We show the existence of Hopf bifurcations of traveling waves and the stability of the limit cycles born at the Hopf bifurcation for some parameter ranges. Using a Lindstedt-type perturbation scheme, we formally construct periodic solutions of the - system near a Hopf bifurcation and show that the periodic solutions superimposed on the original traveling wave have the effect of altering its overall frequency and amplitude. We also study the - system on an annulus ofvariable width, which does not possess reflection symmetry about any axis. We formally construct traveling waves on this variable-width annulus by a perturbation scheme, and find that perturbing the width of the annulus alters the amplitude and frequency of traveling waves on the domain by a small (order ²) amount. For typical parameter values, we find that the speed, frequency, and stability are unaffected by the direction of travel of the wave on the annulus, despite the rotationally asymmetric inhomogeneity. This indicates that the - system on a variable-width domain cannot account for directional preferences of traveling waves in biological systems.     

A simple model of pathogen–immune dynamics including specific and non-specific immunity

We present and analyze a model for the dynamics of the interactions between a pathogen and its host’s immune response. The model consists of two differential equations, one for pathogen load, the other one for an index of specific immunity. Differently from other simple models in the literature, this model exhibits, according to the hosts’ or pathogen’s parameter values, or to the initial infection size, a rich repertoire of behaviours: immediate clearing of the pathogen through aspecific immune response; or acute infection followed by clearing of the pathogen through specific immune response; or uncontrolled infections; or acute infection followed by convergence to a stable state of chronic infection; or periodic solutions with intermittent acute infections. The model can also mimic some features of immune response after vaccination. This model could be a basis on which to build epidemic models including immunological features.     

The effect of intrinsic stochasticity on transmitted HIV drug resistance patterns

Estimates of transmitted HIV drug-resistance prevalence vary widely among and within epidemiological surveys. Interpretation of trends from available survey data is therefore difficult. Because the emergence of drug-resistance involves small populations of infected drug-resistant individuals, the role of stochasticity (chance events) is likely to be important. The question addressed here is: how much variability in transmitted HIV drug-resistance prevalence patterns arises due to intrinsic stochasticity alone, i.e., if all starting conditions in the different epidemics surveyed were identical? This ‘thought experiment’ gives insight into the minimum expected variabilities within and among epidemics. A simple stochastic mathematical model was implemented. Our results show that stochasticity alone can generate a significant degree of variability and that this depends on the size and variation of the pool of new infections when drug treatment is first introduced. The variability in transmitted drug-resistance prevalence within an epidemic (i.e., the temporal variability) is large when the annual pool of all new infections is small (fewer than 200, typical of the HIV epidemics in Central European and Scandinavian countries) but diminishes rapidly as that pool grows. Epidemiological surveys involving hundreds of new infections annually are therefore needed to allow meaningful interpretation of temporal trends in transmitted drug-resistance prevalence within individual epidemics. The stochastic variability among epidemics shows a similar dependence on the pool of new infections if treatment is introduced after endemic equilibrium is established, but can persist even when there are more than 10,000 new infections annually if drug therapy is introduced earlier. Stochastic models may therefore have an important role to play in interpreting differences in transmitted drug-resistance prevalence trends among epidemiological surveys.     

Explicit Separation of Growth and Motility in a New Tumor Cord Model

We investigate a new model of tumor growth in which cell motility is considered an explicitly separate process from growth. Bulk tumor expansion is modeled by individual cell motility in a density-dependent diffusion process. This model is implemented in the context of an in vivo system, the tumor cord. We investigate numerically microscale density distributions of different cell classes and macroscale whole tumor growth rates as functions of the strength of transitions between classes. Our results indicate that the total tumor growth follows a classical von Bertalanffy growth profile, as many in vivo tumors are observed to do. This provides a quick validation for the model hypotheses. The microscale and macroscale properties are both sensitive to fluctuations in the transition parameters, and grossly adopt one of two phenotypic profiles based on their parameter regime. We analyze these profiles and use the observations to classify parameter regimes by their phenotypes. This classification yields a novel hypothesis for the early evolutionary selection of the metastatic phenotype by selecting against less motile cells which grow to higher densities and may therefore induce local collapse of the vascular network.     

Modelling the epidemiology of hepatitis B in New Zealand

Hepatitis B is a vaccine preventable disease caused by the hepatitis B virus (HBV) that can induce potentially fatal liver damage. It has the second highest mortality rate of all vaccine preventable diseases in New Zealand. Vaccination against HBV was introduced in New Zealand in 1988, and the country is now categorised with overall low endemicity but with areas of both high and medium endemic levels. We present an SECIR compartmental mathematical model, with the population divided into age classes, for the transmission of HBV using local data on incidence of infection and vaccination coverage. We estimate the basic reproduction number, R₀, to be 1.53, and show that the vaccination campaign has substantially reduced this below one. However, a large number of carriers remain in the population acting as a source of infection.     

Stability in cyclic epidemic models

A general formulation for a family of cyclic epidemic models with density-dependent feedback mechanisms and removed classes is presented. A parameter, , related to the basic reproductive rate determines the asymptotic behaviour of solutions of the model. It is shown that if <1 the trivial solution is globally stable, and if >1 it is conditionally stable. The results are applied to a set of differential equations that has been used to model the life cycle of a parasite that has two hosts.     

A generalized compartmental model to estimate the fibre mass in the ruminoreticulum: 1. Estimating parameters of digestion

Parameters related to the microbial digestion of nutrients in the ruminoreticulum have been estimated by fitting mathematical models to degradation profiles generated from kinetic studies. In the present paper, we propose a generalized compartmental model of digestion (GCMD) based on implicit theoretical concepts and the gamma probability density function to estimate fibre digestion parameters. The proposed model is consistent to a broader compartmental model presented in a companion paper that integrates aspects of fibre digestion and passage. Different versions of the GCMD were generated by increasing the integer order of time dependency of the gamma function. These versions were fitted to 192 published fibre degradation profiles that were obtained using an in vitro fermentation technique. The quality of fit was evaluated based on the frequency of minimum sum of squares of errors (SSE), the number of runs of signs of residuals, and its likelihood probability calculated according to the Akaike's Information Criterion. The likelihood of the proposed model was also compared to a discrete lag time model (DLT), which is commonly used to interpret fibre degradation profiles. The GCMD had superior quality of fit compared to the DLT and was considered more likely in describing 68.75% of the profiles evaluated. Only 9.38% of the degradation profiles that were fitted to the DLT model had a lower SSE. Even though the degradation profiles studied were generated by incubating feed samples up to 96 h, the true asymptotic limit of fibre degradation can only be achieved by long-term fermentations. This fact leads to questioning the uniformity of the potentially digestible fibre fraction and a further approach based on GCMD-type model was used to account for its heterogeneous nature.     

Variola minor in bragança paulista county,in 1956. time‐interval analysis on disease onsets in two elementary schools

Abstract

In order to investigate the rhythmicity of intermale aggression in mice, and the potential influence of the pineal gland, we maintained 20 male SJL mice who were singly housed in LD 10:14.10 animals were pinealectomized and 10 animals were given sham operations. Pairs of subjects within each group were placed in a testing arena and allowed to fight for 6 minute encounters. Each animal encountered a different member of his group every 27 h so that after 8 days there had been an aggression test for every 3 h of the 24‐h cycle. Beginning on the ninth day the entire test cycle was replicated. During testing five agonistic behaviors were scored for each animal: tail lashing, offensive posture, biting, sniffing and defensive posture. Dominance was also scored on a 5‐point scale. Data were analyzed using factor analysis, multiple regression and analysis of covariance and Mests. Results of the study confirm the hypothesis that agonistic behaviors vary rhythmically in male albino mice. Although pinealectomized mice were significantly rhythmic in fewer behavioral measures than were sham‐operated animals, analysis of covariance did not yield any significant effects of surgical treatment.     

Stability in a two host epidemic model

An epidemic model is derived for a two host infectious disease. It is shown that if a non-trivial equilibrium solution exists, it is globally stable. This result is also proved for a similar one host model.     

Ecological balance in the native population dynamics may cause the paradox of pest control with harvesting

We analyze a time-discrete mathematical model of host-parasite population dynamics with harvesting, in which the host can be regarded as a pest. We harvest a portion of the host population at a moment in each parasitism season. The principal target of the harvesting is the host; however, the parasite population may also be affected and reduced by a portion. Our model involves the Beverton-Holt type density effect on the host population. We investigate the condition in which the harvesting of the host results in an eventual increase of its equilibrium population size, analytically proving that the paradoxical increase could occur even when the harvesting does not directly affect the parasite population at all. We show that the paradox of pest control could be caused essentially by the interspecific relationship and the intraspecific density effect.     

Contact switching as a control strategy for epidemic outbreaks

We study the effects of switching social contacts as a strategy to control epidemic outbreaks. Connections between susceptible and infective individuals can be broken by either individual, and then reconnected to a randomly chosen member of the population. It is assumed that the reconnecting individual has no previous information on the epidemiological condition of the new contact. We show that reconnection can completely suppress the disease, both by continuous and discontinuous transitions between the endemic and the infection-free states. For diseases with an asymptomatic phase, we analyze the conditions for the suppression of the disease, and show that—even when these conditions are not met—the increase of the endemic infection level is usually rather small. We conclude that, within some simple epidemiological models, contact switching is a quite robust and effective control strategy. This suggests that it may also be an efficient method in more complex situations.     

Models to understand the population-level impact of mixed strain M. tuberculosis infections

Over the past decade, numerous studies have identified tuberculosis patients in whom more than one distinct strain of Mycobacterium tuberculosis is present. While it has been shown that these mixed strain infections can reduce the probability of treatment success for individuals simultaneously harboring both drug-sensitive and drug-resistant strains, it is not yet known if and how this phenomenon impacts the long-term dynamics for tuberculosis within communities. Strain-specific differences in immunogenicity and associations with drug resistance suggest that a better understanding of how strains compete within hosts will be necessary to project the effects of mixed strain infections on the future burden of drug-sensitive and drug-resistant tuberculosis. In this paper, we develop a modeling framework that allows us to investigate mechanisms of strain competition within hosts and to assess the long-term effects of such competition on the ecology of strains in a population. These models permit us to systematically evaluate the importance of unknown parameters and to suggest priority areas for future experimental research. Despite the current scarcity of data to inform the values of several model parameters, we are able to draw important qualitative conclusions from this work. We find that mixed strain infections may promote the coexistence of drug-sensitive and drug-resistant strains in two ways. First, mixed strain infections allow a strain with a lower basic reproductive number to persist in a population where it would otherwise be outcompeted if has competitive advantages within a co-infected host. Second, some individuals progressing to phenotypically drug-sensitive tuberculosis from a state of mixed drug-sensitive and drug-resistant infection may retain small subpopulations of drug-resistant bacteria that can flourish once the host is treated with antibiotics. We propose that these types of mixed infections, by increasing the ability of low fitness drug-resistant strains to persist, may provide opportunities for compensatory mutations to accumulate and for relatively fit, highly drug-resistant strains of M. tuberculosis to emerge.     

Mathematical analysis of a basic model for epidermal wound healing

The stimuli for the increase in epidermal mitosis during wound healing are not fully known. We construct a mathematical model which suggests that biochemical regulation of mitosis is fundamental to the process, and that a single chemical with a simple regulatory effect can account for the healing of circular epidermal wounds. The numerical results of the model compare well with experimental data. We investigate the model analytically by making biologically relevant approximations. We then obtain travelling wave solutions which provide information about the accuracy of these approximations and clarify the roles of the various model parameters.     

Network theory and SARS: predicting outbreak diversity

Many infectious diseases spread through populations via the networks formed by physical contacts among individuals. The patterns of these contacts tend to be highly heterogeneous. Traditional "compartmental" modeling in epidemiology, however, assumes that population groups are fully mixed, that is, every individual has an equal chance of spreading the disease to every other. Applications of compartmental models to Severe Acute Respiratory Syndrome (SARS) resulted in estimates of the fundamental quantity called the basic reproductive number R0--the number of new cases of SARS resulting from a single initial case--above one, implying that, without public health intervention, most outbreaks should spark large-scale epidemics. Here we compare these predictions to the early epidemiology of SARS. We apply the methods of contact network epidemiology to illustrate that for a single value of R0, any two outbreaks, even in the same setting, may have very different epidemiological outcomes. We offer quantitative insight into the heterogeneity of SARS outbreaks worldwide, and illustrate the utility of this approach for assessing public health strategies.