In this paper we analyze the stochastic version of a minimalistic multi-strain model, which captures essential differences between primary and secondary infections in dengue fever epidemiology, and investigate the interplay between stochasticity, seasonality and import. The introduction of stochasticity is needed to explain the fluctuations observed in some of the available data sets, revealing a scenario where noise and complex deterministic skeleton strongly interact. For large enough population size, the stochastic system can be well described by the deterministic skeleton gaining insight on the relevant parameter values purely on topological information of the dynamics, rather than classical parameter estimation of which application is in general restricted to fairly simple dynamical scenarios.

We present a model for describing the spread of an infectious disease with public screening measures to control the spread. We want to address the problem of determining an optimal screening strategy for a disease characterized by appreciable duration of the infectiveness period and by variability of the transmission risk. The specific disease we have in mind is the HIV infection. However the model will apply to a disease for which class-age structure is significant and should not be disregarded.

In this paper we present an epidemic model affecting an age-structured population. We show by numerical simulations that this demographic structure can induce persistent oscillations in the epidemic. The model is then extended to encompass a stage-structured disease within an age-dependent population. In this case as well, persistent oscillations are observed in the infected as well as in the whole population.

About twenty five years ago the first discrete mathematical model of the immune system was proposed. It was very simple and stylized. Later, many other computational models have been proposed each one adding a certain level of sophistication and detail to the description of the system. One of these, the Celada-Seiden model published back in 1992, was already mature at its birth, setting apart from the topic-specific nature of the other models. This one was not just a model but rather a framework with which one could implement his own immunological theories.

Here we describe this computational framework, developed to perform simulations of different pathologies that are directly or indirectly connected to the immune system. We briefly describe the system first, then we report on few applications so to give the reader a clear idea of its practical utility in clinical research problems.

Influenza has been responsible for human suffering and economic burden worldwide. Isolation is one of the most effective means to control the disease spread. In this work, we incorporate isolation into a two-strain model of influenza. We find that whether strains of influenza die out or coexist, or only one of them persists, it depends on the basic reproductive number of each influenza strain, cross-immunity between strains, and isolation rate. We propose criteria that may be useful for controlling influenza. Furthermore, we investigate how effective isolation is by considering the host’s mean age at infection and the invasion rate of a novel strain. Our results suggest that isolation may help to extend the host’s mean age at infection and reduce the invasion rate of a new strain. When there is a delay in isolation, we show that it may lead to more serious outbreaks as compared to no delay.

A decision analytical model is presented and analysed to assess the effectiveness and cost-effectiveness of routine vaccination against varicella and herpes-zoster, or shingles. These diseases have as common aetiological agent the varicella-zoster virus (VZV). Zoster can more likely occur in aged people with declining cell-mediated immunity. The general concern is that universal varicella vaccination might lead to more cases of zoster: with more vaccinated children exposure of the general population to varicella infectives become smaller and thus a larger proportion of older people will have weaker immunity to VZV, leading to more cases of reactivation of zoster. Our compartment model shows that only two possible equilibria exist, one without varicella and the other one where varicella and zoster both thrive. Threshold quantities to distinguish these cases are derived. Cost estimates on a possible herd vaccination program are discussed indicating a possible tradeoff choice.

National policies regarding the BCG vaccine for tuberculosis vary greatly throughout the international community and several countries are currently considering discontinuing universal vaccination. Detractors of BCG point to its uncertain effectiveness and its interference with the detection and treatment of latent tuberculosis infection (LTBI).

In order to quantify the trade-off between vaccination and treatment of LTBI, a mathematical model was designed and calibrated to data from Brazil, Ghana, Germany, India, Mexico, Romania, the United Kingdom and the United States. Country-specific thresholds for when LTBI treatment outperforms mass vaccination were found and the consequences of policy changes were estimated.

Our results suggest that vaccination outperforms LTBI treatment in all settings but with greatly reduced efficiency in low incidence countries. While national policy statements emphasize BCG’s interference with LTBI detection, we find that reinfection should be more determinant of a country’s proper policy choice.

An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103–112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopf-bifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.

Tuberculosis (TB) remains a major global health problem. A possible risk factor for TB is diabetes (DM), which is predicted to increase dramatically over the next two decades, particularly in low and middle income countries, where TB is widespread. This study aimed to assess the strength of the association between TB and DM. We present a deterministic model for TB in a community in order to determine the impact of DM in the spread of the disease. The important mathematical features of the TB model are thoroughly investigated. The epidemic threshold known as the basic reproduction number and equilibria for the model are determined and stabilities analyzed. The model is numerically analyzed to assess the impact of DM on the transmission dynamics of TB. We perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Numerical simulations suggest that DM enhances the TB transmission and progression to active TB in a community. The results suggest that there is a need for increased attention to intervention strategies such as the chemoprophylaxis of TB latent individuals and treatment of active TB in people with DM, which may include testing for suspected diabetes, improved glucose control, and increased clinical and therapeutic monitoring in order to reduce the burden of the disease.

We present a unified mathematical approach to epidemiological models with parametric heterogeneity, i.e., to the models that describe individuals in the population as having specific parameter (trait) values that vary from one individuals to another. This is a natural framework to model, e.g., heterogeneity in susceptibility or infectivity of individuals. We review, along with the necessary theory, the results obtained using the discussed approach. In particular, we formulate and analyze an SIR model with distributed susceptibility and infectivity, showing that the epidemiological models for closed populations are well suited to the suggested framework. A number of known results from the literature is derived, including the final epidemic size equation for an SIR model with distributed susceptibility. It is proved that the bottom up approach of the theory of heterogeneous populations with parametric heterogeneity allows to infer the population level description, which was previously used without a firm mechanistic basis; in particular, the power law transmission function is shown to be a consequence of the initial gamma distributed susceptibility and infectivity. We discuss how the general theory can be applied to the modeling goals to include the heterogeneous contact population structure and provide analysis of an SI model with heterogeneous contacts. We conclude with a number of open questions and promising directions, where the theory of heterogeneous populations can lead to important simplifications and generalizations.

In this paper we are interested in a mathematical model of migration of grass eels in an estuary. We first revisit a previous model proposed by O. Arino and based on a degenerate convection-diffusion equation of parabolic-hyperbolic type with time-varying subdomains. Then, we propose an adapted mathematical framework for this model, we prove a result of existence of a weak solution and we propose some numerical simulations.

The immune system is able to protect the host from tumor onset, and immune deficiencies are accompanied by an increased risk of cancer. Immunology is one of the fields in biology where the role of computational and mathematical modeling and analysis were recognized the earliest, beginning from 60s of the last century. We introduce the two most common methods in simulating the competition among the immune system, cancers and tumor immunology strategies: differential equations and rule-based models. Several specific implementations are presented, describing in details how they work and how they advance or contribute the field of tumor immunology.

Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour in response to changes in the ambient conditions can be described by models involving switching. In most cases, this switching is assumed to depend on the system state, and thus it disregards the history and, therefore, memory. Memory can be introduced into a mathematical model using a phenomenon known as hysteresis. We illustrate this idea using a simple SIR compartmental model that is applicable in epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and how it may be applied to describe a variety of memory effects. Our other objective is to introduce a unified paradigm for mathematical modelling with memory effects in epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow related to large ensembles of elementary exchange operations that recently has been successfully applied in a number of other areas, such as terrestrial hydrology, and macroeconomics. For the purposes of illustrating these ideas in an application to biology, we consider a rather simple case study and develop a model from first principles. We accompany the model with extensive numerical simulations which exhibit interesting qualitative effects.

Background

Independence models among variables is one of the most relevant topics in epidemiology, particularly in molecular epidemiology for the study of gene-gene and gene-environment interactions. They have been studied using three main kinds of analysis: regression analysis, data mining approaches and Bayesian model selection. Recently, methods of algebraic statistics have been extensively used for applications to biology. In this paper we present a synthetic, but complete description of independence models in algebraic statistics and a new method of analyzing interactions, that is equivalent to the correction by Markov bases of the Fisher’s exact test.

Methods

We identified the suitable algebraic independence model for describing the dependence of two genetic variables from the occurrence of cancer and exploited the theory of toric varieties and Gröbner basis for developing an exact independence test based on the Diaconis-Sturmfels algorithm. We implemented it in a Maple routine and we applied it to the study of gene-gene interaction in Gen-Air, an European case-control study. We computed the p-value for each pair of genetic variables interacting with disease status and we compared our results with the standard asymptotic chi-square test.

Results

We found an association among COMT Val158Met, APE1 Asp148Glu and bladder cancer (p-value: 0.009). We also found the interaction among TP53 Arg72Pro, GSTP1 Ile105Val and lung cancer (p-value: 0.00035). Leukaemia was observed to significantly interact with the pairs ERCC2 Lys751Gln and RAD51 172 G > T (p-value 0.0072), ERCC2 Lys751Gln and LIG4Thr9Ile (p-value: 0.0095) and APE1 Asp148Glu and GSTP1 Ala114Val (p-value: 0.0036).

Conclusion

Taking advantage of results from theoretical and computational algebra, the method we propose was more selective than other methods in detecting new interactions, and nevertheless its results were consistent with previous epidemiological and functional findings. It also helped us in controlling the multiple comparison problem. In the light of our results, we believe that the epidemiologic study of interactions can benefit of algebraic methods based on properties of toric varieties and Gröbner bases.

Contact behavior plays an important role in influenza transmission. In the progression of influenza spread, human population reduces mobility to decrease infection risks. In this paper, a mathematical model is proposed to include adaptive mobility. It is shown that the mobility response does not affect the basic reproduction number that characterizes the invasion threshold, but reduces dramatically infection peaks, or removes the peaks. Numerical calculations indicate that the mobility response can provide a very good protection to susceptible individuals, and a combination of mobility response and treatment is an effective way to control influenza outbreak.