Epidemiological Compartment Research Articles

A novel mathematical model for the early detection of Dengue Fever using SIR Infectious Disease Epidemiological Compartments, ordinary differential equations, and statistical computing

A novel mathematical model for the early detection of Dengue Fever using SIR Infectious Disease Epidemiological Compartments, ordinary differential equations, and statistical computing

Stochastic modeling approach of infectious disease with sir epidemiological compartment model

The aim of this paper is to present a review on the stochastic version of the deterministic SIR (Susceptible – Infectious - Recovery) epidemiological compartment model through the branching process approximation. The stochastic process (branching process) approximation was developed using the Continuous Time Markov Chains where the time variable is continuous and the state variable is discrete. The state random variables are the compartments: S(t), I(t) and R(t). In this review two ways of estimating the state transition probability has been provided and some stochastic thresholds of the branching process (basic reproduction number, Malthusian parameter and the average number of infections produced by an infectious individual in a single generation) have also been deduced. Finally, the probability of major and minor outbreak of the branching process (epidemic process) has been presented. The theoretical methods have also been validated with some examples of numerical simulations.

Stability Analysis in a Mathematical Model of Corruption in Kenya

The term corruption refers to the process that involves the abuse of a public trust or office for some private benet. Corruption becomes a threat to national development and growth especially when there is no political will to fight it. Prevention and disengagement initiatives are part of EACC strategies used to fight corruption. Prevention strategies aim to stop or discourage citizens from engaging in corruption. Disengagement strategies attempt to reform corrupt individuals and to reclaim the stolen resources back to the public kitty. We describe prevention and disengagement strategies mathematically using an epidemiological compartment model. The prevention and disengagement strategies are modeled using model parameters. The population at risk of adopting corrupt ideology was divided into three compartments: S(t) is the susceptible class, C(t) is the Corrupted class, and M(t) is the corrupt political/sympathersizer class. The model exhibits a threshold dynamics characterised by the basic reproduction number R0. When R0 < 1 the system has a unique equilibrium point that is asymptotically stable. For R0 > 1, the system has additional equilibrium point known as endemic, which is globally asymptotically stable. These results are established by applying lyapunov functions and the LaSalles invariance principle. Based on our model we assess strategies to counter corruption vice.

Características clínico-epidemiológicas de la infección respiratoria aguda (ira) por Virus Sincitial Respiratorio (VSR) en niños menores de dos años: admitidos en el hospital Infantil Dr. Robert Reid Cabral (HIRRC), enero-diciembre 2006

Antecedentes: El VSR es el principal responsable de bronquiolitis y neumonía intersticial en los primeros 2 años de vida y se asocia a complicaciones bacterianas e incremento de hospitalizaciones por infecciones respiratorias.Los casos ocurren usualmente en otoño e invierno, en República Dominicana se desconoce el comportamiento clínico y epidemiológico de la IRA causada por VRS. Objetivos: Determinar las características clínicas y epidemiológicas de los casos de IRA por VRS en niños menores de 2 años durante el periodo de estudio.Objetivo: Determinar las características clínicas y epidemiológicas de los casos de IRA por VRS en niños menores de 2 años durante el período de estudio.Métodos: Estudio prospectivo, descriptivo y observacional donde se estudiaron todos los niños de 0 a 24 meses ingresados con diagnóstico de bronquiolitis durante el período de estudio y radiografías de tórax reportadas como bronquiolitis o neumonía alveolo-intersticial, que fueron confirmadas por la prueba Binax Now® VRS.Resultados: De 114 niños de 0-24 meses de edad y de ambos sexos, admitidos con diagnóstico de bronquiolitis en el HIRRC durante el período de estudio, en 54 casos (48%) se confirmó el VRS a través de la prueba Binax Now®VRS, 62% correspondió al sexo masculino, el 76% a los menores de 4 meses; tos y dificultad respiratoria fueron las manifestaciones clínicas más frecuentes, con 52 casos (94.5%), seguidas por sibilancias, con 46 (83.6%) y retracción subcostal, con 41 (74.5%); la radiografía de tórax reportó bronquiolitis en 36 casos (66.7%), neumonía alveolo-intersticial en 16 casos (29.6%), neumonía más neumotórax y bronquiolitis más atelectasia en 1 caso (1.8%), respectivamente. En el 76.3% la estadía fue menor de 4 días. La totalidad de los casos se presentaron en los meses de octubre (19 casos), noviembre (22 casos), diciembre (12 casos) y enero (1 caso); el 87% presentó anemia, hallazgo no reportado en otros estudios.Conclusión:VSR tiene comportamiento estacional; es el responsable de la mayoría (48%) de los casos de bronquiolitis y neumonía intersticial en lactantes menores; tos y dificultad respiratoria fueron las manifestaciones clínicas más frecuentes; la anemia estuvo presente en el 87% de los casos.

A Model for Temporal Dynamics of Brown Rot Spreading in Fruit Orchards.

Brown rot, caused by Monilinia spp., is a major disease of stone fruit and, in favorable environmental conditions and in the absence of fungicide treatments, it causes important economic losses. In the present work, we propose a modification of classical susceptible, exposed, infectious and removed compartmental models to grasp the peculiarities of the progression of brown rot epidemics in stone fruit orchards in the last stage of the fruit growth (i.e., from the end of the pit hardening to harvest time). Namely, we took into account (i) the lifespan of airborne spores; (ii) the dependence of the latent period on the cuticle crack surface area, which itself varies in time with fruit growth; (iii) the impossibility of recovery in infectious fruit; and (iv) the abrupt interruption of disease development by the elimination of the host fruit at harvest time. We parametrized the model by using field data from a peach Prunus persica orchard infected by Monilinia laxa and M. fructicola in Avignon (southern France). The basic reproduction number indicates that the environmental conditions met in the field were extremely favorable to disease development and the model closely fitted the temporal evolution of the fruit abundance in the different epidemiological compartments. The model permits us to highlight crucial mechanisms undergoing brown rot build up and to evaluate the consequences of different agricultural practices on the quantity and quality of the yield. We found that winter sanitation practices (which decrease the initial infection incidence) and the control of the fruit load (which affects the host fruit density and the single fruit growth trajectory) can be effective in controlling brown rot in conjunction with or in place of fungicide treatments.

A Fractional-Order Infectivity and Recovery SIR Model

The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model.

Detecting a trend change in cross-border epidemic transmission

A method for a system of Langevin equations is developed for detecting a trend change in cross-border epidemic transmission. The equations represent a standard epidemiological SIR compartment model and a meta-population network model. The method analyzes a time series of the number of new cases reported in multiple geographical regions. The method is applicable to investigating the efficacy of the implemented public health intervention in managing infectious travelers across borders. It is found that the change point of the probability of travel movements was one week after the WHO worldwide alert on the SARS outbreak in 2003. The alert was effective in managing infectious travelers. On the other hand, it is found that the probability of travel movements did not change at all for the flu pandemic in 2009. The pandemic did not affect potential travelers despite the WHO alert.

Optimal sampling strategies for detecting zoonotic disease epidemics.

The early detection of disease epidemics reduces the chance of successful introductions into new locales, minimizes the number of infections, and reduces the financial impact. We develop a framework to determine the optimal sampling strategy for disease detection in zoonotic host-vector epidemiological systems when a disease goes from below detectable levels to an epidemic. We find that if the time of disease introduction is known then the optimal sampling strategy can switch abruptly between sampling only from the vector population to sampling only from the host population. We also construct time-independent optimal sampling strategies when conducting periodic sampling that can involve sampling both the host and the vector populations simultaneously. Both time-dependent and -independent solutions can be useful for sampling design, depending on whether the time of introduction of the disease is known or not. We illustrate the approach with West Nile virus, a globally-spreading zoonotic arbovirus. Though our analytical results are based on a linearization of the dynamical systems, the sampling rules appear robust over a wide range of parameter space when compared to nonlinear simulation models. Our results suggest some simple rules that can be used by practitioners when developing surveillance programs. These rules require knowledge of transition rates between epidemiological compartments, which population was initially infected, and of the cost per sample for serological tests.

Modelling the Effect of DOTS and Isolation on TB Transmission Dynamics

Abstract: A deterministic model for the transmission dynamics of Tuberculosis (TB) under Direct Observation Therapy Strategy (DOTS) and Isolation in Nigeria is developed and rigorously analysed. The model, consisting of mutually-exclusive epidemiological compartments representing the number of undetected, detected and isolated individuals who are treated under DOTS programme and those who developed Multi-drug resistance. The model has a disease free equilibrium (DFE), which is locally asymptotically stable, whenever the maximum of the associated reproduction numbers of the model (denoted by Rc) is less than unity. Furthermore, the model undergoes a backward bifurcation, where the disease-free equilibrium co-exists with a stable endemic equilibrium. Numerical simulations, using epidemiological and demographic data relevant to Nigeria obtained from WHO and USAID [35,36,38], shows that provided the rate at which the undetected individuals with active TB recovered exceeded a critical values, then DOTS, the STOP TB initiative programme of WHO can lead to effective elimination of TB in Nigeria. This suggest that the detection rate plays significant role in the elimination of TB. Furthermore, it is shown that if the progress or rate of individuals who are susceptible to TB is low, it can also lead to elimination of the disease in Nigeria. The results also shows that if the effective contact rate (  ) for TB infection remains below certain critical value (0.187), the disease can be eliminated.

Vector preference and host defense against infection interact to determine disease dynamics

Vector preference based on host infection status has long been recognized for its importance in disease dynamics. Prior theoretical work has assumed that all hosts are uniformly susceptible to the pathogen. Here we investigated disease dynamics when this assumption is relaxed using a series of vector–host epidemiological compartment models with variable levels of host resistance or tolerance to infection – collectively termed defense. In our models, vectors cannot acquire the infection from resistant hosts but can acquire from tolerant hosts. Specifically, we investigated the interacting effects of vector preference and host defense in a series of single‐ and two‐patch models. Results indicate that resistant host types generally reduce disease prevalence and pathogen spillover, independent of vector preference. The epidemiological consequences of host tolerance, however, depended on vector preference. When vectors preferred diseased hosts, tolerance reduced incidence compared to susceptible hosts; when vectors avoided diseased hosts, tolerance enhanced disease prevalence. Finally, a variation of the model that included preference‐based vector patch leaving rates suggests that both resistance and tolerance can promote pathogen spillover if vectors prefer diseased hosts, because of increased vector dispersal into susceptible patches. Collectively, we found complex, context‐dependent effects of vector preference and host defense on disease dynamics. In the context of management programs for vector‐borne diseases, managers should consider both the precise form of host defense present in a population, breed, or cultivar, as well as vector feeding behavior.

Tau-leaped Particle Learning

Development of effective policy interventions to stem disease outbreaks requires knowledge of the current state of affairs, e.g. how many individuals are currently infected, a strain's virulence, etc, as well as our uncertainty of these values. A Bayesian inferential approach provides this information, but at a computational expense. We develop a sequential Bayesian approach based on an epidemiological compartment model and noisy count observations of the transitions between compartments.

On the infinitesimal dispersion of multivariate Markov counting systems

We provide a multivariate extension of a recent result for univariate Markov counting processes: necessity and sufficiency of compoundness for infinitesimal over-dispersion. As an illustration, we show that infinitesimally over-dispersed epidemiological SIR-type compartment models must rely on compound counting processes.

Mathematical models of malaria--a review.

Mathematical models have been used to provide an explicit framework for understanding malaria transmission dynamics in human population for over 100 years. With the disease still thriving and threatening to be a major source of death and disability due to changed environmental and socio-economic conditions, it is necessary to make a critical assessment of the existing models, and study their evolution and efficacy in describing the host-parasite biology. In this article, starting from the basic Ross model, the key mathematical models and their underlying features, based on their specific contributions in the understanding of spread and transmission of malaria have been discussed. The first aim of this article is to develop, starting from the basic models, a hierarchical structure of a range of deterministic models of different levels of complexity. The second objective is to elaborate, using some of the representative mathematical models, the evolution of modelling strategies to describe malaria incidence by including the critical features of host-vector-parasite interactions. Emphasis is more on the evolution of the deterministic differential equation based epidemiological compartment models with a brief discussion on data based statistical models. In this comprehensive survey, the approach has been to summarize the modelling activity in this area so that it helps reach a wider range of researchers working on epidemiology, transmission, and other aspects of malaria. This may facilitate the mathematicians to further develop suitable models in this direction relevant to the present scenario, and help the biologists and public health personnel to adopt better understanding of the modelling strategies to control the disease

Actuarial Applications of Epidemiological Models

The risk of a global avian flu or influenza A (H1 N1) pandemic and the emergence of the worldwide SARS epidemic in 2002–2003 have led to a revived interest in the study of infectious diseases. Mathematical models have become important tools in analyzing the transmission dynamics and in measuring the effectiveness of controlling strategies. Research on infectious diseases in the actuarial literature goes only so far in setting up epidemiological models that better reflect the transmission dynamics. This paper attempts to build a bridge between epidemiological and actuarial modeling and set up an actuarial model that provides financial arrangements to cover the expenses resulting from the medical treatments of infectious diseases. Based on classical epidemiological compartment models, the first part of this paper proposes insurance policies and models to quantify the risk of infection and formulates financial arrangements, between an insurer and insureds, using actuarial methodology. For practical purposes, the second part employs a variety of numerical methods to calculate premiums and reserves. The last part illustrates the methods by designing insurance products for two well-known epidemics: the Great Plague in England and the SARS epidemic in Hong Kong.

Effect of cross-immunity on the transmission dynamics of two strains of dengue

A deterministic model for the transmission dynamics of two strains of dengue disease is presented. The model, consisting of mutually exclusive epidemiological compartments representing the human and vector dynamics, has a locally asymptotically stable, disease-free equilibrium whenever the maximum of the associated reproduction numbers of the two strains is less than unity. The model can have infinitely many co-existence equilibria if infection with one strain confers complete cross-immunity against the other strain and the associated reproduction number of each strain exceeds unity. On the other hand, if infection with one strain confers partial immunity against the other strain, disease elimination, competitive exclusion or co-existence of the two strains can occur. The effect of seasonality on dengue transmission dynamics is explored using numerical simulations, where it is shown that the oscillation pattern differs between the strains, depending on the degree of the cross-immunity between the strains.

Discovering network behind infectious disease outbreak

Stochasticity and spatial heterogeneity are of great interest recently in studying the spread of an infectious disease. The presented method solves an inverse problem to discover the effectively decisive topology of a heterogeneous network and reveal the transmission parameters which govern the stochastic spreads over the network from a dataset on an infectious disease outbreak in the early growth phase. Populations in a combination of epidemiological compartment models and a meta-population network model are described by stochastic differential equations. Probability density functions are derived from the equations and used for the maximal likelihood estimation of the topology and parameters. The method is tested with computationally synthesized datasets and the WHO dataset on the SARS outbreak.

Threshold behaviour of a SIR epidemic model with age structure and immigration

We consider a SIR age-structured model with immigration of infectives in all epidemiological compartments; the population is assumed to be in demographic equilibrium between below-replacement fertility and immigration; the spread of the infection occurs through a general age-dependent kernel. We analyse the equations for steady states; because of immigration of infectives a steady state with a positive density of infectives always exists; however, a quasi-threshold theorem is proved, in the sense that, below the threshold, the density of infectives is close to 0, while it is away from 0, above the threshold; furthermore, conditions that guarantee uniqueness of steady states are obtained. Finally, we present some numerical examples, inspired by the Italian demographic situation, that illustrate the threshold-like behaviour, and other features of the stationary solutions and of the transient.

Modeling malaria vaccines I: New uses for old ideas

Starting from a modification of the model of malaria transmission developed for the Garki project, this paper develops a model containing variables relevant to the simulation of malaria vaccination programs. Modifications include (1) integration of maintenance of immunity dependent on boosting and the possibility of loss immunity; (2) introduction of a boosting factor distinct from susceptibility to infection; (3) reinterpretation of the epidemiological compartments of positive immunes and nonimmunes in terms of severity of disease rather than just infection; (4) interpretation of the different stage-specific levels of immunity; (5) discrimination between different susceptibilities for the immune and nonimmune classes; (6) reformulation of the expression for acquisition of immunity to be biologically more acceptable. Simulations using the Garki model, Nedelman's modification of it, and our Basic model compare the similarities and differences in the predictive behavior of the models. Simulations using the Basic model reproduce observed periodic fluctuations of malaria attributed to the interplay of transmission-blocking immunity and loss of immunity in the absence of boosting in areas of unstable malaria transmission.

The structural simplification of an epidemiological compartment model.

It is shown how a multicompartmental infectious disease model can be systematically examined for reduction of structural complexity. For steadystate situations, four basic rules are proposed for eliminating components of flow-lines, whole flow-lines, and compartments, plus combining compartments. An application to a typhoid fever model allows calculations to be done on a pocket calculator. The approach could be particularly important in developing countries.