Mathematical Epidemiology

We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccineage, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

The weekly number of dengue cases in Peru, South America, stratified by province for the period 1994-2006 were analysed in conjunction with associated demographic, geographic and climatological data. Estimates of the reproduction number, moderately correlated with population size (Spearman r=0. 28, P=0. 03), had a median of 1. 76 (IQR 0. 83-4. 46). The distributions of dengue attack rates and epidemic durations follow power-law (Pareto) distributions (coefficient of determination >85%, P<0. 004). Spatial heterogeneity of attack rates was highest in coastal areas followed by mountain and jungle areas. Our findings suggest a hierarchy of transmission events during the large 2000-2001 epidemic from large to small population areas when serotypes DEN-3 and DEN-4 were first identified (Spearman r=x0. 43, P=0. 03). The need for spatial and temporal dengue epidemic data with a high degree of resolution not only increases our understanding of the dynamics of dengue but will also generate new hypotheses and provide a platform for testing innovative control policies.

In our global world, the increasing complexity of social relations and transport infrastructures are key factors in the spread of epidemics. In recent years, the increasing availability of computer power has enabled both to obtain reliable data allowing one to quantify the complexity of the networks on which epidemics may propagate and to envision computational tools able to tackle the analysis of such propagation phenomena. These advances have put in evidence the limits of homogeneous assumptions and simple spatial diffusion approaches, and stimulated the inclusion of complex features and heterogeneities relevant in the description of epidemic diffusion. In this paper, we review recent progresses that integrate complex systems and networks analysis with epidemic modelling and focus on the impact of the various complex features of real systems on the dynamics of epidemic spreading. To cite this article: V. Colizza et al., C. R. Biologies 330 (2007).

The conjecture of Arino and van den Driessche (2003) that a SIS type model in a mover-stayer epidemic model is globally asymptotically stable is confirmed analytically. If the basic reproduction number R0 <= 1, then the disease-free equilibrium is globally asymptotically stable. If R0 > 1, then there exists a unique endemic equilibrium that is globally asymptotically stable on the nonnegative orthant minus the stable manifold of the disease-free equilibrium.

This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a diseasefree equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is shown, using linear and nonlinear Lyapunov functions, coupled with the LaSalle Invariance Principle, that this equilibrium is globally-asymptotically stable for a special case. The model has a resistant strain-only equilibrium, which is globally-asymptotically stable whenever its reproduction number is greater than unity and exceeds that of the wild strain. In this case, the two strains undergo competitive exclusion, where the strain with the higher reproduction number displaces the other. Further, for the case when treatment does not lead to resistance development, the model can have no coexistence equilibrium or a continuum of coexistence equilibria. When treatment leads to resistance development, the model can have a unique coexistence equilibrium or a resistant-only equilibrium. This coexistence equilibrium is shown to be locally-asymptotically stable, using a technique based on Krasnoselskii sub-linearity argument. Numerical simulations of the model show that for high treatment rates, the resistant strain can dominate, and drive out, the wild strain. Finally, when the two strains co-exist, the proportion of individuals with the resistant strain at steady-state decreases with increasing rate of resistance development.

The computational epidemiology is the development and use of computational models that aims to understand the proliferation of diseases of the dynamic point of view. The computational models are capable to simulate the behavior of an epidemic and its effects on the population of a region and develop strategies of control and prevention of diseases. The computational epidemiology proposes models that classify individuals as susceptible, infected and recovered (SIR) and based on individual (IBM). The SIR model considers the homogeneous distribution of individuals in space and time and it is not capable to tell the persistence or eradication of infectious diseases. The IBM reproduces the premises of the SIR model to the individual analyse. The model computational cost to simulate the behavior of an epidemic grows up with the size of the population considered. This work implements a IBM using parallel programming through the use of GPU’s (Graphics Processor Unit) compatible with CUDA (Compute Unified Device Architecture). The goal is to gain performance compared to the traditional implementation. The results show that in some cases the reduction of computational time is around 170% and the growth rate with the increase of the population size is, in some cases, two times lower in a parallel implementation when compared to a sequencial implementation.

A mathematical model of malaria dynamics with naturally acquired transient immunity in the presence of protected travellers is presented. The qualitative analysis carried out on the autonomous model reveals the existence of backward bifurcation, where the locally asymptotically stable malaria-free and malaria-present equilibria coexist as the basic reproduction number crosses unity. The increased fraction of protected travellers is shown to reduce the basic reproduction number significantly. Particularly, optimal control theory is used to analyse the non-autonomous model, which incorporates four control variables. The existence result for the optimal control quadruple, which minimizes malaria infection and costs of implementation, is explicitly proved. Effects of combining at least any three of the control variables on the malaria dynamics are illustrated. Furthermore, the cost-effectiveness analysis is carried out to reveal the most cost-effective strategy that could be implemented to prevent and control the spread of malaria with limited resources.

A mathematical model of malaria dynamics with naturally acquired transient immunity in the presence of protected travellers is presented. The qualitative analysis carried out on the autonomous model reveals the existence of backward bifurcation, where the locally asymptotically stable malaria-free and malaria-present equi-libria coexist as the basic reproduction number crosses unity. The increased fraction of protected travellers is shown to reduce the basic reproduction number significantly. Particularly, optimal control theory is used to analyse the non-autonomous model, which incorporates four control variables. The existence result for the optimal control quadruple, which minimizes malaria infection and costs of implementation, is explicitly proved. Effects of combining at least any three of the control variables on the malaria dynamics are illustrated. Furthermore, the cost-effectiveness analysis is carried out to reveal the most cost-effective strategy that could be implemented to prevent and control the spread of malaria with limited resources.

Las enfermedades anemia falciforme y malaria han sido ampliamente estudiadas y modeladas por aparte. Sin embargo, en este estudio se pretende estudiar la estrecha relación que tiene la enfermedad  genética con la infecciosa. La malaria, es una enfermedad causada por los mosquitos hembra del género Anopheles portadores del parásito Plasmodium. Por otra parte, la anemia falciforme es una enfermedad genética donde los portadores tienen glóbulos rojos  defectuosos con una hemoglobina anormal. La relación que ambas tienen,  es que la malformación de los eritrocitos en los portadores de anemia impiden el óptimo desarrollo del parásito, por lo cual están relativamente protegidos de la muerte por malaria. Se desarrolló un modelo compartimental SEIRS de ecuaciones diferenciales para la malaria en conjunto con la anemia falciforme (2 clases: portadores y no portadores), donde la población de mosquitos fuera proporcional a la población humana, similar a lo que ocurre en el territorio subsahariano del continente africano. Después de realizar el análisis para el punto de equilibrio libre de enfermedad (DFE), se estableció que el número reproductivo básico dependía proporcionalmente del aporte de cada población de humanos, los portadores y los no portadores. Así, finalmente se pudo concluir que las dinámicas de la enfermedad infecciosa, se ven altamente afectadas con una alta presencia de la enfermedad genética, pues la última presenta un impedimento para  el óptimo desarrollo del parásito en el cuerpo del huésped.

"""We present a review of models of swine flu A H1N1. We discuss how control of epidemic critically depends on the value of the Basic Reproduction Number (R_0). The R_0 for new influenza estimated 1.5. By means of suitable Volterra-type Lyapunov functions, we establish the global stability of the steady state of an disease transmission models with control measures type SIR. We conclude that R_0 plays an important role in the designing disease control strategies.

Keywords: SIR epidemic model; influenza; basic reproduction number; control measures; global stability; direct Lyapunov method."""

A transmission model for dengue fever is discussed here. Restricting the dynamics for the constant host and vector populations, the model is reduced to a two-dimensional planar equation. In this model the endemic state is stable if the basic reproductive number of the disease is greater than one. A trapping region containing the heteroclinic orbit connecting the origin (as a saddle point) and the endemic fixed point occurs. By the use of the heteroclinic orbit, we estimate the time needed for an initial condition to reach a certain number of  infectives. This estimate is shown to agree with the numerical results computed directly from the dynamics of the populations.

For a class of multigroup SIR epidemic models with varying subpopulation sizes, we establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasible region. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory.

This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a diseasefree equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is shown, using linear and nonlinear Lyapunov functions, coupled with the LaSalle Invariance Principle, that this equilibrium is globally-asymptotically stable for a special case. The model has a resistant strain-only equilibrium, which is globally-asymptotically stable whenever its reproduction number is greater than unity and exceeds that of the wild strain. In this case, the two strains undergo competitive exclusion, where the strain with the higher reproduction number displaces the other. Further, for the case when treatment does not lead to resistance development, the model can have no coexistence equilibrium or a continuum of coexistence equilibria. When treatment leads to resistance development, the model can have a unique coexistence equilibrium or a resistant-only equilibrium. This coexistence equilibrium is shown to be locally-asymptotically stable, using a technique based on Krasnoselskii sub-linearity argument. Numerical simulations of the model show that for high treatment rates, the resistant strain can dominate, and drive out, the wild strain. Finally, when the two strains co-exist, the proportion of individuals with the resistant strain at steady-state decreases with increasing rate of resistance development.

In this paper, we proposed two deterministic models; MSIR and MSEIR Tuberculosis models. The dynamics of the compartments were described by a system of ODEs and existence & uniqueness theorem was formulated. Theorem on existence and uniqueness of solution was also proved

We formulate a susceptible-vaccinated-infected-recovered (SVIR) model by incorporating the vaccination of newborns, vaccine-age, and mortality induced by the disease into the SIR epidemic model. It is assumed that the period of immunity induced by vaccines varies depending on the vaccine-age. Using the direct Lyapunov method with Volterra-type Lyapunov function, we show the global asymptotic stability of the infection-free and endemic steady states.

In this paper an eco-epidemiological model incorporating a prey refuge and prey harvesting with disease in the prey-population is considered. Predators are assumed to consume both the susceptible and infected prey at different rates. The positivity and boundedness of the solution of the system are discussed. The existence and stability of the biologically feasible equilibrium points are investigated. Numerical simulations are performed to support our analytical findings.

We investigate an SIR compartmental epidemic model in a patchy environment where individuals in each compartment can travel among n patches. We derive the basic reproduction number R 0 and prove that, if R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable. In the case of R 0 > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable.

El objetivo de este trabajo es estudiar la reducción de un modelo epidemiológico simple SI planteado originalmente en ecuaciones diferenciales parciales a otro modelo en ecuaciones diferenciales con retardo. Originalmente, la población es dividida en jóvenes y adultos. Se asume que sólo los adultos son sexualmente activos y que esposible que adultos infectados tengan bebes sanos. La estabilidad global del modelo SI planteado en Ecuaciones con Retardo es analizado en López-Cruz [5].

Protecting children from vaccine-preventable diseases, such as measles, is among primary goals of health administrators worldwide. Since vaccination turned out to be the most effective strategy against childhood diseases, developing a framework that would predict an optimal vaccine coverage level needed to control the spread of these diseases is crucial. In this article, we use a compartmental mathematical model of the dynamics of measles spread within a population with variable size to provide this framework.

The UN [United Nations Office on Drugs and Crime (UNODC): World Drug Report, 2005, vol. 1: Analysis. UNODC, 2005.], EU [European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Annual Report, 2005. http://annualreport.emcdda.eu.int/en/home-en.html.] and WHO [World Health Organisation (WHO): Biregional Strategy for Harm Reduction, 2005-2009. HIV and Injecting Drug Use. WHO, 2005.] have consistently highlighted in recent years the ongoing and persistent nature of opiate and particularly heroin use on a global scale. While this is a global phenomenon, authors have emphasised the significant impact such an epidemic has on individual lives and on society. National prevalence studies have indicated the scale of the problem, but the drug-using career, typically consisting of initiation, habitual use, a treatment-relapse cycle and eventual recovery, is not well understood. This paper presents one of the first ODE models of opiate addiction, based on the principles of mathematical epidemiology. The aim of this model is to identify parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness. An epidemic threshold value, R 0 , is proposed for the drug-using career. Sensitivity analysis is performed on R 0 and it is then used to examine the stability of the system. A condition under which a backward bifurcation may exist is found, as are conditions that permit the existence of one or more endemic equilibria. A key result arising from this model is that prevention is indeed better than cure.

For a class of multigroup SIR epidemic models with varying subpopulation sizes, we establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasible region. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory.

In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and a discrete intracellular delay: the delay corresponds to the time between infection of a infected target hepatocytes and production of new HCV particles. We establish the global stability of the infection-free equilibrium and existence, uniqueness, local and global stabilities of the infected equilibrium, also we establish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. The unique infected equilibrium is globally-asymptotically stable for a special case, where the hepatotropic virus is non-cytopathic. We present a sensitivity analysis for the basic reproductive number. Numerical simulations are carried out to illustrate the analytical results.

We study the spread of disease in an SIS model for a structured population. The model considered is a time-varying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a threshold parameter for this model in the spirit of the basic reproduction number for time-invariant models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model.

Protecting children from vaccine-preventable diseases, such as measles, is among primary goals of health administrators worldwide. Since vaccination turned out to be the most effective strategy against childhood diseases, developing a framework that would predict an optimal vaccine coverage level needed to control the spread of these diseases is crucial. In this article, we use a compartmental mathematical model of the dynamics of measles spread within a population with variable size to provide this framework.

"""In this paper, we study the global properties of classic SIS epidemic model with constant recruitment, disease-induced death and standard incidence term. We apply the Poincaré-Bendixson theorem, Dulac’s criterion, and the method of Lyapunov function to establish conditions for global stability. For this system, three Dulac functions and two Lyapunov functions are constructed for the endemic steady state.

Keywords. SIS epidemic model, Standard incidence term, Global stability, Direct Lyapunov method, Dulac’s criterion, Poincaré-Bendixson theorem."""

For a class of multigroup SIR epidemic models with varying subpopulation sizes, we establish that the global dynamics are completely determined by the basic reproduction number R 0. More specifically, we prove that, if R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasible region. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory.

The SEIR model with nonlinear incidence rates in epidemiology is studied. Global stability of the endemic equilibrium is proved using a general criterion for the orbital stability of periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations.

a b s t r a c t SIR and SIS epidemic models with information-related changes in contact patterns are introduced. The global stability analysis of the endemic equilibrium is performed by means of the Li-Muldowney geometric approach. Biological implications of the stability conditions are given.

Mathematical models for disease spread are mostly based on differential equations with an in-built threshold that determines the behaviour of the system. This study investigates student-lecturer-sex (SELEX) on campus by an epidemiological model. The dynamics of these activities among female students and male lecturers are analyzed by the usual Susceptible-Infected-Recovered (SIR) model. The model suggests that, admitting and recruitment of new members play a significant role in reduction of SELEX on campus. It was revealed that, the basic reproductive numbers are not enough to predict whether or not SELEX will persist on university campus, but minimizing the admitting and recruitment of infected students and lecturers reduces the spread of the disease-SELEX.

A transmission model for dengue fever is discussed here. Restricting the dynamics for the constant host and vector populations, the model is reduced to a two-dimensional planar equation. In this model the endemic state is stable if the basic reproductive number of the disease is greater than one. A trapping region containing the heteroclinic orbit connecting the origin (as a saddle point) and the endemic fixed point occurs. By the use of the heteroclinic orbit, we estimate the time needed for an initial condition to reach a certain number of infectives. This estimate is shown to agree with the numerical results computed directly from the dynamics of the populations.

This paper is devoted to present and study a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, that consists in examining each individual one has been able to identify on the basis of the information collected within a certain time period, is expected to reinforce efficiently the standard random-screening based detection and slack considerably the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size roughly speaking, becomes large. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relation of the model to the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.

The article deals with a system of nonlinear differential equations of tumor growth cancer model under the influence of white noise. This system can be used as mathematical tools for analyzing of various real problems of tumor growth associated with cancer. Necessary and sufficient conditions for the asymptotic mean square stability of the zero solution of this system are derived in this article. The article introduces a new approach to studying such problems through construction of a suitable deterministic system with the use of Lyapunov function. Recently we observed that cellmediated immunity plays a vital role in immune responses against cancer. Cancer cell development and survival is a multifactor process involving genetic mutation of normal cells with physiological changes within both cancer cells and the body&#39;s defence mechanisms. In this paper we considered the special impact of tumor-immune interaction along with the two immune components– resting (helper) T-cells which...

In this paper, a mathematical model which considers population dynamics among infected and uninfected cancer tumor cells has been proposed. Delay differential equations have been utilized to demonstrate the framework to consider the periods of the cell cycle. We examine the steadiness of the framework and demonstrate a hypothesis dependent on the contention standard to decide the dependability of a fixed point and show that the solidness may rely upon the delay. We show hypothetically as well as through numerical results that periodic oscillations may arise through Hopf bifurcations. In this paper we study a stochastic model for the conduct of malignancy tumors, depicted by a stochastic differential condition with multiplicative noise term.
We study the existence of the solution process, as well as its behavior in the framework of stochastic inclusion problems and long time behavior.

In this paper we discuss a mathematical model for the transmission of Lymphatic Filariasis disease in Jati Sampurna, West Java Indonesia. The model assumes that acute infected humans are infectious and treatment is given to a certain number of acute infected humans found from screening process. The treated acute individuals are assumed to be remain susceptible to the disease. The model is analyzed and it is found a condition for the existence and stability of the endemic equilibrium. A well known rule of thumb in epidemiological model, that is, the endemic equilibrium exists and stable if the basic reproduction number is greater than one, is shown. Moreover, it is also shown that if the level of screening n is sufficiently large, current medical treatment strategy will be able to reduce the long-term level of incidences. However, in practice it is not realistic and cannot eliminate the disease, in terms of reducing the basic reproduction number. The reproduction number can be reduced by giving additional treatments, such as reducing the biting rate and mosquito's density. This suggests that there should be a combination of treatment to eliminate the disease.

For a class of multigroup SIR epidemic models with varying subpopulation sizes, we establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1, then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasible region. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory.