Saturation recovery leads to multiple endemic equilibria and backward bifurcation

The number of patients need to be treated may exceed the carry capacity of local hospitals during the spreading of a severe infectious disease. We propose an epidemic model with saturation recovery from infective individuals to understand the effect of limited resources for treatment of infectives on the emergency disease control. It is shown that saturation recovery from infective individuals leads to vital dynamics, such as bistability and periodicity, when the basic reproduction number R₀ is less than unity. An interesting dynamical behavior of the model is a backward bifurcation which raises many new challenges to effective infection control.     

Reproduction numbers for infections with free-living pathogens growing in the environment

The basic reproduction number ?₀ for a compartmental disease model is often calculated by the next generation matrix (NGM) approach. When the interactions within and between disease compartments are interpreted differently, the NGM approach may lead to different ?₀ expressions. This is demonstrated by considering a susceptible–infectious–recovered–susceptible model with free-living pathogen (FLP) growing in the environment. Although the environment could play different roles in the disease transmission process, leading to different ?₀ expressions, there is a unique type reproduction number when control strategies are applied to the host population. All ?₀ expressions agree on the threshold value 1 and preserve their order of magnitude. However, using data for salmonellosis and cholera, it is shown that the estimated ?₀ values are substantially different. This study highlights the utility and limitations of reproduction numbers to accurately quantify the effects of control strategies for infections with FLPs growing in the environment.     

A two-age-classes dengue transmission model

In this paper, we discuss a two-age-classes dengue transmission model with vaccination. The reason to divide the human population into two age classes is for practical purpose, as vaccination is usually concentrated in one age class. We assume that a constant rate of individuals in the child-class is vaccinated. We analyze a threshold number which is equivalent to the basic reproduction number. If there is an undeliberate vaccination to infectious children, which worsens their condition as the time span of being infectious increases, then paradoxically, vaccination can be counter productive. The paradox, stating that vaccination makes the basic reproduction number even bigger, can occur if the worsening effect is greater than a certain threshold, a function of the human demographic and epidemiological parameters, which is independent of the level of vaccination. However, if the worsening effect is to increase virulence so that one will develop symptoms, then the vaccination is always productive. In both situations, screening should take place before vaccination. In general, the presence of class division has obscured the known rule of thumb for vaccination.     

Modeling the transmission dynamics and control of hepatitis B virus in China

Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV) and is a major global health problem. HBV is the most common serious viral infection and a leading cause of death in mainland China. Around 130 million people in China are carriers of HBV, almost a third of the people infected with HBV worldwide and about 10% of the general population in the country; among them 30 million are chronically infected. Every year, 300,000 people die from HBV-related diseases in China, accounting for 40-50% of HBV-related deaths worldwide. Despite an effective vaccination program for newborn babies since the 1990s, which has reduced chronic HBV infection in children, the incidence of hepatitis B is still increasing in China. We propose a mathematical model to understand the transmission dynamics and prevalence of HBV infection in China. Based on the data reported by the Ministry of Health of China, the model provides an approximate estimate of the basic reproduction number R₀=2.406. This indicates that hepatitis B is endemic in China and is approaching its equilibrium with the current immunization program and control measures. Although China made a great progress in increasing coverage among infants with hepatitis B vaccine, it has a long and hard battle to fight in order to significantly reduce the incidence and eventually eradicate the virus.     

A qualitative analysis of a model for the transmission of varicella-zoster virus

Varicella-zoster virus (VZV) is a herpesvirus which is the known agent for causing varicella (chickenpox) in its initial manifestation and zoster (shingles) in a reactivated state. The standard SEIR compartmental model is modified to include the cycle of shingles acquisition, recovery, and possible reacquisition. The basic reproduction number R(0) shows the influence of the zoster cycle and how shingles can be important in maintaining VZV in populations. The model has the typical threshold behavior in the sense that when R(0)1, the virus persists over time and so chickenpox and shingles remain endemic.     

A habitat-based model for the spread of hantavirus between reservoir and spillover species

New habitat-based models for spread of hantavirus are developed which account for interspecies interaction. Existing habitat-based models do not consider interspecies pathogen transmission, a primary route for emergence of new infectious diseases and reservoirs in wildlife and man. The modeling of interspecies transmission has the potential to provide more accurate predictions of disease persistence and emergence dynamics. The new models are motivated by our recent work on hantavirus in rodent communities in Paraguay. Our Paraguayan data illustrate the spatial and temporal overlaps among rodent species, one of which is the reservoir species for Jabora virus and others which are spillover species. Disease transmission occurs when their habitats overlap. Two mathematical models, a system of ordinary differential equations (ODE) and a continuous-time Markov chain (CTMC) model, are developed for spread of hantavirus between a reservoir and a spillover species. Analysis of a special case of the ODE model provides an explicit expression for the basic reproduction number, , such that if , then the pathogen does not persist in either population but if , pathogen outbreaks or persistence may occur. Numerical simulations of the CTMC model display sporadic disease incidence, a new behavior of our habitat-based model, not present in other models, but which is a prominent feature of the seroprevalence data from Paraguay. Environmental changes that result in greater habitat overlap result in more encounters among various species that may lead to pathogen outbreaks and pathogen establishment in a new host.     

The role of screening and treatment in the transmission dynamics of HIV/AIDS and tuberculosis co-infection: a mathematical study

In this paper, we present a deterministic non-linear mathematical model for the transmission dynamics of HIV and TB co-infection and analyze it in the presence of screening and treatment. The equilibria of the model are computed and stability of these equilibria is discussed. The basic reproduction numbers corresponding to both HIV and TB are found and we show that the disease-free equilibrium is stable only when the basic reproduction numbers for both the diseases are less than one. When both the reproduction numbers are greater than one, the co-infection equilibrium point may exist. The co-infection equilibrium is found to be locally stable whenever it exists. The TB-only and HIV-only equilibria are locally asymptotically stable under some restriction on parameters. We present numerical simulation results to support the analytical findings. We observe that screening with proper counseling of HIV infectives results in a significant reduction of the number of individuals progressing to HIV. Additionally, the screening of TB reduces the infection prevalence of TB disease. The results reported in this paper clearly indicate that proper screening and counseling can check the spread of HIV and TB diseases and effective control strategies can be formulated around ‘screening with proper counseling’.     

A model for influenza with vaccination and antiviral treatment

Compartmental models for influenza that include control by vaccination and antiviral treatment are formulated. Analytic expressions for the basic reproduction number, control reproduction number and the final size of the epidemic are derived for this general class of disease transmission models. Sensitivity and uncertainty analyses of the dependence of the control reproduction number on the parameters of the model give a comparison of the various intervention strategies. Numerical computations of the deterministic models are compared with those of recent stochastic simulation influenza models. Predictions of the deterministic compartmental models are in general agreement with those of the stochastic simulation models.     

The effect of transmission route on plant virus epidemic development and disease control

A model for indirect vector transmission and epidemic development of plant viruses is extended to consider direct transmission through vector mating. A basic reproduction number is derived which is the sum of the R₀ values specific for three transmission routes. We analyse the model to determine the effect of direct transmission on plant disease control directed against indirect transmission. Increasing the rate of horizontal sexual transmission means that vector control rate or indirect transmission rate must be increased/decreased substantially to maintain R₀ at a value less than 1. By contrast, proportionately increasing the probability of transovarial transmission has little effect. Expressions are derived for the steady-state values of the viruliferous vector population. There is clear advantage for an insect virus in indirect transmission to plants, especially where the sexual and transovarial transmission rates are low; however information on virulence-transmissibility relationships is required to explain the evolution of a plant virus from an insect virus.     

Approximation of the Basic Reproduction Number <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> for Vector-Borne Diseases with a Periodic Vector Population

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R ₀ of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p ₀ (1+ε cos (ωt − φ)) with ε ≪ 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p ₀. The maximum correction due to the second term is (ε²/8)% and always tends to decrease R ₀. The basic reproduction number R ₀ is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R ₀ are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality. MSC 92D30 ⋅ 45C05 ⋅ 47A55     

Qualitative analysis of models with different treatment protocols to prevent antibiotic resistance

This paper is concerned with the qualitative analysis of two models [S. Bonhoeffer, M. Lipsitch, B.R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Natl. Acad. Sci. USA 94 (1997) 12106] for different treatment protocols to prevent antibiotic resistance. Detailed qualitative analysis about the local or global stability of the equilibria of both models is carried out in term of the basic reproduction number R₀. For the model with a single antibiotic therapy, we show that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable; if R₀ > 1, then the disease-endemic equilibrium is globally asymptotically stable. For the model with multiple antibiotic therapies, stabilities of various equilibria are analyzed and combining treatment is shown better than cycling treatment. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.     

The control of vector-borne disease epidemics

The theoretical underpinning of our struggle with vector-borne disease, and still our strongest tool, remains the basic reproduction number, R₀, the measure of long term endemicity. Despite its widespread application, R₀ does not address the dynamics of epidemics in a model that has an endemic equilibrium. We use the concept of reactivity to derive a threshold index for epidemicity, E₀, which gives the maximum number of new infections produced by an infective individual at a disease free equilibrium. This index describes the transitory behavior of disease following a temporary perturbation in prevalence. We demonstrate that if the threshold for epidemicity is surpassed, then an epidemic peak can occur, that is, prevalence can increase further, even when the disease is not endemic and so dies out. The relative influence of parameters on E₀ and R₀ may differ and lead to different strategies for control. We apply this new threshold index for epidemicity to models of vector-borne disease because these models have a long history of mathematical analysis and application. We find that both the transmission efficiency from hosts to vectors and the vector-host ratio may have a stronger effect on epidemicity than endemicity. The duration of the extrinsic incubation period required by the pathogen to transform an infected vector to an infectious vector, however, may have a stronger effect on endemicity than epidemicity. We use the index E₀ to examine how vector behavior affects epidemicity. We find that parasite modified behavior, feeding bias by vectors for infected hosts, and heterogeneous host attractiveness contribute significantly to transitory epidemics. We anticipate that the epidemicity index will lead to a reevaluation of control strategies for vector-borne disease and be applicable to other disease transmission models.     

The dilution effect of the domestic animal population on the transmission of P. vivax malaria

The diversion of disease carrying insect from humans to animals may reduce transmission of diseases such as malaria. The use of animals to mitigate mosquito bites on human is called ‘zooprophylaxis’. We introduce a mathematical model for Plasmodium vivax malaria transmission with two bloodmeal hosts (humans and domestic animals) to study the effect of zooprophylaxis. After computing the basic reproduction number from the proposed model, we explore how perturbations in the parameters, sensitive to the effects of control measures, affect its value. Zooprophylaxis is shown to determine whether a basic reproduction becomes bigger than an outbreak threshold value or not. Sensitivity analysis shows that increasing the relative animal population size works better in P. vivax malaria control than decreasing the mosquito population when the relative animal population size is larger than a threshold value.     

A Comparison of Continuous and Discrete-time West Nile Virus Models

The first recorded North American epidemic of West Nile virus was detected in New York state in 1999, and since then the virus has spread and become established in much of North America. Mathematical models for this vector-transmitted disease with cross-infection between mosquitoes and birds have recently been formulated with the aim of predicting disease dynamics and evaluating possible control methods. We consider discrete and continuous time versions of the West Nile virus models proposed by Wonham et al. [Proc. R. Soc. Lond. B 271:501–507, 2004] and by Thomas and Urena [Math. Comput. Modell. 34:771–781, 2001], and evaluate the basic reproduction number as the spectral radius of the next-generation matrix in each case. The assumptions on mosquito-feeding efficiency are crucial for the basic reproduction number calculation. Differing assumptions lead to the conclusion from one model [Wonham, M.J. et al., [Proc. R. Soc. Lond. B] 271:501–507, 2004] that a reduction in bird density would exacerbate the epidemic, while the other model [Thomas, D.M., Urena, B., Math. Comput. Modell. 34:771–781, 2001] predicts the opposite: a reduction in bird density would help control the epidemic.     

Basic Reproduction Number for HIV Model Incorporating Commercial Sex and Behavior Change

The basic reproduction number is obtained for an HIV epidemic model incorporating direct and indirect commercial sex as well as behavior change by the female commercial sex workers (CSWs) and their male customers in response to the proliferation of the disease in the community. A recent result by van den Driessche P., and Watmough J. (Math. Biosci. 180:29–48, 2002) is utilized to compute the threshold parameters for the local asymptotic stability of the Disease-Free Equilibrium (DFE), by considering the transfers in and out of the infective classes. Numerical examples are used to describe the uniqueness and global properties of the endemic equilibrium when DFE is unstable. Biological interpretation of the results obtained in this work is discussed, as are the implications of our results for the design of public health policies such as targeting strategy to target intervention and control measures toward specific high-risk population groups in order to reduce infections. We show that targeting any one sector of the commercial sex alone for prevention will be difficult to have a decided effect on eradicating the epidemic. However, if the aim of the targeted intervention policy is not eradication of the epidemic but decrease in HIV incidence of a particular high-risk group, then concentrated targeting strategy could be sufficient, if properly implemented. This work also demonstrates the usefulness of the theorem of van den Driessche and Watmough (Math. Biosci. 180:29–48, 2002) in obtaining threshold parameters for complicated infectious diseases models.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

具有脉冲出生和脉冲接种的SIR传染病模型

考虑了脉冲出生、脉冲接种、垂直传染、因病死亡等因素,建立了脉冲出生和脉冲接种同时进行的SIR传染病模型,通过分析无病周期解的存在性以及稳定性,得出疾病灭绝的条件.     

Dynamical behavior of a hepatitis B virus transmission model with vaccination

Hepatitis B virus (HBV) infection is a globally health problem. In 2005, the WHO Western Pacific Regional Office set a goal of reducing chronic HBV infection rate to less than 2% among children five years of age by 2012, as an interim milestone towards the final goal of less than 1%. Many countries made some plans (such as free HBV vaccination program for all neonates in China now) to control the transmission HBV. We develop a model to explore the impact of vaccination and other controlling measures of HBV infection. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R₀≤1, and a globally asymptotically stable endemic equilibrium when R₀>1. Numerical simulation results show that the vaccination is a very effective measure to control the infection and they also give some useful comments on controlling the transmission of HBV.