Jia Li

• University of Alabama in Huntsville

We formulate simple differential equation models to study the impact of releases of transgenic sterile mosquitoes carrying a dominant lethal on mosquito control based on the modified sterile insects technique. The early acting bisex, late acting bisex, early acting female-killing, and late acting female-killing lethality strategies are all consider...

We incorporate a viral stimulation delay into a chronic virus infection model and investigate its dynamical behavior by theoretic analysis and numerical simulations. The obtained results show that the viral stimulation delay may result in new specific model phenomena, such as the dependence of the upper bound of the model solutions on the delay, tr...

In this paper, allowing for general transmission and recovery times distributions, we proposed an edge-based age-structured-like compartmental model for STIs (EBACMS) in a coupled network. We considered sexual transmissions between men with also heterosexual contacts. Mathematically, we gave the general approach of proving the nonnegativity of solu...

Sterile insect technique is one of the effective biological measures in mosquitoes control. Since mosquitoes fly around as we release sterile mosquitoes, the mosquito dispersals affect the effectiveness of the releases. To study the impact of mosquito dispersals, we formulate a simple two-patch model where sterile mosquitoes are released in only on...

We develop a delay differential equation model for the interactive wild and sterile mosquitoes. Different from the existing modelling studies, we assume that only those sexually active sterile mosquitoes play a role for the interactive dynamics. We consider the cases where the release amount is either constant or described by a given function of ti...

Cannibalism, as a behavioral trait, is prevalent in many species. To have better understanding of their dynamics, we investigate a structured predator-prey system with predator cannibalism, where the prey population follows the logistic growth in the absence of the predator. We study the effects of the cannibalism attack rate and the corresponding...

In this study, we first formulate a baseline discrete-time mathematical model for malaria transmission where the survival function of mosquitoes is of Beverton–Holt type. We then introduce sterile mosquitoes to the baseline model to explore the transmission dynamics with sterile mosquitoes. We derive formulas for the reproductive number of infectio...

The sterile mosquitoes technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population has been used in preventing the malaria transmission. To study the impact of releasing sterile mosquitoes on the malaria transmission, we first formulate a simple SEIR (susceptible-exposed-infected-recovered) malaria transmi...

To study the interactive dynamics of wild mosquitoes and mosquitoes carrying genetically-modified bacteria, we formulate continuous-time homogeneous and stage-structured models in this study. With appropriate transformations, complete results of the existence and stability of all boundary and positive equilibria for the homogeneous model are establ...

In this paper, a malaria transmission model with sterile mosquitoes is considered. We first formulate a simple SEIR malaria transmission model as our baseline model. Then sterile mosquitoes are introduced into the baseline model. We consider the case that the release rate of sterile mosquitoes is proportional to the wild mosquito population size. T...

In this paper, the correlation coefficients between nodes in states are used as dynamic variables, and we construct SIR epidemic dynamic models with correlation coefficients by using the pair approximation method in static networks and dynamic networks, respectively. Considering the clustering coefficient of the network, we analytically investigate...

In this paper, we formulate discrete-time mathematical models for the interactive wild and sterile mosquitoes. Instead of the Ricker-type of nonlinearity for the survival functions, we assume the Beverton–Holt-type in these models. We consider three different strategies for the releases of sterile mosquitoes and investigate the model dynamics. Thre...

To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals' behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number $\Re_0$ for the model, and show t...

The epidemic characteristics of two classic SIS epidemic models, including the epidemic size, peak and turning point, are investigated. The two SIS models are with bilinear and standard incidences, respectively. For the SIS models, the susceptible individuals generally can be divided into two classes. One consists of the individuals who had not bee...

The technique of sterile mosquitoes plays an important role in the control of mosquito-borne diseases such as malaria, dengue, yellow fever, west Nile, and Zika. To explore the interactive dynamics between the wild and sterile mosquitoes, we formulate a delayed mosquito population suppression model with constant releases of sterile mosquitoes. Thro...

To study the impact of releasing sterile mosquitoes on mosquito-borne disease transmissions, we propose two mathematical models with impulsive releases of sterile mosquitoes. We consider periodic impulsive releases in the first model and obtain the existence, uniqueness, and globally stability of a wild-mosquito-eradication periodic solution. We al...

A non-smooth switched harvest on predators is introduced into a simple predator-prey model with logistical growth of the prey and a bilinear functional response. If the density of the predator is below a switched value, the harvesting rate is linear; otherwise, it is constant. The model links the well studied predator-prey model with constant harve...

To study the impact of the sterile insect technique and effects of the mosquitoes' metamorphic stage structure on the transmission dynamics of mosquito-borne diseases, we formulate stage-structured continuous-time mathematical models, based on systems of differential equations, for the interactive dynamics of the wild and sterile mosquitoes. We inc...

In this paper, we include two time delays in a mathematical model for the CD8+ cytotoxic T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection, where one is the intracellular infection delay and the other is the immune delay to account for a series of immunological events leading to the CTL response. We show th...

To prevent the transmissions of malaria, dengue fever, or other mosquito-borne diseases, one effective weapon is the sterile insect technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population. To study the impact of the sterile insect technique on disease transmission, we formulate discrete-time mathematic...

To prevent the transmissions of malaria, dengue fever, or other mosquito-borne diseases, one of the effective weapons is the sterile insect technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population. To study the impact of the sterile insect technique on disease transmission, we formulate continuous-time...

Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transm...

Vector borne diseases spread rapidly in the population. Hence their control intervention must work quickly and target large area as well. A rational approach to combat these diseases is mobilizing people and making them aware through media campaigns. In the present paper, a non-linear mathematical model is proposed to assess the impact of creating...

We formulate a homogeneous model and a stage-structured model for the interactive wild mosquitoes and mosquitoes carrying genetically-modified bacteria. We establish conditions for the existence and stability of fixed points for both models. We show that a unique positive fixed point exists and is asymptotically stable if the two boundary fixed poi...

We investigate dynamics of mosquito population models under two assumptions, respectively, and then formulate simple discrete-time compartmental susceptible-exposed-infective-recovered models for the malaria transmission based on the mosquito population models. We show that the mosquito population models either have robust dynamics or exhibit perio...

We formulate mosquito-stage-structured, continuous-time, compartmental, malaria models which include four distinct metamorphic stages of mosquitoes. We derive a formula for the reproductive number of infection and investigate the existence of endemic equilibria. We determine conditions under which the models undergo either forward or backward bifur...

Transgenic mosquitoes that are resistant to malaria infection become one of the effective weapons to control malaria transmission. To investigate the impact of releasing transgenic mosquitoes on the malaria transmission, we first formulate mathematical models for interactive wild and transgenic mosquitoes based on systems of differential equations....

A simple SEIR model for malaria transmission dynamics is formulated as our baseline model. The metamorphic stages in the mosquito population are then included and a simple stage-structured mosquito population model is introduced, where the mosquito population is divided into two classes, with all three aquatic stages in one class and all adults in...

We formulate discrete-time stage-structured models, based on systems of difference equations, for mosquito populations. We include the four distinct mosquito metamorphic stages, egg, pupa, larva, and adult, in the models. We derive a formula for the inherent net reproductive number, and investigate existence and stability of fixed points. We also s...

To study the impact of releasing transgenic mosquitoes on malaria transmission, we formulate discrete-time models for interacting wild and transgenic mosquitoes populations, based on systems of difference equations. We start with models including all homozygous and heterozygous mosquitoes. We then consider either dominant or recessive transgenes to...

We formulate and study discrete-time stage-structured models, based on systems of difference equations, for wild and transgenic mosquito populations. We divide the mosquito population into two classes: the larvae class which consists of the first three aquatic stages in a mosquito's lifetime, and the adult class. Due to the intraspecific competitio...

We formulate and study epidemic models with differential susceptibilities and staged-progressions, based on systems of ordinary differential equations, for disease transmission where the susceptibility of susceptible individuals vary and the infective individuals progress the disease gradually through stages with different infectiousness in each st...

This timely book covers the basic concepts of the dynamics of epidemic disease, presenting various kinds of models as well as typical research methods and results. It introduces the latest results in the current literature, especially those obtained by highly rated Chinese scholars. Alot of attention is paid to the qualitative analysis of models, t...

In this paper, we formulate a mathematical model for malaria transmission that includes incubation periods for both infected human hosts and mosquitoes. We assume humans gain partial immunity after infection and divide the infected human population into subgroups based on their infection history. We derive an explicit formula for the reproductive n...

We formulate and study continuous-time models, based on systems of ordinary differential equations, for interacting wild and transgenic mosquito populations. We assume that the mosquito mating rate is either constant, proportional to total mosquito population size, or has a Holling-II-type functional form. The focus is on the model with the Holling...

We present continuous-time models for age-structured populations and disease transmission. We show how to use the method of characteristic lines to analyze the model dynamics and to write an age-structured population model as an integral equation model. We then extend to an agestructured SIR epidemic model. As an example we describe an age-structur...

We formulate infection-age structured susceptible-infective-removed (SIR) models with behavior change or treatment of infections. Individuals change their behavior or have treatment after they are infected. Using infection age as a continuous variable, and dividing infectives into discrete groups with different infection stages, respectively, we fo...

We formulate di®erential susceptibility and di®erential infectivity models for disease transmission in this paper. The susceptibles are divided into n groups based on their susceptibilities, and the infectives are divided into m groups according to their infectivities. Both the standard incidence and the bilinear incidence are considered for di®ere...

We formulate differential susceptibility and differential infectivity models for disease transmission in this paper. The susceptibles are divided into n groups based on their susceptibilities, and the infectives are divided into m groups according to their infectivities. Both the standard incidence and the bilinear incidence are considered for diff...

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We formulate compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models by dividing the susceptible population into multiple subgroups according to the susceptibility of individuals in each group. We analyze the impact of disease-induced mortality in the situations where the number of contacts per individual is either...

We formulate an HIV epidemic model with differential infectivity and staged disease progression to account for variations in viral loads and in the rate of disease progression in infected individuals. The stability of the infection-free equilibrium determines the threshold conditions under which the modeled disease either dies out or persists in th...

Two discrete-time models for interacting populations of wild and genetically altered mosquito are presented, where the genetically altered mosquitoes are grouped into a single population without distinguishing their zygosity. The birth and death rates for both populations are density-dependent, and the mating rates between the mosquitoes are assume...

We formulate epidemiological models for the transmission of a pathogen that can mutate in the host to create a second infectious mutant strain. The models account for mutation rates that depend on how long the host has been infected. We derive explicit formulas for the reproductive number of the epidemic based on the local stability of the infectio...

We present a sexually-transmitted disease (STD) model for two strains of pathogen in a one-sex, heterogeneously-mixing population, where the dynamics are of SIS (susceptible/infected/susceptible) type, and there are two different groups of individuals. We analyze all equilibria for the case where contacts are modeled via proportionate (random) mixi...

We present several differential infectivity (DI) epidemic models under different assumptions. As the number of contacts is assumed to be constant or a linear function of the total population size, either standard or bilinear incidence of infection is resulted. We establish global stability of the infection-free equilibrium and the endemic equilibri...

Mathematical models can help predict the effectiveness of control measures on the spread of HIV and other sexually transmitted diseases (STDs) by reducing the uncertainty in assessing the impact of intervention strategies such as random screening and contact tracing. Even though contact tracing is one of the most effective methods used for controll...

We study two multigroup mathematical models of the spread of HIV. In the differential infectivity model, the infected population is divided into groups according to their infectiousness, and HIV is primarily spread by a small, highly infectious, group of superspreaders. In the staged-progression model, every infected individual goes through a serie...

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We define the mean number of contacts, the mean duration of infection, and the mea...

In this paper we give a rather complete analysis for a two-sex, susceptible-infective-susceptible (SIS) sexually transmitted disease (STD) model with two competing strains, where the females are divided into two different groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the bound...

We consider two lumped age-structured models of juvenile against adult competition, one of which is a compartmental ordinary differential equation (ODE) model and one of which is a system of ODEs with delay derived from the McKendrick partial differential equation (PDE) model. Existence and stability of positive equilibria and existence of oscillat...

Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transm...

Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transm...

We formulate and analyze a two-group, selective-mixing, susceptible-infective-suscep-tible (SIS), sexually transmitted disease (STD) model where the infection-dependent desirability and acceptability in partnership formations are zero at high infection levels. We analyze two strategies to limit the spread of the epidemic by avoiding forming partner...

We propose and analyze a heterogeneous, multigroup, susceptible-infective-susceptible (SIS) sexually transmitted disease (STD) model where the desirability and acceptability in partnership formations are functions of the infected individuals. We derive explicit formulas for the epidemic thresholds, prove the existence and uniqueness of the equilibr...

In multi-group epidemiological models with nonrandom mixing between people in the different groups, often artificial constraints have to be imposed in order to satisfy the balance conditions. We present and analyze simple selective mixing models governed by systems of ordinary differential equations, where the balance conditions are automatically s...

We study the dynamics of sexually transmitted pathogens in a heterosexually active population, where females are divided into two different groups based on their susceptibility to two distinct pathogenic strains. It is assumed that a host cannot be invaded simultaneously by both disease agents and that when symptoms appear--a function of the pathog...

The case of juvenile versus adult competitive effects on adult fertility in an agedependent population dynamics model introduced by Cushing and Li [1] is considered. It is shown that stronger juvenile competition stabilises the model, while low sensitivity of adult fertility to juvenile cohort sizes leads to destabilisation and oscillatory solution...

A heterosexually active population is exposed to two competing strains or two distinct sexually transmitted pathogens. It is assumed that a host cannot be invaded simultaneously by both disease agents and that when symptoms appear, a function of the pathogen or strain virulence, individuals recover. We conclude that in a behaviorally and geneticall...

by Carlos Castillo-Chavez, Wenzhang Huang and Jia Li.

We formulate and analyze pair-formation models for multiple groups with general pairing rates and arbitrary mixing probabilities. Under the assumption of constant recruitment rates and equal average duration of all types of partnerships, we have shown that the dynamics are relatively simple because of the monotonicity properties of the dynamical sy...

Lotka-Volterra ecological systems with discrete time delays are studied, where the information about the biological vital parameters, such as the rates of birth, death, and interaction between species, is incomplete, but their upper and lower bounds are known.By using stability theory of the interval dynamical systems and constructing suitable Liap...

The asymptotic equivalence of certain large scale dynamical systems and their isolated subsystems is studied by utilizing different stability degrees of the isolated subsystems to allow a wide range of perturbations to these subsystems so that the unperturbed system and the perturbed system have the same stability behavior. Sufficient conditions ar...

The age-structure of a population, and the distribution of sexual behavior according to age, are significant factors determining the spread of the AIDS epidemic. The threshold conditions for age-structured models account for life-history information, and thus differ significantly from their age-independent counterparts. We examine the threshold con...

A difference equation model for the dynamics of a semelparous size-structured species consisting of juvenile and adult individuals is derived and studied. The adult population consists of two size classes, a smaller class and a larger more fertile class. Negative feedback occurs through slowed juvenile growth due to increased total population level...

A general class of age-structured models based upon the McKendrick/von Foerster equations are used to study intraspecific competition between juveniles and adults. Criteria for the existence and stability of equilibria are obtained and the dependence of equilibrium stability (i.e. equilibrium resilience) on competition coefficients is analyzed for...

A predator-prey model is investigated in which the prey population is assumed to have age structure and is governed by the McKendrick-von Foerster partial differential equation and the predator population is described by the classical Volterra-Lotka ordinary differential equation. Quite general hypotheses are assumed for the mortality function, the...

Motivated by problems where variation among individuals is necessary to explain properties of ecological systems, we develop a mathematical model of an individual organism. The model, based primarily upon energetics, is developed specifically for female daphnids, although with appropriate modifications it shoud be applicable to other aquatic animal...

A method to determine the mortality effects of a hydrophobic chemical on a population is proposed. The ecotoxicological protocol is based on individual organism response and is derived from the static theory of “survival of the fattest.” This study, focusing upon effects of mortality and the effects of toxicant stress on population succession, exam...

In this paper we consider a model of size-structured, intraspecific competition in which increased competition during juvenile growth reduces size at adulthood and thereby reduces adult fertility. Our goal here is to derive a model which is simple enough to be as analytically tractable as possible, and yet still capture these essential features. We...

The main purpose of this article is to describe the formulation of an appropriate mathematical representation of a population based on physiological attributes relevant to the individual species considered and to the problem under investigation. There are two main parts of the article. The first discusses the relationship between model hypotheses a...

A general version of a model of Ebenman for the dynamics of a population consisting of competing juveniles and adults is analyzed using methods of bifurcation theory. A very general existence results is obtained for non-trivial equilibria and non-negative synchronous two-cycles that bifurcate simultaneously at the critical valuer=1 of the inherent...

Survival analyses of populations are developed in a continuous process. Persistence and extinction criteria of a class of continuous age-structured population models with separable mortality function and fertility function are established by investigation of asymptotic behaviors of solutions of McKendrick-von Foerster equation.

The extended McKendrick-von Foerster structured population model is employed to derive a nonautonomous ordinary differential equation model of a population. The derivation assumes that the individual life history can be delineated into several physiological stages. We study the persistence of the population when the model is autonomous and base the...

Survival analyses of populations are developed in dicrete growth processes. Persistence and extinction atributes of age-structured discrete population models are explored on both a finite and infinite time horizon. Conditions for persistence and extinction are found. Decompositions of the initial population size axes into intervals where population...

Perturbations of population structure due to toxic chemical exposure are studied by employing a mathematical model. The risk assessment scheme is composed of an individual model, an exposure model, and a population model. The differential equation model of an individual has components chosen by chemical affinity to organism phase and is based upon...

Persistence and extinction attributes of a discrete population model are explored on both a finite and an infinite time horizon. For a first-order autonomous nonlinear difference equation, a classification is found of when, for each positive integer N, trajectories go to extinction at time N. The dynamic complexity that is known to permeate differe...

Based on lecture notes of two summer schools with a mixed audience from mathematical sciences, epidemiology and public health, this volume offers a comprehensive introduction to basic ideas and techniques in modeling infectious diseases, for the comparison of strategies to plan for an anticipated epidemic or pandemic, and to deal with a disease out...

A heterosexually-active population is invaded by n competing strains of a sexually-transmitted and treatable pathogen such as gonorrhea. It is assumed that a host cannot be invaded simultaneously by more than one strain and that individuals recover. We conclude that in a behaviorally and genetically homogeneous population, coexistence is not possib...

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