M. Martcheva

M. Martcheva
University of Florida | UF · Department of Mathematics

Ph.D.

About

163
Publications
29,136
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3,453
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Introduction
M. Martcheva currently works at the Department of Mathematics, University of Florida. Dr. Martcheva does research in Mathematical Biology.
Additional affiliations
August 2003 - present
University of Florida
Position
  • Professor (Full)

Publications

Publications (163)
Article
In this paper we assess the effectiveness of different non-pharmaceutical interventions (NPIs) against COVID-19 utilizing a compartmental model. The local asymptotic stability of equilibria (disease-free and endemic) in terms of the basic reproduction number have been determined. We find that the system undergoes a backward bifurcation in the case...
Article
Full-text available
COVID-19 is a disease caused by infection with the virus 2019-nCoV, a single-stranded RNA virus. During the infection and transmission processes, the virus evolves and mutates rapidly, though the disease has been quickly controlled in Wuhan by ‘Fangcang’ hospitals. To model the virulence evolution, in this paper, we formulate a new age structured e...
Article
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Human papillomavirus (HPV) infection can spread between regions. What is the impact of disassortative geographical mixing on the dynamics of HPV transmission? Vaccination is effective in preventing HPV infection. How to allocate HPV vaccines between genders within each region and between regions to reduce the total infection? Here we develop a two-...
Article
In ecological systems, the fear of predation risk may assert privilege to the prey species by restricting their exposure to potential predators, and also impose costs by constraining the exploration of optimal resources. In this paper, an eco-epidemic model of the predator-prey system is investigated by considering disease in prey population and se...
Article
Media impact has significant effect on reducing the disease prevalence, meanwhile sanitation and awareness can control the epidemic by reducing the growth rate of bacteria and direct contacts with infected individuals. In this article, we investigate the impacts of media and sanitation coverage on the dynamics of epidemic outbreak. We observe that...
Article
Full-text available
In this paper, we present a multi-scale co-affection model of HIV infection and opioid addiction. The population scale epidemiological model is linked to the within-host model which describes the HIV and opioid dynamics in a co-affected individual. CD4 cells and viral load data obtained from morphine addicted SIV-infected monkeys are used to valida...
Article
Vaccination is effective in preventing human papillomavirus (HPV) infection. It still remains debatable whether males should be included in a vaccination program and unclear how to allocate the vaccine in genders to achieve the maximum benefits. In this paper, we use a two-sex model to assess HPV vaccination strategies and use the data from Guangxi...
Preprint
Full-text available
Vaccination is effective in preventing human papillomavirus (HPV) infection. It still remains debatable whether males should be included in a vaccination program and unclear how to allocate the vaccine in genders to achieve the maximum benefits. In this paper, we use a two-sex model to assess HPV vaccination strategies and use the data from Guangxi...
Article
Full-text available
We propose two models inspired by the COVID-19 pandemic: a coupled disease-human behaviour (or disease-game theoretic), and a coupled disease-human behaviour-economic model, both of which account for the impact of social-distancing on disease control and economic growth. The models exhibit rich dynamical behaviour including multistable equilibria,...
Article
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In this paper, we introduce three within-host and one within-vector models of Zika virus. The within-host models are the target cell limited model, the target cell limited model with natural killer (NK) cells class, and a within-host-within-fetus model of a pregnant individual. The within-vector model includes the Zika virus dynamics in the midgut...
Article
Full-text available
In this paper, we investigate the combined effects of fear, prey refuge and additional food for predator in a predator-prey system with Beddington type functional response. We observe oscillatory behavior of the system in the absence of fear, refuge and additional food whereas the system shows stable dynamics if anyone of these three factors is int...
Article
In this paper, a two-strain model with coinfection that links immunological and epidemiological dynamics across scales is formulated. On the with-in host scale, the two strains eliminate each other with the strain having the larger immunological reproduction number persisting. However, on the population scale coinfection is a common occurrence. Ind...
Article
In this paper, we formulate a new-age structured heroin transmission model with respect to the age of vaccination which structures the vaccine wanes rate [Formula: see text] and infection ratio of vaccination individuals [Formula: see text]. The well-posedness and the basic reproduction number [Formula: see text] of our model are first presented. I...
Article
Full-text available
In this paper, we propose a model to assess the impacts of budget allocation for vaccination and awareness programs on the dynamics of infectious diseases. The budget allocation is assumed to follow logistic growth, and its per capita growth rate increases proportional to disease prevalence. An increment in per-capita growth rate of budget allocati...
Article
Full-text available
The ongoing COVID-19 pandemic has affected most of the countries on Earth. It has become a pandemic outbreak with more than 50 million confirmed infections and above 1 million deaths worldwide. In this study, we consider a mathematical model on COVID-19 transmission with the prosocial awareness effect. The proposed model can have four equilibrium s...
Article
In this paper we formulate a multi-scale nested immuno-epidemiological model of HIV on complex networks. The system is described by ordinary differential equations coupled with a partial differential equation. First, we prove the existence and uniqueness of the immunological model and then establish the well-posedness of the multi-scale model. We d...
Article
Vaccination is effective in preventing human papillomavirus (HPV) infection. It is imperative to investigate who should be vaccinated and what the best vaccine distribution strategy is. In this paper, we use a dynamic model to assess HPV vaccination strategies in a heterosexual population combined with gay, bisexual, and other men who have sex with...
Article
Full-text available
We propose a model of a joint spread of heroin use and HIV infection. The unique disease-free equilibrium always exists and it is stable if the basic reproduction numbers of heroin use and HIV infection are both less than 1. The semi-trivial equilibrium of HIV infection (heroin use) exists if the basic reproduction number of HIV infection (heroin u...
Article
Full-text available
Vector-borne infectious diseases may involve both horizontal transmission between hosts and transmission from infected vectors to susceptible hosts. In this paper, we incorporate these two transmission modes into a vector-borne disease model that includes general nonlinear incidence rates and the age of infection for both hosts and vectors. We show...
Preprint
In this paper, we will demonstrate how to use a nonlinear polyhedral constrained optimization solver called the Polyhedral Active Set Algorithm (PASA) for solving a general singular control problem. We present methods of discretizing a general optimal control problem that involves the use of the gradient of the Lagrangian for computing the gradient...
Preprint
Full-text available
The ongoing COVID-19 pandemic caused by SARS-CoV-2 virus, a highly contagious virus, affected most of the countries of Earth. COVID-19 is causing obstacles for public health organizations and is affecting almost every aspect of human life. It has become an epidemic outbreak with more than 22 million confirmed infections and above 750 thousand death...
Article
In this paper, to understand the impact of spatial heterogeneity of treatment and movement of individuals on the persistence and extinction of heroin spread, we propose a new diffusive heroin transmission model with treatment-dependent age-structure. The basic reproduction number in heterogenous environment R0 of the system is defined, which is con...
Article
Full-text available
Backward bifurcation is an important property of infectious disease models. A centre manifold method has been developed by Castillo-Chavez and Song for detecting the presence of backward bifurca-tion and deriving a necessary and sufficient condition for its occurrence in Ordinary Differential Equations (ODE) models. In this paper, we extend this me...
Article
Full-text available
Visceral leishmaniasis in dogs is believed to have an impact on the prevalence of the disease in human populations. Here, we continue the analysis of the nested immuno-epidemiological model of visceral leishmaniasis in dogs, including a proof of well-posedness using functional analytical methods. Once well-posedness is established, we continue stab...
Book
This book introduces advanced mathematical methods and techniques for analysis and simulation of models in mathematical epidemiology. Chronological age and class-age play an important role in the description of infectious diseases and this text provides the tools for the analysis of this type of partial differential equation models. This book prese...
Chapter
Game theory is the process of modeling strategic interactions between two or more players in a situation containing set rules and outcomes. Game theory is used in many disciplines, but we are interested in introducing here its application to infectious diseases. Vaccination against all childhood diseases poses an interesting dilemma to the parents:...
Chapter
In this section we present some basic and advanced analysis of age-structured epidemic models. In the next subsection, we discuss well posedness.
Chapter
Richard Levins [111] used the term “metapopulation” to describe a population of populations. Metapopulations naturally occur in fragmented habitats where each component of the habitat is occupied by one population and the populations are connected by migration. Levins proposed a very simple equation model to investigate the dynamics of the metapopu...
Chapter
In most epidemiological models for the transmission of infectious diseases, the infectious individuals are assumed to have the same infectivity. This assumption is reasonable in modeling communicable diseases such as influenza [24] and sexually transmitted diseases such as gonorrhea [82]. However, in the study of diseases such as HIV/AIDS, tubercul...
Chapter
Human behaviors affect epidemic spread. One of the main components of the disease transmission is the contact rate which in all models so far has been assumed constant or varying by age (age-since-infection) only. However, the contact rate is not constant from individual to individual; in particular some individuals have high contact rate while oth...
Chapter
Age-structured epidemic models are based on age-structured population models. Typically the linear age-structured population model, called the Lotka-McKendrick model, is used as a baseline population model in epidemic systems.
Chapter
Chapters 1 and 3 introduce and analyze several age-structured or class age-structured epidemic models. These models describe the diseases transmitted from human to human directly. However, there are many diseases, such as malaria, Chagas disease, dengue fever, which are transmitted via a vector. In this chapter, we formulate and analyze age-structu...
Chapter
Immuno-epidemiological modeling of infectious diseases is a technique that links a within-host mathematical model with a between-host mathematical model. In the simplest nested models the within-host model is an ODE structured by the time since infection. The between-host model is a time-since-infection and chronological time structured PDE. The ne...
Article
Full-text available
Vaccination and treatment are both effective methods of preventing the spread of infectious diseases. In this paper, we propose an SIRVS epidemic model with ages of vaccination and recovery, involving vaccine‐induced immunity and infection‐induced immunity. The basic reproduction number of the epidemic model, R0, is first obtained. If R0<1, the loc...
Article
Despite centuries of enormous control efforts, mosquito-borne diseases continue to show upward trend of morbidity. According to WHO reports, malaria caused 438000 deaths in the year 2015 and dengue cases has been increased 30-fold over the last five decades. To control these diseases, it is necessary to understand the transmission dynamics of them...
Article
Full-text available
During aging humans lose midbrain dopamine neurons, but not all dopamine regions exhibit vulnerability to neurodegeneration. Microglia maintain tissue homeostasis and neuronal support, but microglia become senescent and likely lose some of their functional abilities. Since aging is the greatest risk factor for Parkinson's disease, we hypothesized t...
Book
This book introduces advanced mathematical methods and techniques for analysis and simulation of models in mathematical epidemiology. Chronological age and class-age play an important role in the description of infectious diseases and this text provides the tools for the analysis of this type of partial differential equation models. This book prese...
Article
Full-text available
Backward bifurcation has significant implications for disease control. Deriving necessary and sufficient conditions for backward bifurcation is of paramount importance to understand the reasons for its occurrence and devise effective control strategies. In this paper, we review the methods that lead to necessary and sufficient conditions for backwa...
Article
The transmission of avian influenza between humans is extremely rare, and it mostly affects individuals who are in contact with infected family member. Although this scenario is uncommon, there have been multiple outbreaks that occur in small infection clusters in Asia with relatively low transmissibility, and thus are too weak to cause an epidemic...
Article
Full-text available
This paper considers an eco-epidemiological model with disease in the prey population. The disease in the prey divides the total prey population into two subclasses, susceptible prey and infected prey. The model also incorporates fear of predator that reduces the growth rate of the prey population. Furthermore, fear of predator lowers the activity...
Article
Full-text available
Leishmaniasis is a neglected and emerging disease prevalent in Mediterranean and tropical climates. As such, the study and development of new models are of increasing importance. We introduce a new immuno-epidemiological model of visceral leishmaniasis in dogs. The within-host system is based on previously collected and published data, showing the...
Article
Full-text available
Based on the development of heroin vaccine, in this paper, we propose an age structured heroin transmission model with treatment and vaccination. The model allows the drug reuse rate of the individuals in treatment to depend on a treatment-age and the vaccine waning rate of the vaccinated to depend on a vaccination age. Meanwhile, the model allows...
Article
Full-text available
This paper introduces a novel partial differential equation immuno-eco-epidemiological model of competition in which one species is affected by a disease while another can compete with it directly and by lowering the first species' immune response to the infection, a mode of competition termed stress-induced competition. When the disease is chronic...
Article
Full-text available
The largest outbreak of Ebola to date is the 2014 West Africa Ebola outbreak, with more than 10,000 cases and over 4000 deaths reported in Liberia alone. To control the spread of the outbreak, multiple interventions were implemented: identification and isolation of cases, contact tracing, quarantining of suspected contacts, proper personal protecti...
Article
Full-text available
Global eradication of Guinea worm disease (GWD) is in the final stage but a mysterious epidemic of the parasite in dog population makes the elimination programme challenging. There is neither a vaccine nor an effective treatment against the disease and therefore intervention strategies rely on the current epidemiological understandings to control t...
Article
The Zika virus (ZIKV) epidemic has caused an ongoing threat to global health security and spurred new investigations of the virus. Use of epidemiological models for arbovirus diseases can be a powerful tool to assist in prevention and control of the emerging disease. In this article, we introduce six models of ZIKV, beginning with a general vector-...
Article
Migration is a key ecological process that enables connections between spatially separated populations. Previous studies have indicated that migration can stabilize chaotic ecosystems. However, the role of migration for two similar types of ecosystems, one chaotic and the other stable, has not yet been studied properly. In the present paper, we inv...
Article
In this paper, a multi-strain virus dynamic model with spatial diffusion, age of infection and general incidence function is formulated. The well-posedness of the initial-boundary value problem of the model in the bounded domain Ω⊂Rn is analyzed. By constructing a suitable Lyapunov functional, the global stability of the uninfected steady state is...
Article
Full-text available
Leishmaniasis is a vector-borne disease of worldwide distribution, currently present in 98 countries. Since late 2010, an unusual increase of human visceral and cutaneous leishmaniasis cases has been observed in the south-western Madrid region, totaling more than 600 cases until 2015. Some hosts, such as human, domestic dog and cat, rabbit (Oryctol...
Data
Model of the leishmaniasis dynamic in dogs with and without insecticide impregnated collar. Legend: Vector: V1) Non infected, V2) Infected but not infective; V3) Infected and infective; Dogs: Sd) Susceptible; Ld) Latent with visceral leishmaniasis, Ad) Asymptomatic; Dd) Sick with visceral leishmaniasis; Rd) Recovered; Dogs with collar: Sdc) Suscept...
Data
Equations of dogs and vectors adapted for using vaccine and insecticide impregnated collar in dogs. (DOCX)
Article
Full-text available
Malaria is one of the most common mosquito-borne diseases widespread in the tropical and subtropical regions. Few models coupling the within-host malaria dynamics with the between-host mosquito-human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formula...
Article
Full-text available
In this paper, a partial differential equation (PDE) model is proposed to explore the transmission dynamics of vector-borne diseases. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts which describe incubation-age dependent removal rates in the latent period and the variable infectiousness in the...
Article
In this paper, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If...
Article
Over the last decade, the epidemiology of avian influenza has undergone a significant transformation. Not only have we seen an increase in the number of outbreaks of the deadly strain known as highly pathogenic avian influenza (HPAI), but the number of birds infected, and the cost of control has risen drastically. Live poultry markets play a huge r...
Article
In this paper, an eco-epidemiological model where prey disease is structured as a susceptible-infected model is investigated. Thresholds that control disease spread and population persistence are obtained. Existence, stability and instability of the system are studied. Hopf bifurcation is shown to occur where a periodic solution bifurcates from the...
Chapter
Full-text available
This chapter introduces an avian influenza model which includes the dynamics of low pathogenic avian influenza (LPAI) and high pathogenic avian influenza (HPAI). The model structures the LPAI-recovered individuals by time-since-recovery and involves the crossimmunity that LPAI infection generates toward the HPAI. Reproduction numbers (R0LW, R0HW) a...
Article
Full-text available
Vector-borne disease transmission is a common dissemination mode used by many pathogens to spread in a host population. Similar to directly transmitted diseases, the within-host interaction of a vector-borne pathogen and a host's immune system influence the pathogen's transmission potential between hosts via vectors. Yet there are few theoretical s...
Article
Full-text available
This paper introduces a time-since-recovery structured, multi-strain, multi-population model of avian influenza. Influenza A viruses infect many species of wild and domestic birds and are classified into two groups based on their ability to cause disease: low pathogenic avian influenza (LPAI) and high pathogenic avian influenza (HPAI). Prior infect...
Article
Full-text available
This paper introduces a novel immuno-eco-epidemiological model of competition in which one of the species is affected by a pathogen. The infected individuals from species one are structured by time-since-infection and the within-host dynamics of the pathogen and the immune response is also modelled. A novel feature of the model is the impact of the...
Article
Full-text available
In this paper, a multi-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the n strains eliminate each other with the strain having the largest immunological reproduction number persisting. However, on the population scale, we extend the competitive exclusion principle to a mult...
Article
Full-text available
This paper investigates a two strain SIS model with diffusion, spatially heterogeneous coefficients of the reaction part and distinct diffusion rates of the separate epidemiological classes. First, it is shown that the model has bounded classical solutions. Next, it is established that the model with spatially homogeneous coefficients leads to comp...
Article
In this article, we discuss the structural and practical identifiability of a nested immuno-epidemiological model of arbovirus diseases, where host–vector transmission rate, host recovery, and disease-induced death rates are governed by the within-host immune system. We incorporate the newest ideas and the most up-to-date features of numerical meth...
Chapter
Full-text available
In this study, we consider a simple mathematical model to describe the behavior of both high pathogenic avian influenza (HPAI) and low pathogenic avian influenza (LPAI) strains in a domestic bird population with culling. We compute the strain reproduction numbers $\mathcal R_1, \ \mathcal R_2$. We find that besides the disease-free equilibrium, the...
Article
This paper considers global asymptotic properties for an age-structured model of heroin use based on the principles of mathematical epidemiology where the incidence rate depends on the age of susceptible individuals. The basic reproduction number of the heroin spread is obtained. It completely determines the stability of equilibria. By using the di...