Invasion Reproduction Numbers Research Articles

A Host-Parasite System with Multiple Parasite Strains and Superinfection Revisited: The Global Dynamics.

In this paper, we revisit a host-parasite system with multiple parasite strains and superinfection proposed by Nowak and May (Proc R Soc Lond B 255(1342):81-89, 1994), and study its global dynamics when we relax the two strict conditions assumed therein. As for system with two parasite strains, we derive that the basic reproduction number [Formula: see text] is the threshold condition for parasite extinction and the invasion reproduction number [Formula: see text] is the subthreshold condition for coexistence of two parasite strains. As for system with three parasite strains, we are surprised to discover the global stability of parasite-free and coexistence equilibrium, which is distinct from the previous result. Furthermore, for system with n strains, we obtain the global asymptotical stability of the parasite-free equilibrium, conjecture a general result on the global stability of coexistence equilibrium and provide two numerical examples to testify our conjecture.

Dynamics analysis of a Zika–dengue co-infection model with dengue vaccine and antibody-dependent enhancement

Mounting clinical and experimental evidence indicates that antibodies against dengue can enhance Zika infection through the antibody-dependent enhancement (ADE) mechanism. ADE is an alarming evolutionary development that makes it difficult to develop a vaccination campaign against dengue. To better understand the effect of dengue vaccine and ADE on a Zika and dengue outbreak, we formulate a Zika–dengue co-infection model that describes the joint dynamics of Zika and dengue. We derive the basic and invasion reproduction numbers, which are threshold values to identify the existence and stability of a disease-free state and two boundary equilibrium points, where only one strain (Zika or dengue) is present, and to identify the persistence of disease. Our analysis and numerical simulations suggest that although vaccination against dengue has a positive effect on the control of dengue, it has a negative effect on the control of Zika, and the increasing level of ADE induces a large number of accumulated Zika cases. Later, in this paper, criteria of vaccination against dengue strategies are established for disease eradication. Finally, the model is applied to simulate the accumulated cases of Zika and dengue in Brazil from February 19, 2016 to April 14, 2018. Based on the parameters estimated, we obtain that in Brazil the dengue epidemic is endemic whereas the Zika epidemic will eventually disappear.

Invasion reproductive numbers for discrete-time models.

Although invasion reproductive numbers (IRNs) are utilized frequently in continuous-time models with multiple interacting pathogens, they are yet to be explored in discrete-time systems. Here, we extend the concept of IRNs to discrete-time models by showing how to calculate them for a set of two-pathogen SIS models with coinfection. In our exploration, we address how sequencing events impacts the basic reproductive number (BRN) and IRN. As an illustrative example, our models are applied to rhinovirus and respiratory syncytial virus co-circulation. Results show that while the BRN is unaffected by variations in the order of events, the IRN differs. Furthermore, our models predict copersistence of multiple pathogen strains under cross-immunity, which is atypical of analogous continuous-time models. This investigation shows that sequencing events has important consequences for the IRN and can drastically alter competition dynamics.

Invasion reproductive numbers for periodic epidemic models.

There are many cases within epidemiology where infections compete to persist within a population. In studying models for such cases, one of the goals is to determine which infections can invade a population and persist when other infections are already resident within the population. Invasion reproductive numbers (IRN), which are tied to the stability of boundary endemic equilibria, can address this question. By reinterpreting resident infections epidemiologically, this study extends methods for finding IRNs to periodic systems, and presents some examples which illustrate the often complex computations required. Results identify conditions under which a simple time-average can be used to derive IRNs, and apply the methods to examine how seasonal fluctuations in influenza incidence facilitate the year-round persistence of bacterial respiratory infections.

Intermittent Preventive Treatment (IPT): Its Role in Averting Disease-Induced Mortality in Children and in Promoting the Spread of Antimalarial Drug Resistance

We develop an age-structured ODE model to investigate the role of intermittent preventive treatment (IPT) in averting malaria-induced mortality in children, and its related cost in promoting the spread of antimalarial drug resistance. IPT, a malaria control strategy in which a full curative dose of an antimalarial medication is administered to vulnerable asymptomatic individuals at specified intervals, has been shown to reduce malaria transmission and deaths in children and pregnant women. However, it can also promote drug resistance spread. Our mathematical model is used to explore IPT effects on drug resistance and deaths averted in holoendemic malaria regions. The model includes drug-sensitive and drug-resistant strains as well as human hosts and mosquitoes. The basic reproduction, and invasion reproduction numbers for both strains are derived. Numerical simulations show the individual and combined effects of IPT and treatment of symptomatic infections on the prevalence of both strains and the number of lives saved. Our results suggest that while IPT can indeed save lives, particularly in high transmission regions, certain combinations of drugs used for IPT and to treat symptomatic infection may result in more deaths when resistant parasite strains are circulating. Moreover, the half-lives of the treatment and IPT drugs used play an important role in the extent to which IPT may influence spread of the resistant strain. A sensitivity analysis indicates the model outcomes are most sensitive to the reduction factor of transmission for the resistant strain, rate of immunity loss, and the natural clearance rate of sensitive infections.

Threshold dynamics of the Transmission of Antibiotic-Resistant Infections

Despite numerous studies conducted, multidrug-resistant infections such as methicillin-resistant Staphylococcus aureus (MRSE) and vancomycin-resistant enterococci (VRE) are still increasing in hospitals, and yet continued to be the important challenges of worldwide health. Mathematical modeling gives an insightful information in policy and decision making to control the transmission and spread of these infections globally. We formulated and analyse a mathematical model to characterise the transmission co-dynamics of hospital-acquired MRSE (HA-MRSA) and community-acquired MRSE (CA-MRSA) and to investigate the long run competitiveness of the two strains in hospital. Numerical simulations are carried out to explore the basic reproduction numbers for the two strains so as to determine the dominant strain in the future in hospital setting. Under some conditions, invasion reproduction numbers are also applied to determine the uniform persistence of the two strains. We further performed sensitivity analysis to examine the influence of model parameters on the transmission and spread of the the strains, thereby determine the effective intervention strategies that reduce the overflow of the infections in hospital setting. To support theoretical findings qualitatively, graphical representations are provided.

Bifurcation analysis and global dynamics of a mathematical model of antibiotic resistance in hospitals.

Antibiotic-resistant bacteria have posed a grave threat to public health by causing a number of nosocomial infections in hospitals. Mathematical models have been used to study transmission dynamics of antibiotic-resistant bacteria within a hospital and the measures to control antibiotic resistance in nosocomial pathogens. Studies presented in Lipstich et al. (Proc Natl Acad Sci 97(4):1938-1943, 2000) and Lipstich and Bergstrom (Infection control in the ICU environment. Kluwer, Boston, 2002) have provided valuable insights in understanding the transmission of antibiotic-resistant bacteria in a hospital. However, their results are limited to numerical simulations of a few different scenarios without analytical analyses of the models in broader parameter regions that are biologically feasible. Bifurcation analysis and identification of the global stability conditions can be very helpful for assessing interventions that are aimed at limiting nosocomial infections and stemming the spread of antibiotic-resistant bacteria. In this paper we study the global dynamics of the mathematical model of antibiotic resistance in hospitals considered in Lipstich etal. (2000) and Lipstich and Bergstrom (2002). The invasion reproduction number [Formula: see text] of antibiotic-resistant bacteria is derived, and the relationship between [Formula: see text] and two control reproduction numbers of sensitive bacteria and resistant bacteria ([Formula: see text] and [Formula: see text]) is established. More importantly, we prove that a backward bifurcation may occur at [Formula: see text] when the model includes superinfection, which is not mentioned in Lipstich and Bergstrom (2002). More specifically, there exists a new threshold [Formula: see text], such that if [Formula: see text], then the system can have two positive interior equilibria, which leads to an interesting bistable phenomenon. This may have critical implications for controlling the antibiotic-resistance in a hospital.

Dynamical properties and thresholds of an HIV model with super-infection.

Super-infection by multiple HIV-1 subtypes, previously thought restricted to high risk groups, has now been reported in the general heterosexual populations at relatively the same incidence rate as in high risk groups. We present a simple deterministic HIV model with super-infection by two HIV-1 subtypes. Mathematical characteristics including the basic reproductive number $(\mathcal{R}_0)$, invasion threshold $(\mathcal{R}_{21},\mathcal{R}_{12})$ and conditions for asymptotic stability are derived. In the absence of super-infection the model exhibits competitive exclusion, and all equilibria are globally attracting if they exist except for the disease free which is a saddle for $\mathcal{R}_0>1.$ The results show that the subtype with the dominant reproductive number exceeding unity dominates the weaker subtype forcing it to extinction regardless of the size of the reproductive number. On the other end, super-infection may promote subtype co-existence whenever the minimum of the subtype specific reproductive numbers $(\mathcal{R}_1,\mathcal{R}_2)$ and the invasion reproductive numbers $(\mathcal{R}_{12},\mathcal{R}_{21})$ exceed unity. Our results demonstrate that if the partial reproductive numbers $(\mathcal{R}_1~\mbox{and}~\mathcal{R}_2 )$ and the invasion reproductive number for the weaker subtype $(\mathcal{R}_{21})$ satisfy $\mathcal{R}_2<1,~\mathcal{R}_1>1~\mbox{and}~\mathcal{R}_{21}>1,$ then primary infection by subtype $1$ may stay the extinction of subtype $2$ despite its relatively low reproductive fitness. For certain parameter ranges, hysteresis (including backward bifurcation) occurs with possible differences in the asymptotic level of disease prevalence. Super-infection may thus facilitate the continued re-generation of reproductively noncompetent subtypes whose subtype specific reproductive numbers will be less than unity while at the same time allowing for the mutual coexistence and persistence of multiple strains. Persistence and co-existence of multiple strains has detrimental effect on vaccine design and development and administration of ART where one or more of the strains are drug resistant.

Stability analysis of a two-strain epidemic model on complex networks with latency

In this paper, a two-strain epidemic model on a complex network is proposed. The two strains are the drug-sensitive strain and the drug-resistant strain. The related basic reproduction numbers $R_s$ and $R_r$ are obtained. If $R_0=\max\{R_s,R_r\} 1$, then there is a unique drug-resistant strain dominated equilibrium $E_r$, which is locally asymptotically stable if the invasion reproduction number $R_r^s 1$ and $R_s>R_r$, then there is a unique coexistence equilibrium $E^*$. The persistence of the model is also proved. The theoretical results are supported with numerical simulations.

Invasion fitness, inclusive fitness, and reproductive numbers in heterogeneous populations.

How should fitness be measured to determine which phenotype or "strategy" is uninvadable when evolution occurs in a group-structured population subject to local demographic and environmental heterogeneity? Several fitness measures, such as basic reproductive number, lifetime dispersal success of a local lineage, or inclusive fitness have been proposed to address this question, but the relationships between them and their generality remains unclear. Here, we ascertain uninvadability (all mutant strategies always go extinct) in terms of the asymptotic per capita number of mutant copies produced by a mutant lineage arising as a single copy in a resident population ("invasion fitness"). We show that from invasion fitness uninvadability is equivalently characterized by at least three conceptually distinct fitness measures: (i) lineage fitness, giving the average individual fitness of a randomly sampled mutant lineage member; (ii) inclusive fitness, giving a reproductive value weighted average of the direct fitness costs and relatedness weighted indirect fitness benefits accruing to a randomly sampled mutant lineage member; and (iii) basic reproductive number (and variations thereof) giving lifetime success of a lineage in a single group, and which is an invasion fitness proxy. Our analysis connects approaches that have been deemed different, generalizes the exact version of inclusive fitness to class-structured populations, and provides a biological interpretation of natural selection on a mutant allele under arbitrary strength of selection.

Coinfection dynamics of two diseases in a single host population

A susceptible-infectious-susceptible (SIS) epidemic model that describes the coinfection and cotransmission of two infectious diseases spreading through a single population is studied. The host population consists of two subclasses: susceptible and infectious, and the infectious individuals are further divided into three subgroups: those infected by the first agent/pathogen, the second agent/pathogen, and both. The basic reproduction numbers for all cases are derived which completely determine the global stability of the system if the presence of one agent/pathogen does not affect the transmission of the other. When the constraint on the transmissibility of the dually infected hosts is removed, we introduce the invasion reproduction number, compare it with two other types of reproduction number and show the uniform persistence of both diseases under certain conditions. Numerical simulations suggest that the system can display much richer dynamics such as backward bifurcation, bistability and Hopf bifurcation.

Mathematical analysis of a model for AVL–HIV co-endemicity

A model for the transmission dynamics of Anthroponotic Visceral Leishmaniasis (AVL) and human immunodeficiency virus (HIV) in a population is developed and used to assess the impact of the spread of each disease on the overall transmission dynamics. As for other vector-borne disease models, the AVL component of the model undergoes backward bifurcation when the associated reproduction number of the AVL-only sub-model (denoted by RL) is less than unity. Uncertainty and sensitivity analyzes of the model, using data relevant to the dynamics of the two diseases in Ethiopia, show that the top three parameters that drive the AVL infection (with respect to the associated response function, RL) are the average number of times a sandfly bites humans per unit time (σV), carrying capacity of vectors (KV) and transmission probability from infected humans to susceptible sandflies (β2). The distribution of RL is RL∈[0.06,3.94] with a mean of RL=1.08. Furthermore, the top three parameters that affect HIV dynamics (with respect to the response function RH) are the transmission rate of HIV (βH), HIV-induced death rate (δH), and the modification parameter for the increase in infectiousness of AIDS individuals in comparison to HIV infected without clinical symptoms of AIDS (ωH). The distribution of RH is RH∈[0.88,2.79] with a mean of RH=1.46. The dominant parameters that affect the dynamics of the full VL-HIV model (with respect to the associated reproduction number, RLH, as the response function) are the transmission rate of HIV (βH), the average number of times a sandfly bites humans per unit time (σV), and HIV-induced death rate (δH) (the distribution of RLH is RLH∈[0.88,3.94] with a mean of RLH=1.64). Numerical simulations of the model show that the two diseases co-exist (with AVL dominating, but not driving HIV to extinction) whenever the reproduction number of each disease exceeds unity. It is shown that AVL can invade a population at HIV-endemic state if a certain threshold quantity, known as invasion reproduction number, exceeds unity.

Modelling HIV superinfection among men who have sex with men.

Superinfection, a phenomenon that an individual infected by one HIV strain is re-infected by the second heterologous HIV strain, occurs in HIV infection. A mathematical model is formulated to examine how superinfection affects transmission dynamics of drug sensitive/resistant strains. Three reproduction numbers are defined: reproduction numbers Rr and Rs for drug-resistant and drug-sensitive strains, respectively, and the invasion reproduction number R (r)s. The disease-free equilibrium always exists and is locally stable when the larger of Rs and Rr is less than one. The drug resistant strain-only equilibrium is locally stable when Rr > 1 and R (r)s < 1. Numerical studies show that as the superinfection coefficient of the sensitive strain increases the system may (1) change to bistable states of disease-free equilibrium and the coexistence state from the stable disease-free equilibrium under no superinfection; (2) experience the stable resistant-strain only equilibrium, the bistable states of resistant-strain only equilibrium and the coexistence state, and the stable coexistence state in turn. This implies that superinfection of the sensitive strain is beneficial for two strains to coexist. While superinfection of the resistant strain makes resistant strain more likely to be sustained. The findings suggest that superinfection induces the complicated dynamics, and brings more difficulties in antiretroviral therapy.

A model for the coupled disease dynamics of HIV and HSV-2 with mixing among and between genders

Evidence indicates that those with genital herpes (HSV-2) infections have greater risks of infection by HIV; and, once co-infected, are more likely to transmit HIV. To better understand the interactions between HIV and HSV-2, we construct a mathematical model that describes the joint dynamics. A new feature of this model is the inclusion of both heterosexual and homosexual interactions. We derive and interpret the basic and invasion reproduction numbers for HIV and HSV-2 using the approach of next-generation matrices. We then perform scenario analyses and conduct a sensitivity analysis to investigate the impact of the model parameters on the reproduction numbers and disease prevalences. We conclude that homosexual transmission drastically changes the disease prevalences; hence, it is important to account for this interaction as models that ignore homosexuality may greatly underestimate the disease burden.

An age-structured model for the coupled dynamics of HIV and HSV-2.

Evidence suggests a strong correlation between the prevalence of HSV-2 (genital herpes) and the perseverance of the HIV epidemic. HSV-2 is an incurable viral infection, characterized by periodic reactivation. We construct a model of the co-infection dynamics between the two diseases by incorporating a time-since-infection variable to track the alternating periods of infectiousness of HSV-2. The model considers only heterosexual relationships and distinguishes three population groups: males, general population females, and female sex workers. We calculate the basic reproduction numbers for each disease that provide threshold conditions, which determine whether a disease dies out or becomes endemic in the absence of the other disease. We also derive the invasion reproduction numbers that determine whether or not a disease can invade into a population in which the other disease is endemic. The calculations of the invasion reproduction numbers suggest a new aspect in their interpretation - the class from which the initial disease carrier arises is important for understanding the invasion dynamics and biological interpretation of the expressions of the reproduction numbers. Sensitivity analysis is conducted to examine the role of model parameters in influencing the model outcomes. The results are discussed in the last section.

A double age-structured model of the co-infection of tuberculosis and HIV.

After decades on the decline, it is believed that the emergence of HIV is responsible for an increase in the tuberculosis prevalence. The leading infectious disease in the world, tuberculosis is also the leading cause of death among HIV-positive individuals. Each disease progresses through several stages. The current model suggests modeling these stages through a time-since-infection tracking transmission rate function, which, when considering co-infection, introduces a double-age structure in the PDE system. The basic and invasion reproduction numbers for each disease are calculated and the basic ones established as threshold for the disease progression. Numerical results confirm the calculations and a simple treatment scenario suggests the importance of time-since-infection when introducing disease control and treatment in the model.

An Age-Structured Model For the Coupled Dynamics of HIV and HSV-2

Evidence suggests a strong correlation between the prevalence of HSV-2 (genital herpes) and the perseverance of the HIV epidemic. HSV-2 is an incurable viral infection, characterized by periodic reactivation. We construct a model of the co-infection dynamics between the two diseases by incorporating a time-since-infection variable to track the alternating periods of infectiousness of HSV-2. The model considers only heterosexual relationships and distinguishes three population groups: males, general population females, and female sex workers. We calculate the basic reproduction numbers for each disease that provide threshold conditions, which determine whether a disease dies out or becomes endemic in the absence of the other disease. We also derive the invasion reproduction numbers that determine whether or not a disease can invade into a population in which the other disease is endemic. The calculations of the invasion reproduction numbers suggest a new aspect in their interpretation - the class from which the initial disease carrier arises is important for understanding the invasion dynamics and biological interpretation of the expressions of the reproduction numbers. Sensitivity analysis is conducted to examine the role of model parameters in inuencing the model outcomes.

Dynamics of Low and High Pathogenic Avian Influenza in Wild and Domestic Bird Populations

Avian influenza H5N1 is at present the most dangerous zoonotic disease infecting wild and domestic birds. Should the virus mutate and become efficiently human-to-human transmittable, a pandemic will occur with high mortality. Avian influenza H5N1 exists in two forms: Low pathogenic (LPAI) and high pathogenic (HPAI). In this talk we build a model of LPAI and HPAI in wild and domestic birds. Birds, wild and domestic, who have been priorly infected with LPAI are partially protected against HPAI. We compute the relevant reproduction numbers and invasion reproduction numbers. We find that the systems has a disease-free equilibrium, LPAI-only equilibrium, HPAI-only equilibrium and at least one coexistence equilibrium. Furthermore, the LPAI-only equilibrium and HPAI-only equilibrium are locally asymptotically stable under appropriate conditions on the reproduction numbers. In contrast, the coexistence equilibrium can lose stability and oscillations are possible. We show that the oscillations are caused by the cross-immunity and can exist in the wild bird system, separate from the domestic bird system. For a pathogen circulating in a multi-species system, species A is called a sink (source), if the pathogen cannot (can) sustain itself in species A without the inflow of infectives from other species. We investigate the sink/source status of LPAI and HPAI in wild and domestic birds.

Dynamics of Low and High Pathogenic Avian Influenza in Wild and Domestic Bird Populations

Avian influenza H5N1 is at present the most dangerous zoonotic disease infecting wild and domestic birds. Should the virus mutate and become efficiently human-to-human transmittable, a pandemic will occur with high mortality. Avian influenza H5N1 exists in two forms: Low pathogenic (LPAI) and high pathogenic (HPAI). In this talk we build a model of LPAI and HPAI in wild and domestic birds. Birds, wild and domestic, who have been priorly infected with LPAI are partially protected against HPAI. We compute the relevant reproduction numbers and invasion reproduction numbers. We find that the systems has a disease-free equilibrium, LPAI-only equilibrium, HPAI-only equilibrium and at least one coexistence equilibrium. Furthermore, the LPAI-only equilibrium and HPAI-only equilibrium are locally asymptotically stable under appropriate conditions on the reproduction numbers. In contrast, the coexistence equilibrium can lose stability and oscillations are possible. We show that the oscillations are caused by the cross-immunity and can exist in the wild bird system, separate from the domestic bird system. For a pathogen circulating in a multi-species system, species A is called a sink (source), if the pathogen cannot (can) sustain itself in species A without the inflow of infectives from other species. We investigate the sink/source status of LPAI and HPAI in wild and domestic birds.

Dynamics analysis of a quarantine model in two patches

A new deterministic model for assessing the impact of quarantine on the transmission dynamics of a communicable disease in a two‐patch community is designed. Rigorous analysis of the model shows that the imperfect nature of quarantine (in the two patches) could induce the phenomenon of backward bifurcation when the associated reproduction number of the model is less than unity. For the case when quarantined susceptible individuals do not acquire infection during quarantine, the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model has a unique Patch i‐only boundary equilibrium (i = 1,2) whenever the associated reproduction number for Patch i is greater than unity. The unique Patch i‐only boundary equilibrium is locally asymptotically stable whenever the invasion reproduction number of Patch 3 − i is less than unity (and the associated reproduction number for Patch i exceeds unity). The model has at least one endemic equilibrium when its reproduction number exceeds unity (and the disease persists in both patches in this case). It is shown that adding multi‐patch dynamics to a single‐patch quarantine model (which allow the quarantine of susceptible individuals) in a single patch does not alter its quantitative dynamics (with respect to the existence and asymptotic stability of its associated equilibria as well as its backward bifurcation property). Copyright © 2014 John Wiley &amp; Sons, Ltd.