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’.     

Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion

This article introduces a two-strain spatially explicit SIS epidemic model with space-dependent transmission parameters. We define reproduction numbers of the two strains, and show that the disease-free equilibrium will be globally stable if both reproduction numbers are below one. We also introduce the invasion numbers of the two strains which determine the ability of each strain to invade the single-strain equilibrium of the other strain. The main question that we address is whether the presence of spatial structure would allow the two strains to coexist, as the corresponding spatially homogeneous model leads to competitive exclusion. We show analytically that if both invasion numbers are larger than one, then there is a coexistence equilibrium. We devise a finite element numerical method to numerically confirm the stability of the coexistence equilibrium and investigate various competition scenarios between the strains. Finally, we show that the numerical scheme preserves the positive cone and converges of first order in the time variable and second order in the space variables.     

An evolutionary model of influenza A with drift and shift

Influenza A virus evolves through two types of evolutionary mechanisms – drift and shift. These two evolutionary mechanisms allow the pathogen to infect us repeatedly, as well as occasionally create pandemics with large morbidity and mortality. Here we introduce a novel model that incorporates both evolutionary mechanisms. This necessitates the modelling of three types of strains – seasonal human strains, bird-to-human transmittable H5N1 strains and evolved pandemic H5N1 strain. We define reproduction and invasion reproduction numbers and use them to establish the presence of dominant and coexistence equilibria. We find that the amino acid substitution structure of human influenza can destabilize the human influenza equilibrium and sustained oscillations are possible. We find that for low levels of infection in domestic birds, these oscillations persist, inducing oscillations in the number of humans infected with the avian flu strain. The oscillations have a period of 365 days, similar to the one that can be observed in the cumulative number of human H5N1 cases reported by the World Health Organization (WHO). Furthermore, we establish some partial global results on the competition of the strains.     

Modeling the synergy between HSV-2 and HIV and potential impact of HSV-2 therapy

Mounting evidence indicates that genital HSV-2 infection may increase susceptibility to HIV infection and that co-infection may increase infectiousness. Accordingly, antiviral treatment of people with HSV-2 may mitigate the incidence of HIV in populations where both pathogens occur. To better understand the epidemiological synergy between HIV and HSV-2, we formulate a deterministic compartmental model that describes the transmission dynamics of these pathogens. Unlike earlier models, ours incorporates gender and heterogeneous mixing between activity groups. We derive explicit expressions for the reproduction numbers of HSV-2 and HIV, as well as the invasion reproduction numbers via next generation matrices. A qualitative analysis of the system includes the local and global behavior of the model. Simulations reinforce these analytical results and demonstrate epidemiological synergy between HSV-2 and HIV. In particular, numerical results show that HSV-2 favors the invasion of HIV, may dramatically increase the peak as well as reducing the time-to-peak of HIV prevalence, and almost certainly has exacerbated HIV epidemics. The potential population-level impact of HSV-2 on HIV is demonstrated by calculating the fraction of HIV infections attributable to HSV-2 and the difference between HIV prevalence in the presence and absence of HSV-2. The potential impact of treating people with HSV-2 on HIV control is demonstrated by comparing HIV prevalence with and without HSV-2 therapy. Most importantly, we illustrate that the aforementioned aspects of the population dynamics can be significantly influenced by the sexual structure of the population.     

Mathematical Analysis of a Two Strain HIV/AIDS Model with Antiretroviral Treatment

A two strain HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear ordinary differential equations. The disease-free equilibrium is shown to be globally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The centre manifold theory is used to show that the sensitive HIV-strain only and resistant HIV-strain only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The model is numerically analysed to assess the effects of treatment with amelioration on the dynamics of a two strain HIV/AIDS model. Numerical simulations of the model show that the two strains co-exist whenever the reproduction numbers exceed unity. Further, treatment with amelioration may result in an increase in the total number of infective individuals (asymptomatic) but results in a decrease in the number of AIDS patients. Further, analysis of the reproduction numbers show that antiretroviral resistance increases with increase in antiretroviral use.     

Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs

One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R t 0) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where R t 0 > 1, it is shown that the model can have two co-existing endemic equilibria, and competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (R t r) is greater than that of the wild strain (R t w). Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold (R(0)) exceeds unity. For the case when [Formula in text], it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make R t w < R t r.     

Theoretical Assessment of Public Health Impact of Imperfect Prophylactic HIV-1 Vaccines with Therapeutic Benefits

This paper presents a number of deterministic models for theoretically assessing the potential impact of an imperfect prophylactic HIV-1 vaccine that has five biological modes of action, namely “take,” “degree,” “duration,” “infectiousness,” and “progression,” and can lead to increased risky behavior. The models, which are of the form of systems of nonlinear differential equations, are constructed via a progressive refinement of a basic model to incorporate more realistic features of HIV pathogenesis and epidemiology such as staged progression, differential infectivity, and HIV transmission by AIDS patients. The models are analyzed to gain insights into the qualitative features of the associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction numbers and a measure for vaccine impact or efficacy. The key findings of the study include the following (i) if the vaccinated reproduction number is greater than unity, each of the models considered has a locally unstable disease-free equilibrium and a unique endemic equilibrium; (ii) owing to the vaccine-induced backward bifurcation in these models, the classical epidemiological requirement of vaccinated reproduction number being less than unity does not guarantee disease elimination in these models; (iii) an imperfect vaccine will reduce HIV prevalence and mortality if the reproduction number for a wholly vaccinated population is less than the corresponding reproduction number in the absence of vaccination; (iv) the expressions for the vaccine characteristics of the refined models take the same general structure as those of the basic model.     

Eliminating bovine tuberculosis in cattle and badgers: insight from a dynamic model

Bovine tuberculosis (BTB) is a multi-species infection that commonly affects cattle and badgers in Great Britain. Despite years of study, the impact of badgers on BTB incidence in cattle is poorly understood. Using a two-host transmission model of BTB in cattle and badgers, we find that published data and parameter estimates are most consistent with a system at the threshold of control. The most consistent explanation for data obtained from cattle and badger populations includes within-host reproduction numbers close to 1 and between-host reproduction numbers of approximately 0.05. In terms of controlling infection in cattle, reducing cattle-to-cattle transmission is essential. In some regions, even large reductions in badger prevalence can have a modest impact on cattle infection and a multi-stranded approach is necessary that also targets badger-to-cattle transmission directly. The new perspective highlighted by this two-host approach provides insight into the control of BTB in Great Britain.     

Evaluating the importance of within- and between-host selection pressures on the evolution of chronic pathogens

Infectious pathogens compete and are subject to natural selection at multiple levels. For example, viral strains compete for access to host resources within an infected host and, at the same time, compete for access to susceptible hosts within the host population. Here we propose a novel approach to study the interplay between within- and between-host competition. This approach allows for a single host to be infected by and transmit two strains of the same pathogen. We do this by nesting a model for the host-pathogen dynamics within each infected host into an epidemiological model. The nesting of models allows the between-host infectivity and mortality rates suffered by infected hosts to be functions of the disease progression at the within-host level. We present a general method for computing the basic reproduction ratio of a pathogen in such a model. We then illustrate our method using a basic model for the within-host dynamics of viral infections, embedded within the simplest susceptible-infected (SI) epidemiological model. Within this nested framework, we show that the virion production rate at the level of the cell-virus interaction leads, via within-host competition, to the presence or absence of between-host level competitive exclusion. In particular, we find that in the absence of mutation the strain that maximizes between-host fitness can outcompete all other strains. In the presence of mutation we observe a complex invasion landscape showing the possibility of coexistence. Although we emphasize the application to human viral diseases, we expect this methodology to be applicable to be many host-parasite systems.     

Evaluating the importance of within- and between-host selection pressures on the evolution of chronic pathogens

Infectious pathogens compete and are subject to natural selection at multiple levels. For example, viral strains compete for access to host resources within an infected host and, at the same time, compete for access to susceptible hosts within the host population. Here we propose a novel approach to study the interplay between within- and between-host competition. This approach allows for a single host to be infected by and transmit two strains of the same pathogen. We do this by nesting a model for the host–pathogen dynamics within each infected host into an epidemiological model. The nesting of models allows the between-host infectivity and mortality rates suffered by infected hosts to be functions of the disease progression at the within-host level. We present a general method for computing the basic reproduction ratio of a pathogen in such a model. We then illustrate our method using a basic model for the within-host dynamics of viral infections, embedded within the simplest susceptible–infected (SI) epidemiological model. Within this nested framework, we show that the virion production rate at the level of the cell–virus interaction leads, via within-host competition, to the presence or absence of between-host level competitive exclusion. In particular, we find that in the absence of mutation the strain that maximizes between-host fitness can outcompete all other strains. In the presence of mutation we observe a complex invasion landscape showing the possibility of coexistence. Although we emphasize the application to human viral diseases, we expect this methodology to be applicable to be many host–parasite systems.     

Superinfections can induce evolutionarily stable coexistence of pathogens

Parasites reproduce and are subject to natural selection at several different, but intertwined, levels. In the recent paper, Gilchrist and Coombs (Theor. Popul. Biol. 69:145–153, 2006) relate the between-host transmission in the context of an SI model to the dynamics within a host. They demonstrate that within-host selection may lead to an outcome that differs from the outcome of selection at the host population level. In this paper we combine the two levels of reproduction by considering the possibility of superinfection and study the evolution of the pathogen’s within-host reproduction rate p. We introduce a superinfection function φ = φ(p,q), giving the probability with which pathogens with trait q, upon transmission to a host that is already infected by pathogens with trait p, “take over” the host. We consider three cases according to whether the function q → φ(p,q) (i) has a discontinuity, (ii) is continuous, but not differentiable, or (iii) is differentiable in q = p. We find that in case (i) the within-host selection dominates in the sense that the outcome of evolution at the host population level coincides with the outcome of evolution in a single infected host. In case (iii), it is the transmission to susceptible hosts that dominates the evolution to the extent that the singular strategies are the same as when the possibility of superinfections is ignored. In the biologically most relevant case (ii), both forms of reproduction contribute to the value of a singular trait. We show that when φ is derived from a branching process variant of the submodel for the within-host interaction of pathogens and target cells, the superinfection functions fall under case (ii). We furthermore demonstrate that the superinfection model allows for steady coexistence of pathogen traits at the host population level, both on the ecological, as well as on the evolutionary time scale.      

Two optimal mutation rates in obligate pathogens subject to deleterious mutation

Pathogen species with high mutation rates are likely to accumulate deleterious mutations that reduce their reproductive potential within the host. By altering the within-host growth rate of the pathogen, the deleterious mutation load has the potential to affect epidemiological properties such as prevalence, mean pathogen load, and the mean duration of infections. Here, I examine an epidemiological model that allows for multiple segregating mutations that affect within-host replication efficiency. The model demonstrates a complex range of outcomes depending on pathogen mutation rate, including two distinct, widely separated mutation rates associated with high pathogen prevalence. The low mutation rate prevalence peak is associated with small amounts of genetic diversity within the pathogen population, relatively stable prevalence and infection dynamics, and genetic variation partitioned between hosts. The high mutation rate peak is characterized by considerable genetic diversity both within and between hosts, relatively frequent invasions by more virulent types, and is qualitatively similar to an RNA virus quasispecies. The two prevalence peaks are separated by a valley where natural selection favors evolution toward the optimal within-host state, which is associated with high virulence and relatively rapid host mortality. Both chronic and acute infections are examined using stochastic forward simulations.     

The transmission dynamics of a within-and between-hosts malaria model

In this paper, we developed a novel deterministic coupled model tying together the effects of within-host and population level dynamics on malaria transmission dynamics. We develop within-host and within-vector dynamic models, population level between-hosts models, and a nested coupled model combining these levels. The unique feature of this work is the way the coupling and feedback for the model use the various life stages of the malaria parasite both in the human host and the mosquito vector. Analysis of the coupled and the within-human host models indicate the existence of locally asymptotically stable infection- and parasite-free equilibria when the associated reproduction numbers are less than one. The population-level model, on the other hand, exhibits backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium. A global sensitivity analysis was carried out to measure the effects of the sensitivity and uncertainty in the various model parameters estimates. The results indicate that the most important parameters driving the pathogen level within an infected human are the production rate of the red blood cells from the bone marrow, the infection rate, the immunogenicity of the infected red blood cells, merozoites and gametocytes, and the immunosensitivity of the merozoites and gametocytes. The key parameters identified at the population level are the human recovery rate, the death rate of the mosquitoes, the recruitment rate of susceptible humans into the population, the mosquito biting rate, the transmission probabilities per contact in mosquitoes and in humans, and the parasite production and clearance rates in the mosquitoes. Defining the feedback functions as a linear function of the mosquito biting rate, numerical exploration of the coupled model reveals oscillations in the parasite populations within a human host in the presence of the host immune response. These oscillations dampen as the mosquito biting rate increases. We also observed that the oscillation and damping effect seen in the within-human host dynamics fed back into the population level dynamics; this in turn amplifies the oscillations in the parasite population within the mosquito-host.     

Backward bifurcations and multiple equilibria in epidemic models with structured immunity

Many disease pathogens stimulate immunity in their hosts, which then wanes over time. To better understand the impact of this immunity on epidemiological dynamics, we propose an epidemic model structured according to immunity level that can be applied in many different settings. Under biologically realistic hypotheses, we find that immunity alone never creates a backward bifurcation of the disease-free steady state. This does not rule out the possibility of multiple stable equilibria, but we provide two sufficient conditions for the uniqueness of the endemic equilibrium, and show that these conditions ensure uniqueness in several common special cases. Our results indicate that the within-host dynamics of immunity can, in principle, have important consequences for population-level dynamics, but also suggest that this would require strong non-monotone effects in the immune response to infection. Neutralizing antibody titer data for measles are used to demonstrate the biological application of our theory.     

Modelling the evolution and spread of HIV immune escape mutants

During infection with human immunodeficiency virus (HIV), immune pressure from cytotoxic T-lymphocytes (CTLs) selects for viral mutants that confer escape from CTL recognition. These escape variants can be transmitted between individuals where, depending upon their cost to viral fitness and the CTL responses made by the recipient, they may revert. The rates of within-host evolution and their concordant impact upon the rate of spread of escape mutants at the population level are uncertain. Here we present a mathematical model of within-host evolution of escape mutants, transmission of these variants between hosts and subsequent reversion in new hosts. The model is an extension of the well-known SI model of disease transmission and includes three further parameters that describe host immunogenetic heterogeneity and rates of within host viral evolution. We use the model to explain why some escape mutants appear to have stable prevalence whilst others are spreading through the population. Further, we use it to compare diverse datasets on CTL escape, highlighting where different sources agree or disagree on within-host evolutionary rates. The several dozen CTL epitopes we survey from HIV-1 gag, RT and nef reveal a relatively sedate rate of evolution with average rates of escape measured in years and reversion in decades. For many epitopes in HIV, occasional rapid within-host evolution is not reflected in fast evolution at the population level.     

Mathematical Study of a Staged-Progression HIV Model with Imperfect Vaccine

A staged-progression HIV model is formulated and used to investigate the potential impact of an imperfect vaccine. The vaccine is assumed to have several desirable characteristics such as protecting against infection, causing bypass of the primary infection stage, and offering a disease-altering therapeutic effect (so that the vaccine induces reversal from the full blown AIDS stage to the asymptomatic stage). The model, which incorporates HIV transmission by individuals in the AIDS stage, is rigorously analyzed to gain insight into its qualitative features. Using a comparison theorem, the model with mass action incidence is shown to have a globally-asymptotically stable disease-free equilibrium whenever a certain threshold, known as the vaccination reproduction number, is less than unity. Furthermore, the model with mass action incidence has a unique endemic equilibrium whenever this threshold exceeds unity. Using the Li-Muldowney techniques for a reduced version of the mass action model, this endemic equilibrium is shown to be globally-asymptotically stable, under certain parameter restrictions. The epidemiological implications of these results are that an imperfect vaccine can eliminate HIV in a given community if it can reduce the reproduction number to a value less than unity, but the disease will persist otherwise. Furthermore, a future HIV vaccine that induces the bypass of primary infection amongst vaccinated individuals (who become infected) would decrease HIV prevalence, whereas a vaccine with therapeutic effect could have a positive or negative effect at the community level.     

Malaria Drug Resistance: The Impact of Human Movement and Spatial Heterogeneity

Human habitat connectivity, movement rates, and spatial heterogeneity have tremendous impact on malaria transmission. In this paper, a deterministic system of differential equations for malaria transmission incorporating human movements and the development of drug resistance malaria in an \(n\) patch system is presented. The disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. For a two patch case, the boundary equilibria (drug sensitive-only and drug resistance-only boundary equilibria) when there is no movement between the patches are shown to be locally asymptotically stable when they exist; the co-existence equilibrium is locally asymptotically stable whenever the reproduction number for the drug sensitive malaria is greater than the reproduction number for the resistance malaria. Furthermore, numerical simulations of the connected two patch model (when there is movement between the patches) suggest that co-existence or competitive exclusion of the two strains can occur when the respective reproduction numbers of the two strains exceed unity. With slow movement (or low migration) between the patches, the drug sensitive strain dominates the drug resistance strain. However, with fast movement (or high migration) between the patches, the drug resistance strain dominates the drug sensitive strain.     

Lethal mutagenesis and evolutionary epidemiology

The lethal mutagenesis hypothesis states that within-host populations of pathogens can be driven to extinction when the load of deleterious mutations is artificially increased with a mutagen, and becomes too high for the population to be maintained. Although chemical mutagens have been shown to lead to important reductions in viral titres for a wide variety of RNA viruses, the theoretical underpinnings of this process are still not clearly established. A few recent models sought to describe lethal mutagenesis but they often relied on restrictive assumptions. We extend this earlier work in two novel directions. First, we derive the dynamics of the genetic load in a multivariate Gaussian fitness landscape akin to classical quantitative genetics models. This fitness landscape yields a continuous distribution of mutation effects on fitness, ranging from deleterious to beneficial (i.e. compensatory) mutations. We also include an additional class of lethal mutations. Second, we couple this evolutionary model with an epidemiological model accounting for the within-host dynamics of the pathogen. We derive the epidemiological and evolutionary equilibrium of the system. At this equilibrium, the density of the pathogen is expected to decrease linearly with the genomic mutation rate U. We also provide a simple expression for the critical mutation rate leading to extinction. Stochastic simulations show that these predictions are accurate for a broad range of parameter values. As they depend on a small set of measurable epidemiological and evolutionary parameters, we used available information on several viruses to make quantitative and testable predictions on critical mutation rates. In the light of this model, we discuss the feasibility of lethal mutagenesis as an efficient therapeutic strategy.     

An age-structured within-host HIV-1 infection model with virus-to-cell and cell-to-cell transmissions

In this paper, a within-host HIV-1 infection model with virus-to-cell and direct cell-to-cell transmission and explicit age-since-infection structure for infected cells is investigated. It is shown that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number. By analysing the corresponding characteristic equations, the local stability of an infection-free steady state and a chronic-infection steady state of the model is established. By using the persistence theory in infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if the basic reproduction number is less than unity, the infection-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the chronic-infection steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.