Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis

To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.     

The Role of Viral Infection in Pest Control: A Mathematical Study

In this paper, we propose a mathematical model of viral infection in pest control. As the viral infection induces host lysis which releases more virus into the environment, on the average ‘κ’ viruses per host, κ∈(1,∞), so the ‘virus replication parameter’ is chosen as the main parameter on which the dynamics of the infection depends. There exists a threshold value κ ₀ beyond which the infection persists in the system. Still for increasing the value of κ, the endemic equilibrium bifurcates towards a periodic solution, which essentially indicates that the viral pesticide has a density-dependent ‘numerical response’ component to its action. Investigation also includes the dependence of the process on predation of natural enemy into the system. A concluding discussion with numerical simulation of the model is also presented.     

A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay

 We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases. We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1). Received: 18 November 2000 / Published online: 28 February 2003 RID="*" ID="*" Research was partially supported by the NSERC and MITACS of Canada and a start-up fund from the College of Arts and Sciences at the University of Miami. On leave from Dalhousie University, Halifax, Nova Scotia, Canada. Current address: Department of Mathematics, Clarke College, Dubuque, Iowa 52001, USA Key words or phrases: HIV-1 – Cell-to-cell spread – Time delay – Stability – Hopf bifurcation – Periodicity     

Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R ₀ for the viral infection, and establish that the global dynamics are completely determined by the values of R ₀. If R ₀≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R ₀>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R ₀>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.     

Understanding drug resistance for monotherapy treatment of HIV infection

The purpose of this study was to investigate strategies in the monotherapy treatment of HIV infection in the presence of drug-resistant (mutant) strains. A mathematical system is developed to model resistance in HIV chemotherapy. It includes the key players in the immune response to HIV infection: virus and both uninfected CD4⁺ and infected CD4⁺ T-cell populations. We model the latent and progressive stages of the disease, and then introduce monotherapy treatment. The model is a system of differential equations describing the interaction of two distinct classes of HIV—drug-sensitive (wild type) and drug-resistant (mutant)—with lymphocytes in the peripheral blood. We then introduce chemotherapy effects. In the absence of treatment, the model produces the three types of qualitative clinical behavior—anuninfected steady state, andinfected steady state (latency), andprogression to AIDS. Simulation of treatment is provided for monotherapy, during theprogression to AIDS state, in the consideration of resistance effects. Treatment benefit is based on an increase or retention in CD4⁺ T-cell counts together with a low viral titer. We explore the following treatment approaches: an antiviral drug which reduces viral infectivity that is administered early—when the CD4⁺ T-cell count is ≥300/mm³, and late—when the CD4⁺ T-cell count is less than 300/mm³. We compare all results with data. When treatment is initiated during the progression to AIDS state, treatment prevents T-cell collapse, but gradually loses effectiveness due to drug resistance. We hypothesize that it is the careful balance of mutant and wild-type HIV strains which provides the greatest prolonged benefit from treatment. This is best achieved when treatment is initiated when the CD4⁺ T-cell counts are greater than 250/mm³, but less than 400/mm³ in this model (i.e. not too early, not too late). These results are supported by clinical data. The work is novel in that it is the first model to accurately simultate data before, during and after monotherapy treatment. Our model also provides insight into recent clinical results, as well as suggests plausible guidelines for clinical testing in the monotherapy of HIV infection.     

The onset of oscillatory dynamics in models of multiple disease strains

 We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the Takens–Bogdanov bifurcation associated with the presence of oscillatory dynamics observed by previous authors. Received: 5 December 2001 / Revised version: 5 May 2002 / Published online: 17 October 2002 Keywords or phrases: Infection – Pathogen – Epidemiology – Multiple strains – Cross-immunity – Oscillations – Dynamics – Bifurcations     

Conventional and combinatorial chlorophyll fluorescence imaging of tobamovirus-infected plants

We compared by chlorophyll (Chl) fluorescence imaging the effects of two strains of the same virus (Italian and Spanish strains of the Pepper mild mottle virus — PMMoV-I and-S, respectively) in the host plant Nicotiana benthamiana. The infection was visualized either using conventional Chl fluorescence parameters or by an advanced statistical approach, yielding a combinatorial set of images that enhances the contrast between control and PMMoV-infected plants in the early infection steps. Among the conventional Chl fluorescence parameters, the non-photochemical quenching parameter NPQ was found to be an effective PMMoV infection reporter in asymptomatic leaves of N. benthamiana, detecting an intermediate infection phase. The combinatorial imaging revealed the infection earlier than any of the standard Chl fluorescence parameters, detecting the PMMoV-S infection as soon as 4 d post-inoculation (dpi), and PMMoV-I infection at 6 dpi; the delay correlates with the lower virulence of the last viral strain.     

Competitive exclusion in a vector-host model for the dengue fever

 We study a system of differential equations that models the population dynamics of an SIR vector transmitted disease with two pathogen strains. This model arose from our study of the population dynamics of dengue fever. The dengue virus presents four serotypes each induces host immunity but only certain degree of cross-immunity to heterologous serotypes. Our model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. Dengue is in the Americas an epidemic disease and our model reproduces this kind of dynamics. We consider two viral strains and temporary cross-immunity. Our analysis shows the existence of an unstable endemic state (‘saddle’ point) that produces a long transient behavior where both dengue serotypes cocirculate. Conditions for asymptotic stability of equilibria are discussed supported by numerical simulations. We argue that the existence of competitive exclusion in this system is product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates. Received 4 December 1995; received in revised form 5 March 1996     

Polymorphisms in eight host genes associated with control of HIV replication do not mediate elite control of viral replication in SIV-infected Indian rhesus macaques

Polymorphisms in several host genes in HIV-infected individuals facilitate slow progression to AIDS. We have identified several SIV-infected Indian rhesus macaques that naturally control viral replication. We investigated whether spontaneous control of SIV in any of these animals could be explained by mutations in host genes. Such variables could confound studies of associations between MHC class I alleles and control of viral replication. We searched for polymorphisms in CCR5, CXCR6, GPR15, RANTES, IL-10, APOBEC3G, TNF-α, and TSG101 and looked for associations with decreased viral replication. We did not detect any correlations between plasma viral concentration and polymorphisms in host genes examined in this study. In addition, we did not find the polymorphisms present in humans in any of our macaques.Nucleotide sequence data reported are available in the GenBank database under accession numbers DQ890030–DQ890063, DQ887987–DQ888038, DQ902356–DQ902543, and DQ913647–DQ913733.     

The minimum effort required to eradicate infections in models with backward bifurcation

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R ₀ is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.Dedicated to Karl Peter Hadeler on the occasion of his 70th birthday.     

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.     

Effect of mutation on helper T-cells and viral population: A computer simulation model for HIV

Summary A Monte Carlo simulation is proposed to study the dynamics of helper T-cells (N _H) and viral (N _V) populations in an immune response model relevant to HIV. Cellular states are binary variables and the interactions are described by logical expressions. Viral population shows a nonmonotonic growth before reaching a constant value while helper T-cells grow to a constant after a relaxation/reaction time. Initially, the population of helper cells grows with time with a power-law, N _H ∼t ^β, before reaching the steady-state; the growth exponent β increases systematically (β ≈ 1 – 2) with the mutation rate (P _mut≈0.1–0.4). The critical recovery time (t _c) increases exponentially with the viral mutation, t _c≈Ae ^αP _mut , with α=4.52±0.29 in low mutation regime and α=15.21±1.41 in high mutation regime. The equilibrium population of helper T-cell declines slowly with P _mut and collapses at ∼ 0.40; the viral population exhibits a reverse trend, i.e., a slow increase before the burst around the same mutation regime.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multiple transmission pathways exist for many waterborne diseases, including cholera, Giardia, Cryptosporidium, and Campylobacter. Theoretical work exploring the effects of multiple transmission pathways on disease dynamics is incomplete. Here, we consider a simple ODE model that extends the classical SIR framework by adding a compartment (W) that tracks pathogen concentration in the water. Infected individuals shed pathogen into the water compartment, and new infections arise both through exposure to contaminated water, as well as by the classical SIR person–person transmission pathway. We compute the basic reproductive number (ℛ₀), epidemic growth rate, and final outbreak size for the resulting “SIWR” model, and examine how these fundamental quantities depend upon the transmission parameters for the different pathways. We prove that the endemic disease equilibrium for the SIWR model is globally stable. We identify the pathogen decay rate in the water compartment as a key parameter determining when the distinction between the different transmission routes in the SIWR model is important. When the decay rate is slow, using an SIR model rather than the SIWR model can lead to under-estimates of the basic reproductive number and over-estimates of the infectious period.     

Fuzzy modeling in symptomatic HIV virus infected population

This paper introduces a model for the evolution of positive HIV population and manifestation of AIDS (acquired immunideficiency syndrome). The focus is on the nature of the transference rate of HIV to AIDS. Expert knowledge indicates that the transference rate is uncertain and depends strongly on the viral load and the CD4+ level of the infected individuals. Here, we suggest to view the transference rate as a fuzzy set of the viral load and CD4+ level values. In this case the dynamic model results in a fuzzy model that preserves the biological meaning and nature of the transference rate λ. Its behavior fits the natural history of HIV infection reported in the medical science domain. The paper also includes a comparison between the fuzzy model and a classic Anderson’s model using data reported in the literature.     

Bacterial Growth Rate and Marine Virus–Host Dynamics

Abstract The dynamics of a marine virus–host system were investigated at different steady state growth rates in chemostat cultures and the data were analyzed using a simple model. The virus–host interactions showed strong dependence on host cell growth rate. The duration of the infection cycle and the virus burst size were found to depend on bacterial growth rate, and the rate of cell lysis and virus production were positively correlated with steady state growth rate in the cultures (r ² > 0.96, p < 0.05). At bacterial growth rates of 0.02 to 0.10 h⁻¹ in the chemostats the virus burst size increased from 12 ± 4 to 56 ± 4, and the latent period decreased from 2.0 to 1.7 h. Resistant clones of the host strain were present in the cultures from the beginning of the experiment and replaced the sensitive host cells following viral lysis in the cultures. Regrowth of resistant cells correlated significantly (r ²= 1.000, p < 0.02) with the lysis rate of sensitive cells, indicating that release of viral lysates stimulated growth of the non-infected, resistant cells. The constructed model was suitable for simulating the observed dynamics of the sensitive host cells, viruses and resistant clones in the cultures. The model was therefore used in an attempt to predict the dynamics of this virus–host interaction in a natural marine environment during a certain set of growth conditions. The simulation indicated that a steady state relationship between the specific viruses and sensitive and resistant bacterial clones may occur at densities that are reasonable to assume for natural environments. The study demonstrates that basic characterization and modeling of specific virus–host interactions may improve our understanding of the behavior of bacteria and viruses in natural systems. Received: 12 November 1999; Accepted: 2 May 2000; Online Publication: 11 August 2000     

Age-dependent pathogenesis of murine gammaherpesvirus 68 infection of the central nervous system

Gammaherpesvirus infection of the central nervous system (CNS) has been linked to various neurological diseases, including meningitis, encephalitis, and multiple sclerosis. However, little is known about the interactions between the virus and the CNS in vitro or in vivo. Murine gammaherpesvirus 68 (MHV-68 or _γHV-68) is genetically related and biologically similar to human gammaherpesviruses, thereby providing a tractable animal model system in which to study both viral pathogenesis and replication. In the present study, we show the successful infection of cultured neuronal cells, microglia, and astrocytes with MHV-68 to various extents. Upon intracerebroventricular injection of a recombinant virus (MHV-68/LacZ) into 4–5-week-old and 9–10-week-old mice, the 4–5-week-old mice displayed high mortality within 5–7 days, while the majority of the 9–10-week-old mice survived until the end of the experimental period. Until a peak at 3–4 days post-infection, viral DNA replication and gene expression were similar in the brains of both mouse groups, but only the 9–10-week-old mice were able to subdue viral DNA replication and gene expression after 5 days post-infection. Pro-inflammatory cytokine mRNAs of tumor necrosis factor-α, interleukin 1β, and interleukin 6 were highly induced in the brains of the 4–5-week-old mice, suggesting their possible contributions as neurotoxic factors in the agedependent control of MHV-68 replication of the CNS. These authors contributed equally to this work.     

Global dynamics of a chemostat competition model with distributed delay

 We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctuation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the population that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in L ¹-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentrations are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. Received: 9 August 1997 / Revised version: 2 July 1998     

Transient dynamics and early diagnostics in infectious disease

To date, mathematical models of the dynamics of infectious disease have consistently focused on understanding the long-term behavior of the interacting components, where the steady state solutions are paramount. However for most acute infections, the long-term behavior of the pathogen population is of little importance to the host and population health. We introduce the notion of transient pathology, where the short-term dynamics of interaction between the immune system and pathogens is the principal focus. We identify the amplifying effect of the absence of a fully operative immune system on the pathogenesis of the initial inoculum, and its implication for the acute severity of the infection. We then formalize the underlying dynamics, and derive two measures of transient pathogenicity: the peak of infection (maximum pathogenic load) and the time to peak of infection, both crucial to understanding the early dynamics of infection and its consequences for early intervention. Received: 25 January 2000 / Revised version: 30 November 2000 / Published online: 12 October 2001     

Empirical Modelling of the Population Dynamics of a Small Population of the Threespine Stickleback, <Emphasis Type="Italic">Gasterosteus aculeatus</Emphasis>

Synopsis We estimated the abundance of a small population of threespine stickleback, Gasterosteus aculeatus, by mark-recapture over a 21 year period. Length-frequency analysis showed that the population in October consisted almost entirely of young-of-the-year. The per capita annual rate of increase was inversely related to abundance in October. Time series analysis suggested the presence of a cycle of abundance with a period of about 6 years. There was a significant inverse relationship between abundance in year t and in year t + 3. A simple, empirical, deterministic model based on this inverse relationship and run for 100 years predicted that population abundance showed damped oscillations leading to a stable abundance. When a stochastic component was added to the model, seven of 10 runs included a component with a period of about 6 years. These simulations suggest that the dynamics of this population are driven by an interaction between a deterministic (density-dependent) component and a stochastic component. We compare these results with time series of abundance of threespine stickleback obtained from the Thames Estuary in south-east England and Loch Lomond in Scotland.