Seven challenges for model-driven data collection in experimental and observational studies.

Infectious disease models are both concise statements of hypotheses and powerful techniques for creating tools from hypotheses and theories. As such, they have tremendous potential for guiding data collection in experimental and observational studies, leading to more efficient testing of hypotheses and more robust study designs. In numerous instances, infectious disease models have played a key role in informing data collection, including the Garki project studying malaria, the response to the 2009 pandemic of H1N1 influenza in the United Kingdom and studies of T-cell immunodynamics in mammals. However, such synergies remain the exception rather than the rule; and a close marriage of dynamic modeling and empirical data collection is far from the norm in infectious disease research. Overcoming the challenges to using models to inform data collection has the potential to accelerate innovation and to improve practice in how we deal with infectious disease threats.

The wMel Wolbachia strain blocks dengue and invades caged Aedes aegypti populations.

Dengue fever is the most important mosquito-borne viral disease of humans with more than 50 million cases estimated annually in more than 100 countries. Disturbingly, the geographic range of dengue is currently expanding and the severity of outbreaks is increasing. Control options for dengue are very limited and currently focus on reducing population abundance of the major mosquito vector, Aedes aegypti. These strategies are failing to reduce dengue incidence in tropical communities and there is an urgent need for effective alternatives. It has been proposed that endosymbiotic bacterial Wolbachia infections of insects might be used in novel strategies for dengue control. For example, the wMelPop-CLA Wolbachia strain reduces the lifespan of adult A. aegypti mosquitoes in stably transinfected lines. This life-shortening phenotype was predicted to reduce the potential for dengue transmission. The recent discovery that several Wolbachia infections, including wMelPop-CLA, can also directly influence the susceptibility of insects to infection with a range of insect and human pathogens has markedly changed the potential for Wolbachia infections to control human diseases. Here we describe the successful transinfection of A. aegypti with the avirulent wMel strain of Wolbachia, which induces the reproductive phenotype cytoplasmic incompatibility with minimal apparent fitness costs and high maternal transmission, providing optimal phenotypic effects for invasion. Under semi-field conditions, the wMel strain increased from an initial starting frequency of 0.65 to near fixation within a few generations, invading A. aegypti populations at an accelerated rate relative to trials with the wMelPop-CLA strain. We also show that wMel and wMelPop-CLA strains block transmission of dengue serotype 2 (DENV-2) in A. aegypti, forming the basis of a practical approach to dengue suppression.

Quantitative models for germination and infection of Pseudoperonospora cubensis in response to temperature and duration of leaf wetness.

The influence of temperature and leaf wetness duration on germination of sporangia and infection of cantaloupe leaves by Pseudoperonospora cubensis was examined in three independent controlled-environment experiments by inoculating plants with a spore suspension and exposing them to a range of leaf wetness durations (2 to 24 h) at six fixed temperatures (5 to 30 degrees C). Germination of sporangia was assessed at the end of each wetness period and infection was evaluated from assessments of disease severity 5 days after inoculation. Three response surface models based on modified forms of the Weibull function were evaluated for their ability to describe germination of sporangia and infection in response to temperature and leaf wetness duration. The models estimated 15.7 to 17.3 and 19.5 to 21.7 degrees C as the optimum temperature (t) range for germination and infection, respectively, with little germination or infection at 5 or 30 degrees C. For wetness periods of 4 to 8 h, a distinct optimum for infection was observed at t = 20 degrees C but broader optimum curves resulted from wetness periods >8 h. Model 1 of the form f(w,t) = f(t) x (1 - exp{-[B x w](D)}) resulted in smaller asymptotic standard errors and yielded higher correlations between observed and predicted germination and infection data than either model 2 of the form f(w,t) = A{1 - exp[- f(t) x (w - C)](D)} or model 3 of the form f(w,t) = [1 - exp{-(B x w)(2)}]/cosh[(t - F)G/2]. Models 1 and 2 had nonsignificant lack-of-fit test statistics for both germination and infection data, whereas a lack-of-fit test was significant for model 3. The models accounted for approximately 87% (model 3) to 98% (model 1) of the total variation in the germination and infection data. In the validation of the models using data generated with a different isolate of P. cubensis, slopes of the regression line between observed and predicted germination and infection data were not significantly different (P > 0.2487) and correlation coefficients between observed and predicted values were high (r(2) > 0.81). Models 1 and 2 were used to construct risk threshold charts that can be used to estimate the potential risk for infection based on observed or forecasted temperature and leaf wetness duration.

Stochastic and deterministic models for agricultural production networks.

An approach to modeling the impact of disturbances in an agricultural production network is presented. A stochastic model and its approximate deterministic model for averages over sample paths of the stochastic system are developed. Simulations, sensitivity and generalized sensitivity analyses are given. Finally, it is shown how diseases may be introduced into the network and corresponding simulations are discussed.

Infection dynamics on scale-free networks.

We discuss properties of infection processes on scale-free networks, relating them to the node-connectivity distribution that characterizes the network. Considering the epidemiologically important case of a disease that confers permanent immunity upon recovery, we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual's degree of connectivity within the population. As in an earlier study [R. Pastor-Satorras and A. Vesipignani, Phys. Rev. Lett. 86, 3200 (2001); Phys. Rev. E. 63, 006117 (2001)] for an infection that did not confer immunity upon recovery, the epidemic process--in contrast with many traditional epidemiological models--does not exhibit threshold behavior, and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale-free network. Finally, we discuss effects that arise from finite population sizes, showing that networks of finite size do exhibit threshold effects: infections cannot spread for arbitrarily low transmission probabilities.

Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics.

Most mathematical models used to study the epidemiology of childhood viral diseases, such as measles, describe the period of infectiousness by an exponential distribution. The effects of including more realistic descriptions of the infectious period within SIR (susceptible/infectious/recovered) models are studied. Less dispersed distributions are seen to have two important epidemiological consequences. First, less stable behaviour is seen within the model: incidence patterns become more complex. Second, disease persistence is diminished: in models with a finite population, the minimum population size needed to allow disease persistence increases. The assumption made concerning the infectious period distribution is of a kind routinely made in the formulation of mathematical models in population biology. Since it has a major effect on the central issues of population persistence and dynamics, the results of this study have broad implications for mathematical modellers of a wide range of biological systems.

The dependence of viral parameter estimates on the assumed viral life cycle: limitations of studies of viral load data.

Estimation of viral parameters, such as the basic reproductive number (R0) and infected cell life span, is central to the quantitative study of the within-host dynamics of viral diseases such as human immunodeficiency virus, hepatitis B or hepatitis C. As these parameters can rarely be determined directly, they are usually estimated indirectly by fitting mathematical models to viral load data. This paper investigates how parameter estimates obtained by such procedures depend on the assumptions made concerning the viral life cycle. It finds that estimates of the basic reproductive number obtained using viral load data collected during the initial stages of infection can depend quite sensitively on these assumptions. The use of models which neglect the intracellular delay before virion production can lead to severe underestimates of R0 and, hence, to overly optimistic predictions of how efficacious treatment must be in order to prevent or eradicate the disease. These results are also of importance for attempts at estimating R0 from similar epidemiological data as there is a correspondence between within-host and between-host models. Estimates of the life span of infected cells obtained from viral load data collected during drug treatment studies also depend on the assumptions made in modelling the virus life cycle. The use of more realistic descriptions of the life cycle is seen to increase estimates of infected cell life span, in addition to providing a new explanation for the shoulder phase seen during drug treatment. This study highlights the limitations of what can be learnt by fitting mathematical models to infectious disease data without detailed independent knowledge of the life cycle of the infectious agent.

Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods.

Most mathematical models used to understand the dynamical patterns seen in the incidence of childhood viral diseases, such as measles, employ a simple, but epidemiologically unrealistic, description of the infection and recovery process. The inclusion of more realistic descriptions of the recovery process is shown to cause a significant destabilization of the model. When there is seasonal variation in discase transmission this destabilization leads to the appearance of complex dynamical patterns with much lower levels of seasonality than previously predicted. More generally this study illustrates how detailed dynamical properties of a model may depend in an important way on the assumptions made in the formulation of the model.

The influence of HLA class I alleles and heterozygosity on the outcome of human T cell lymphotropic virus type I infection.

The inflammatory disease human T cell lymphotropic virus type I (HTLV-I)-associated myelopathy (HAM/TSP) occurs in only 1-2% of HTLV-I-infected individuals and is associated with a high provirus load of HTLV-I. We hypothesize that a person's risk of developing HAM/TSP depends upon the efficiency of their immune response to the virus, which differs between individuals because of polymorphism in genes that influence this response. Previously we showed that the possession of HLA-A*02 was associated with a lower risk of HAM/TSP, and with a lower provirus load in healthy carriers of HTLV-I. However, HLA-A*02 did not account for all the observed difference in the risk of HAM/TSP. Here we present evidence, in the same study population in Japan, that HLA-Cw*08 was also associated with disease protection (probability value, two-tailed test = 0.002) and with a lower proviral load in healthy carriers. Possession of the A*02 and/or Cw*08 genes prevented 36% of potential HAM/TSP cases. In contrast, HLA-B*5401 was associated with a higher susceptibility to HAM/TSP (probability value, two-tailed test = 0.0003) and with a higher provirus load in HAM/TSP patients. At a given provirus load, B*5401 appeared to increase the risk of disease. The fraction of HAM/TSP cases attributable to B*5401 was 17%. Furthermore, individuals who were heterozygous at all three HLA class I loci have a lower HTLV-I provirus load than those who were homozygous at one or more loci. These results are consistent with the proposal that a strong class I-restricted CTL response to HTLV-I reduces the proviral load and hence the risk of disease.

Local stability analysis of spatially homogeneous solutions of multi-patch systems.

Multi-patch systems, in which several species interact in patches connected by dispersal, offer a general framework for the description and analysis of spatial ecological systems. This paper describes how to analyse the local stability of spatially homogeneous solutions in such systems. The spatial arrangement of the patches and their coupling is described by a matrix. For a local stability analysis of spatially homogeneous solutions it turns out to be sufficient to know the eigenvalues of this matrix. This is shown for both continuous and discrete time systems. A bookkeeping scheme is presented that facilitates stability analyses by reducing the analysis of a k-species, n-patch system to that of n uncoupled k-dimensional single-patch systems. This is demonstrated in a worked example for a chain of patches. In two applications the method is then used to analyse the stability of the equilibrium of a predator-prey system with a pool of dispersers and of the periodic solutions of the spatial Lotka-Volterra model.

Fluctuations in HIV-1 viral load are correlated to CD4+ T-lymphocyte count during the natural course of infection.

Viral load fluctuates during the natural course of asymptomatic HIV-1 infection. It is often assumed that these fluctuations are random around a set point or underlying growth trend. Using longitudinal data, we tested whether fluctuations in viral load can be better explained by changes in CD4+ T-cell count than by a set point or trend of exponential growth. The correspondence between viral load and CD4+ T-cell count could be described by a simple mathematical relation. Using a bootstrapping approach, the hypothesis that viral load fluctuations are random around a set point was rejected with p < .00005. The hypothesis that viral load fluctuations are random around a trend of exponential growth was rejected with p < .005. Viral load data was explained better by changes in CD4+ T-cell counts than by a set point or by a trend of exponential growth. The implications of this finding for improved prognostication are discussed.

Direct quantitation of rapid elimination of viral antigen-positive lymphocytes by antiviral CD8(+) T cells in vivo.

Lysis of infected cells by CD8(+) T cells is an important mechanism for the control of virus infections, but remains difficult to quantify in vivo. Here, we study the elimination kinetics of viral antigen-positive lymphocytes by antiviral CD8(+) T cells using flow cytometry and mathematical analysis. In mice acutely infected with lymphocytic choriomeningitis virus, more than 99.99 % of target cells were eliminated each day, corresponding to a half-life of 1.4 h. Even in mice exposed to virus 300 days previously, and with no ex vivo killing activity, 84 % of the target cells were eliminated per day. Unexpectedly, the elimination kinetics of antigen-positive lymphocytes was not significantly impaired in mice deficient in either perforin-, CD95 ligand- or TNF-mediated cytotoxicity. For viruses with a particular tropism for lymphocytes, such as Epstein-Barr virus or HIV, our results illustrate how effectively CD8(+) T cell-mediated elimination of target cells can potentially contribute to virus control and immunosuppression.

A simple relationship between viral load and survival time in HIV-1 infection.

Despite important recent insights into the short-term dynamics of HIV-1 infection, our understanding of the long-term pathogenesis of AIDS remains unclear. Using an approach that places rapid progressors, typical progressors, and nonprogressors on a single clinical spectrum of disease progression, we quantitate the previously reported relationship between viral load and survival time. We introduce the concept of viral constant, present evidence that this quantity is conserved across patients, and explore the immunopathological implications of this finding. We conclude with a quantitative approach for assessing the benefits of a given regime of antiviral therapy.

HLA alleles determine human T-lymphotropic virus-I (HTLV-I) proviral load and the risk of HTLV-I-associated myelopathy.

The risk of disease associated with persistent virus infections such as HIV-I, hepatitis B and C, and human T-lymphotropic virus-I (HTLV-I) is strongly determined by the virus load. However, it is not known whether a persistent class I HLA-restricted antiviral cytotoxic T lymphocyte (CTL) response reduces viral load and is therefore beneficial or causes tissue damage and contributes to disease pathogenesis. HTLV-I-associated myelopathy (HAM/TSP) patients have a high virus load compared with asymptomatic HTLV-I carriers. We hypothesized that HLA alleles control HTLV-I provirus load and thus influence susceptibility to HAM/TSP. Here we show that, after infection with HTLV-I, the class I allele HLA-A*02 halves the odds of HAM/TSP (P < 0.0001), preventing 28% of potential cases of HAM/TSP. Furthermore, HLA-A*02(+) healthy HTLV-I carriers have a proviral load one-third that (P = 0.014) of HLA-A*02(-) HTLV-I carriers. An association of HLA-DRB1*0101 with disease susceptibility also was identified, which doubled the odds of HAM/TSP in the absence of the protective effect of HLA-A*02. These data have implications for other persistent virus infections in which virus load is associated with prognosis and imply that an efficient antiviral CTL response can reduce virus load and so prevent disease in persistent virus infections.

Dynamics of macrophage and T cell infection by HIV.

We analyse mathematical models comparing the in vivo dynamics of macrophage- and T cell infection by HIV. Experiments suggest that HIV can only replicate in activated T cells whereas cell activation may not be required for successful replication in macrophages. These assumptions lead to fundamentally different conditions required to establish a persistent infection in the two cell types. While persistent replication in macrophages is achieved if the basic reproductive ratio of the virus, R0, exceeds unity, the establishment of T cell infection may depend on a complex balance between host and viral parameters as well as initial conditions. More specifically, the replication rate of HIV needs to lie above a threshold level and the immune responsiveness of the host below a certain threshold for persistent T cell infection to be possible. In addition, initial virus load has to be intermediate and the initial abundance of CTLs low. Mathematical models predict that macrophage infection may be essential for the successful establishment of HIV in the primary phase of the infection. Acting as a reservoir, they allow the virus to evolve towards increased replication kinetics as well as away from immune recognition, thus paving the way for the rise of exclusively T cell tropic strains using the CXCR4-coreceptor.

Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy.

Although highly active antiretroviral therapy (HAART) in the form of triple combinations of drugs including protease inhibitors can reduce the plasma viral load of some HIV-1-infected individuals to undetectable levels, it is unclear what the effects of these regimens are on latently infected CD4+ T cells and what role these cells play in the persistence of HIV-1 infection in individuals receiving such treatment. The present study demonstrates that highly purified CD4+ T cells from 13 of 13 patients receiving HAART with an average treatment time of 10 months and with undetectable (<500 copies HIV RNA/ml) plasma viremia by a commonly used bDNA assay carried integrated proviral DNA and were capable of producing infectious virus upon cellular activation in vitro. Phenotypic analysis of HIV-1 produced by activation of latently infected CD4+ T cells revealed the presence in some patients of syncytium-inducing virus. In addition, the presence of unintegrated HIV-1 DNA in infected resting CD4+ T cells from patients receiving HAART, even those with undetectable plasma viremia, suggests persistent active virus replication in vivo.

The extent of early viral replication is a critical determinant of the natural history of simian immunodeficiency virus infection.

Different patterns of viral replication correlate with the natural history of disease progression in humans and macaques infected with human immunodeficiency virus type 1 (HIV-1) and simian immunodeficiency virus (SIV), respectively. However, the viral and host factors influencing these patterns of viral replication in vivo are poorly understood. We intensively studied viral replication in macaques receiving identical inocula of SIV. Marked differences in viral replication patterns were apparent within the first week following inoculation, a time prior to the development of measurable specific immune effector responses to viral antigens. Plasma viral RNA levels measured on day 7 postinoculation correlated with levels measured in the postacute phase of infection. Differences in the susceptibility of host cells from different animals to in vitro SIV infection correlated with the permissiveness of the animals for early in vivo viral replication and hence with the postacute set point level of plasma viremia. These results suggest that host factors that exert their effects prior to full development of specific immune responses are critical in establishing the in vivo viral replication pattern and associated clinical course in subjects infected with SIV and, by extension, with HIV-1.

Viral dynamics of primary viremia and antiretroviral therapy in simian immunodeficiency virus infection.

Mathematical modeling of viral replication dynamics, based on sequential measurements of levels of virion-associated RNA in plasma during antiretroviral treatment, has led to fundamental new insights into human immunodeficiency virus type 1 pathogenesis. We took advantage of the simian immunodeficiency virus (SIV)-infected macaque model to perform detailed measurements and mathematical modeling during primary infection and during treatment of established infection with the antiretroviral drug (R)-9-(2-phosphonylmethoxypropyl)adenine (PMPA). The calculated clearance half-life for productively infected cells during resolution of the peak viremia of primary infection was on the order of 1 day, with slightly shorter clearance half-lives calculated during PMPA treatment. Viral reproduction rates upon discontinuation of PMPA treatment after 2 weeks were approximately twofold greater than those obtained just prior to initiation of treatment in the same animals, likely reflecting accumulation of susceptible target cells during treatment. The basic reproductive ratio (R0) for the spread of SIV infection in vivo, which represents the number of productively infected cells derived from each productively infected cell at the beginning of infection, was also estimated. This parameter quantifies the extent to which antiviral therapy or vaccination must limit the initial spread of virus to prevent establishment of chronic disseminated infection. The results thus provide an important guide for efforts to develop vaccines against SIV and, by extension, human immunodeficiency virus.

Spatial heterogeneity in epidemic models.

Spatial heterogeneity is believed to play an important role in the persistence and dynamics of epidemics of childhood diseases because asynchrony between populations within different regions allows global persistence, even if the disease dies out locally. A simple multi-patch (metapopulation) model for spatial heterogeneity in epidemics is analysed and we examine conditions under which patches become synchronized. We show that the patches in non-seasonal deterministic models often oscillate in phase for all but the weakest between patch coupling. Synchronization is also seen for stochastic models, although slightly stronger coupling is needed to overcome the random effects. We demonstrate that the inclusion of seasonal forcing in deterministic models can lead to the maintenance of phase differences between patches. Complex dynamic behaviour is observed in the seasonally forced spatial model, along with the coexistence of many different behaviours. Compared to the non-spatial model, chaotic solutions are observed for weaker seasonal forcing; these solutions have a more realistic minimum number of infectives.