Epidemiological models for the spread of anti-malarial resistance

BACKGROUND

The spread of drug resistance is making malaria control increasingly difficult. Mathematical models for the transmission dynamics of drug sensitive and resistant strains can be a useful tool to help to understand the factors that influence the spread of drug resistance, and they can therefore help in the design of rational strategies for the control of drug resistance.

METHODS

We present an epidemiological framework to investigate the spread of anti-malarial resistance. Several mathematical models, based on the familiar Macdonald-Ross model of malaria transmission, enable us to examine the processes and parameters that are critical in determining the spread of resistance.

RESULTS

In our simplest model, resistance does not spread if the fraction of infected individuals treated is less than a threshold value; if drug treatment exceeds this threshold, resistance will eventually become fixed in the population. The threshold value is determined only by the rates of infection and the infectious periods of resistant and sensitive parasites in untreated and treated hosts, whereas the intensity of transmission has no influence on the threshold value. In more complex models, where hosts can be infected by multiple parasite strains or where treatment varies spatially, resistance is generally not fixed, but rather some level of sensitivity is often maintained in the population.

CONCLUSIONS

The models developed in this paper are a first step in understanding the epidemiology of anti-malarial resistance and evaluating strategies to reduce the spread of resistance. However, specific recommendations for the management of resistance need to wait until we have more data on the critical parameters underlying the spread of resistance: drug use, spatial variability of treatment and parasite migration among areas, and perhaps most importantly, cost of resistance.

Recruitment times, proliferation, and apoptosis rates during the CD8(+) T-cell response to lymphocytic choriomeningitis virus

The specific CD8(+) T-cell response during acute lymphocytic choriomeningitis virus (LCMV) infection of mice is characterized by a rapid proliferation phase, followed by a rapid death phase and long-term memory. In BALB/c mice the immunodominant and subdominant CD8(+) responses are directed against the NP118 and GP283 epitopes. These responses differ mainly in the magnitude of the epitope-specific CD8(+) T-cell expansion. Using mathematical models together with a nonlinear parameter estimation procedure, we estimate the parameters describing the rates of change during the three phases and thereby establish the differences between the responses to the two epitopes. We find that CD8(+) cell proliferation begins 1 to 2 days after infection and occurs at an average rate of 3 day(-1), reaching the maximum population size between days 5 and 6 after immunization. The 10-fold difference in expansion to the NP118 and GP283 epitopes can be accounted for in our model by a 3.5-fold difference in the antigen concentration of these epitopes at which T-cell stimulation is half-maximal. As a consequence of this 3.5-fold difference in the epitope concentration needed for T-cell stimulation, the rates of activation and proliferation of T cells specific for the two epitopes differ during the response and in combination can account for the large difference in the magnitude of the response. After the peak, during the death phase, the population declines at a rate of 0.5 day(-1), i.e., cells have an average life time of 2 days. The model accounts for a memory cell population of 5% of the peak population size by a reversal to memory of 1 to 2% of the activated cells per day during the death phase.

Why we don't get sick: the within-host population dynamics of bacterial infections

To pathogenic microparasites (viruses, bacteria, protozoa, or fungi), we and other mammals (living organisms at large) are little more than soft, thin-walled flasks of culture media. Almost every time we eat, brush our teeth, scrape our skin, have sex, get bitten by insects, and inhale, we are confronted with populations of microbes that are capable of colonizing the mucosa lining our orifices and alimentary tract and proliferating in fluids and cells within us. Nevertheless, we rarely get sick, much less succumb to these infections. The massive numbers of bacteria and other micro- and not-so-micro organisms that abound and replicate in our alimentary tract and cover our skin and the mucosa lining our orifices normally maintain their communities in seemingly peaceful coexistence with the somatic cells that define us. Why don't these microbes invade and proliferate in the culture media within the soft, thin-walled flask that envelops us? Why don't they cause disease and lead to our rapid demise?

Effects of antiviral usage on transmission dynamics of herpes simplex virus type 1 and on antiviral resistance: predictions of mathematical models

Herpes simplex virus type 1 (HSV-1) causes recurrent herpes labialis (RHL), a common disease afflicting up to 40% of adults worldwide. Mathematical models are used to analyze the effect of antiviral treatment on the transmission of, and the prevalence of drug resistance in, HSV-1 in the United States. Three scenarios are analyzed: no antiviral use, the current level of use, and a substantial increase in nucleoside analogue use, such as might occur if topical penciclovir were available over-the-counter for the treatment of RHL. A basic model predicts that present level of nucleoside analogue use has a negligible effect on HSV-1 transmission and that even if use of topical penciclovir for (RHL) increased substantially, the overall prevalence of infectious HSV-1 is unlikely to be reduced by more than 5%. An expanded model, which allows for acquired resistance and includes immunocompromised hosts and other more realistic features, predicts that current antiviral use is unlikely to lead to any noticeable increase in resistance. If antiviral use increases, the resulting rise in resistance in the population will depend primarily on the probability that immunocompetent hosts will acquire permanent resistance upon treatment. This probability is known to be small, but its exact value remains uncertain. If acquired resistance occurs less than once per 2,500 treated episodes, then in the community at large, the frequency of HSV-1 resistance is predicted to increase slowly, if at all (remaining below 0.5% for >50 years), even with extensive nucleoside analogue use. If acquired resistance emerges in 1 of 625 treated episodes (the maximum of an approximate 95% confidence interval derived from the results of several studies of resistance in treated hosts), then the prevalence of infection with resistant HSV-1 could rise from about 0.2% to 1.5 to 3% within 50 years. The limitations of existing data on acquired resistance and the potential impact of acquired resistance if it occurs are discussed, and strategies are suggested for enhancing information on acquired resistance. The predictions of this model contrast with the more rapid increases in antimicrobial resistance anticipated by models and observed for other pathogenic bacteria and viruses. The reasons for these contrasting predictions are discussed.

Modeling immune responses with handling time

Simple predator-prey type models have brought much insight into the dynamics of both nonspecific and antigen-specific immune responses. However, until now most attention has been focused on examining how the dynamics of interactions between the parasite and the immune system depends on the nature of the function describing the rate of activation or proliferation of immune cells in response to the parasite. In this paper we focus on the term describing the killing of the parasite by cell-mediated immune responses. This term has previously been assumed to be a simple mass-action term dependent solely on the product of the densities of the parasite and the immune cells and does not take into account a handling time (which we define as the time of interaction between an immune cell and its target, during which the immune cell cannot interact with and/or destroy additional targets). We show how the handling time (i) can be incorporated into simple models of nonspecific and specific immunity and (ii) how it affects the dynamics of both nonspecific and antigen-specific immune responses, and in particular the ability of the immune response to control the infection.

Models of immune memory: on the role of cross-reactive stimulation, competition, and homeostasis in maintaining immune memory

There has been much debate on the contribution of processes such as the persistence of antigens, cross-reactive stimulation, homeostasis, competition between different lineages of lymphocytes, and the rate of cell turnover on the duration of immune memory and the maintenance of the immune repertoire. We use simple mathematical models to investigate the contributions of these various processes to the longevity of immune memory (defined as the rate of decline of the population of antigen-specific memory cells). The models we develop incorporate a large repertoire of immune cells, each lineage having distinct antigenic specificities, and describe the dynamics of the individual lineages and total population of cells. Our results suggest that, if homeostatic control regulates the total population of memory cells, then, for a wide range of parameters, immune memory will be long-lived in the absence of persistent antigen (T1/2 > 1 year). We also show that the longevity of memory in this situation will be insensitive to the relative rates of cross-reactive stimulation, the rate of turnover of immune cells, and the functional form of the term for the maintenance of homeostasis.

Mathematical models of parasite responses to host immune defences

We examine the evolution of microparasites in response to the immune system of vertebrate hosts. We first describe a simple model for an acute infection. This model suggests that the within-host dynamics of the microparasite will be a 'race' between parasite multiplication and a clonally expanding response by the host immune system, resulting either in immune-mediated clearance or host death. In this very simple model, in which there is only a single parasite and host genotype, maximum transmission is obtained by parasites with intermediate rates of growth (and virulence). We examine how these predictions depend on key assumptions about the parasite and the host, and consider how this model may be expanded to incorporate the effect of additional complexities such as host-parasite co-evolution, host polymorphism, and multiple infections.

The population genetics of antibiotic resistance. II: Analytic theory for sustained populations of bacteria in a community of hosts

The phenomenon of antibiotic resistance is of practical importance and theoretical interest. As a foundation for further studies by simulation, experiment, and observation, we here develop a mathematical model for the dynamics of resistance among the bacteria resident in a population of hosts. The model incorporates the effects of natural selection within untreated hosts, colonization by bacteria from the environment, and the rapid increase of resistance in hosts who receive antibiotics. We derive explicit formulas for the distribution of resistance among hosts and for the rise or fall of resistance when the frequency of treatment is changed.

Humoral immunity due to long-lived plasma cells

Conventional models suggest that long-term antibody responses are maintained by the continuous differentiation of memory B cells into antibody-secreting plasma cells. This is based on the notion that plasma cells are short-lived and need to be continually replenished by memory B cells. We examined the issue of plasma cell longevity by following the persistence of LCMV-specific antibody and plasma cell numbers after in vivo depletion of memory B cells and by adoptive transfer of virus-specific plasma cells into naive mice. The results show that a substantial fraction of plasma cells can survive and continue to secrete antibody for extended periods of time (>1 year) in the absence of any detectable memory B cells. This study documents the existence of long-lived plasma cells and demonstrates a new mechanism by which humoral immunity is maintained.

Resistance to antimicrobial chemotherapy: a prescription for research and action

The growing problem of resistance to antimicrobial chemotherapy was discussed by participants at the February 1995 workshop at Emory University on population biology, evolution, and control of infectious diseases. They discussed the nature and source of this problem and identified areas of research in which information is lacking for the development of programs to control of the emergence and spread of resistant bacteria. Particular attention was given to theoretical (mathematical modeling) and empirical studies of the within and between-host population biology (epidemiology) and the evolution of microbial resistance to chemotherapeutic agents. Suggestions were made about the kinds of models and data needed, and the procedures that could be employed to stem the ascent and dissemination of resistant bacteria. This article summarizes the observations and recommendations made at the 1995 meeting and in the correspondence between participants that followed. It concludes with an update on the theoretical and empirical research on the between- and within-host population biology and evolution of resistance to antimicrobial chemotherapy most of which has been done since that meeting.

Modeling T-cell proliferation: an investigation of the consequences of the Hayflick limit

Somatic cells, including immune cells such as T-cells have a limited capacity for proliferation and can only replicate for a finite number of generations (known as the Hayflick limit) before dying. In this paper we use mathematical models to investigate the consequences of introducing a Hayflick limit on the dynamics of T-cells stimulated with specific antigen. We show that while the Hayflick limit does not alter the dynamics of T-cell response to antigen over the short term, it may have a profound effect on the long-term immune response. In particular we show that over the long term the Hayflick limit may be important in determining whether an immune response can be maintained to a persistent antigen (or parasite). The eventual outcome is determined by the magnitude of the Hayflick limit, the extent to which antigen reduces the input of T-cells from the thymus, and the rate of antigen-induced proliferation of T-cells. Counter to what might be expected we show that the persistence of an immune response (immune memory) requires the density of persistent antigen to be less than a defined threshold value. If the amount of persistent antigen (or parasite) is greater than this threshold value then immune memory will be relatively short lived. The consequences of this threshold for persistent mycobacterial and HIV infections and for the generation of vaccines are discussed.

The population genetics of antibiotic resistance

Mathematical models are used to ascertain the relationship between the incidence of antibiotic treatment and the frequency of resistant bacteria in the commensal flora of human hosts, as well as the rates at which these frequencies would decline following a cessation of antibiotic use. Recent studies of the population biology of plasmid-encoded and chromosomal antibiotic resistance are reviewed for estimates of the parameters of these models and to evaluate other factors contributing to the fate of antibiotic-resistant bacteria in human hosts. The implications of these theoretical and empirical results to the future of antibacterial chemotherapy are discussed.

Recent developments in theories of pathogenesis of AIDS

Experimental and theoretical progress in HIV research includes an improved resolution of the spatial heterogeneity and the dynamics (time course and turnover rates) of virus and CD4+ cells. Some of these advances have resulted from the joint work of experimental and theoretical groups, demonstrating that interdisciplinary research can be fruitful.

T-cell homeostasis, competition, and drift: AIDS as HIV-accelerated senescence of the immune repertoire

The observation that the density of CD8+ T-lymphocytes increases as the density of CD4+ T-cells declines in adult HIV-1/AIDS patients, together with evidence that the total density of T-cells is regulated (homeostasis) has led to the suggestion that competition between lineages, and classes of T-cells contributes to the pathology of HIV. We use a mathematical model of the interactions between populations of T-cells, HIV, and other parasites to explore the effects of T-cell homeostasis and competition on the progression to AIDS. We demonstrate that as a consequence of parasite-mediated T-cell replication, of competition within and between different T-cell clones, and random processes (T-cell drift), some CD4+ lineages will be represented by relatively few cells, dearths, and some lineages may be lost, leaving holes in the immune repertoire. By killing CD4+ T-lymphocytes, HIV accelerates the rate at which these dearths and holes accumulate and leads to an early breakdown of the immune control of HIV and other parasites, AIDS. When this model allows for intense, but not complete, competition between the CD4+ and CD8+ T-cell populations, it can account for most of the features of an HIV-1 infection in adults, including the gradual decline in CD4+ T-cell densities and concomitant increase in HIV density, as well as the variability in time from infection to AIDS and the decline in the time from infection to AIDS in older patients.

Models of the within-host dynamics of persistent mycobacterial infections

We use mathematical models to investigate the within-host dynamics of mycobacterial infections. In particular, we investigate the mechanisms by which bacteria such as Mycobacterium tuberculosis and Mycobacterium leprae persist at low densities for extended periods, and attain high densities much later. We suggest that the persistence of bacteria in face of immune pressure may result from the bacteria having a very slow growth rate, or having a dormant stage. We show that whereas these mechanisms may lead to long-term persistence, this will be obtained at relatively low densities. We then suggest that the long-term persistence of bacteria may result in the loss of immunity because of the deletion of specific T-cells arriving from the thymus, and the exhaustion of the specific T-cells as these cells reach the Hayflick limit and die. This loss of immunity will allow the bacteria to attain a high density. We propose experiments capable of testing our models and discuss the implications of the models for the treatment of infected hosts.

Antigenic variation and the within-host dynamics of parasites

Many parasites exhibit antigenic variation within their hosts. We use mathematical models to investigate the dynamical interaction between an antigenically varying parasite and the host's immune system. The models incorporate antigenic variation in the parasite population and the generation of immune responses directed against (i) antigens specific to individual parasite variants and (ii) antigens common to all the parasite variants. Analysis of the models allows us to evaluate the relative importance of variant-specific and cross-reactive immune responses in controlling the parasite. Early in the course of infection within the host, when parasite diversity is below a defined threshold value (the value is determined by the biological properties of the parasite and of the host's immune response), the variant-specific immune responses are predominant. Later, when the parasite diversity is high, the cross-reactive immune response is largely responsible for controlling the parasitemia. It is argued that increasing antigenic diversity leads to a switch from variant-specific to cross-reactive immune responses. These simple models mimic various features of observed infections recorded in the experimental literature, including an initial peak in parasitemia, a long and variable duration of infection with fluctuating parasitemia that ends with either the clearance of the parasite or persistent infection.

A model of non-specific immunity

Though the importance of the non-specific immune response is well known, it has often been neglected in theoretical studies. Whereas adaptive or antigen-specific immune responses arise from the proliferation of clones of antigenic-specific cells to form populations sufficiently large to control the parasite, the non-specific response involves the activation of cells such as macrophages from a reservoir consisting of a fixed number of cells. In this paper, we use simple mathematical models to investigate the dynamics of the non-specific immune response to parasites. In particular we describe the conditions under which the non-specific immune response can clear a parasite, control a parasite, or merely reduce the growth rate of a parasite. We also show that non-specific response to concurrent infections of hosts with two parasites can lead to competitive exclusion of one of the parasites. The model incorporating non-specific immunity is then expanded to include specific immune responses. This more complex model, is used to investigate the relative roles of non-specific and specific immunity in dealing with parasites and shows that the non-specific immune system may control the density of parasites prior to the generation of specific immune responses which are capable of clearing them. Finally we show that the predictions of the models conform with results from published experiments on listeria infections.

Binding of perforin to membranes is sensitive to lipid spacing and not headgroup

When triggered, cytolytic effector cells (cytolytic T-lymphocytes (CTL) and large granular lymphocytes (LGL)) release effector molecules from cytoplasmic granules, including the lytic protein perforin. This protein binds and incorporates into the plasma membrane of target cells, where it aggregates to form pores which cause target cell lysis and death. Phosphorylcholine, the headgroup of the ubiquitous phospholipids phosphatidylcholine (PC) and sphingomyelin, has been proposed as the specific receptor for perforin. We report here that any headgroup specificity is outweighed by phospholipid spacing in determining binding of perforin to liposomes. We also find that the spacing of outer leaflet lipids in a natural bilayer, the plasma membrane of the erythrocyte, influences susceptibility of the cell to perforin-mediated lysis. Finally, we demonstrate that the plasma membrane lipids in CTL are more closely spaced than in target cells, suggesting that lipid spacing contributes to the relative resistance of CTL to perforin-mediated lysis.

A quantitative model suggests immune memory involves the colocalization of B and Th cells

A prominent and essential feature of the humoral immune response of vertebrates is immunologic memory: the ability to recall previous exposure to antigen. We present a mathematical model of the growth and interactions of the major cell populations involved in the humoral immune response. Our analysis of this model predicts that the formation of a dynamic association between small numbers of antigen-specific B and Th cells, "colocalization", is sufficient to account for memory and the kinetics of the secondary response--neither specifically differentiated Th or B memory cells nor networks of antigen and anti-idiotypes are required. The colocalization hypothesis explains a number of existing experimental observations and can be tested by straightforward experiments which we describe.

Mechanism of killing by virus-induced cytotoxic T lymphocytes elicited in vivo

The mechanism of lysis by in vivo-induced cytotoxic T lymphocytes (CTL) was examined with virus-specific CTL from mice infected with lymphocytic choriomeningitis virus (LCMV). LCMV-induced T cells were shown to have greater than 10 times the serine esterase activity of T cells from normal mice, and high levels of serine esterase were located in the LCMV-induced CD8+ cell population. Serine esterase was also induced in purified T-cell preparations isolated from mice infected with other viruses (mouse hepatitis, Pichinde, and vaccinia). In contrast, the interferon inducer poly(I.C) only marginally enhanced serine esterase in T cells. Serine esterase activity was released from the LCMV-induced T cells upon incubation with syngeneic but not allogeneic LCMV-infected target cells. Both cytotoxicity and the release of serine esterase were calcium dependent. Serine esterase released from disrupted LCMV-induced T cells was in the form of the fast-sedimenting particles, suggesting its inclusion in granules. Competitive substrates for serine esterase blocked killing by LCMV-specific CTL, but serine esterase-containing granules isolated from LCMV-induced CTL, in contrast to granules isolated from a rat natural killer cell tumor line, did not display detectable hemolytic activity. Fragmentation of target cell DNA was observed during the lytic process mediated by LCMV-specific CTL, and the release of the DNA label [125I]iododeoxyuridine from target cells and the accompanying fragmentation of DNA also were calcium dependent. These data support the hypothesis that the mechanism of killing by in vivo-induced T cells involves a calcium-dependent secretion of serine esterase-containing granules and a target cell death by a process involving nuclear degradation and DNA fragmentation.

Maintenance of membrane phospholipid asymmetry. Lipid-cytoskeletal interactions or lipid pump?

Two models for the mechanism of maintenance of lipid asymmetry in erythrocytes are considered: binding of internal lipids to cytoskeletal proteins, and pumping of internal lipids from the outside to the inside of the cell. Analysis of the kinetics of lipid internalization suggests that the first model is more likely, and that the apparent pumping of lipids represents the activity of an ATP-dependent lipid flip/flop catalyst.