The reinfection threshold

Thresholds in transmission are responsible for critical changes in infectious disease epidemiology. The epidemic threshold indicates whether infection invades a totally susceptible population. The reinfection threshold indicates whether self-sustained transmission occurs in a population that has developed a degree of partial immunity to the pathogen (by previous infection or vaccination). In models that combine susceptible and partially immune individuals, the reinfection threshold is technically not a bifurcation of equilibria as correctly pointed out by Breban and Blower. However, we show that a branch of equilibria to a reinfection submodel bifurcates from the disease-free equilibrium as transmission crosses this threshold. Consequently, the full model indicates that levels of infection increase by two orders of magnitude and the effect of mass vaccination becomes negligible as transmission increases across the reinfection threshold.     

The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy

Population patterns of infection are determined largely by susceptibility to infection. Infection and vaccination induce an immune response that, typically, reduces susceptibility to subsequent infections. With a general epidemic model, we detect a 'reinfection threshold', above which reinfection is the principal type of transmission and, consequently, infection levels are much higher and vaccination fails. The model is further developed to address human tuberculosis (TB) and the impact of vaccination. The bacille Calmette-Guérin (BCG) is the only vaccine in current use against TB, and there is no consensus about its usefulness. Estimates of protection range from 0 to 80%, and this variability is aggravated by an association between low vaccine efficacy and high prevalence of the disease. We propose an explanation based on three postulates: (i) the potential for transmission varies between populations, owing to differences in socio-economic and environmental factors; (ii) exposure to mycobacteria induces an immune response that is partially protective against reinfection; and (iii) this protection is not significantly improved by BCG vaccination. These postulates combine to reproduce the observed trends, and this is attributed to a reinfection threshold intrinsic to the transmission dynamics. Finally, we demonstrate how reinfection thresholds can be manipulated by vaccination programmes, suggesting that they have a potentially powerful role in global control.     

Heterogeneity in susceptibility to infection can explain high reinfection rates

Heterogeneity in susceptibility and infectivity is inherent to infectious disease transmission in nature. Here we are concerned with the formulation of mathematical models that capture the essence of heterogeneity while keeping a simple structure suitable of analytical treatment. We explore the consequences of host heterogeneity in the susceptibility to infection for epidemiological models for which immunity conferred by infection is partially protective, known as susceptible-infected-recovered-infected (SIRI) models. We analyze the impact of heterogeneity on disease prevalence and contrast the susceptibility profiles of the subpopulations at risk for primary infection and reinfection. We present a systematic study in the case of two frailty groups.We predict that the average rate of reinfection may be higher than the average rate of primary infection, which may seem paradoxical given that primary infection induces life-long partial protection. Infection generates a selection mechanism whereby fit individuals remain in S and frail individuals are transferred to R. If this effect is strong enough we have a scenario where, on average, the rate of reinfection is higher than the rate of primary infection even though each individual has a risk reduction following primary infection. This mechanism may explain high rates of tuberculosis reinfection recently reported.Finally, the enhanced benefits of vaccination strategies that target the high-risk groups are quantified.     

Effective degree network disease models

An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.     

Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis

The aim of this paper is to study the impact of introducing a partially protective vaccine on the dynamics of infection in SIRS models where primary and secondary infections are distinguished. We investigate whether a public health strategy based solely on vaccinating a proportion of newborns can lead to an effective control of the disease. In addition to carrying out the qualitative analysis, the findings are further explained by numerical simulations. The model exhibits backward bifurcation for certain values of the parameters. In these cases the standard basic reproduction number (obtained by inspection of the uninfected state) is not significant. The key threshold is the reinfection level which depends on the relative transmissibility (susceptibility) of secondary, with respect to primary, infected (susceptible) individuals and the relative loss of immunity of vaccinated, with respect to recovered, individuals. If one or all of these ratios decrease, then the threshold increases which increases the possibility to contain the infection by vaccination. The analysis shows further that symptomatic infections can be eliminated by vaccination solely.     

No evidence for homosubtypic immunity of influenza H3 in Mallards following vaccination in a natural experimental system

The Mallard (Anas platyrhynchos) is an important reservoir species for influenza A viruses (IAV), and in this host, prevalence and virus diversity are high. Studies have demonstrated the presence of homosubtypic immunity, where individuals are unlikely to be reinfected with the same subtype within an autumn season. Further, evidence for heterosubtypic immunity exists, whereby immune responses specific for one subtype offer partial or complete protection against related HA subtypes. We utilized a natural experimental system to determine whether homo‐ or heterospecific immunity could be induced following experimental vaccination. Thirty Mallards were vaccinated with an inactivated H3, H6 or a sham vaccine and after seroconversion were exposed to naturally infected wild conspecifics. All ducks were infected within 2 days and had both primary and secondary infections. Overall, there was no observable difference between groups; all individuals were infected with H3 and H10 IAV. At the cessation of the experiment, most individuals had anti‐NP antibodies and neutralizing antibodies against H10. Not all individuals had H3 neutralizing antibodies. The isolated H3 IAVs revealed genetic dissimilarity to the H3 vaccine strain, specifically substitutions in the vicinity of the receptor‐binding site. There was no evidence of vaccine‐induced homosubtypic immunity to H3, a likely result of both a poor H3 immune response in the ducks and H3 immune escape. Likewise, there was no observed heterosubtypic protection related to H6 vaccination. This study highlights the need for experimental approaches to assess how exposure to pathogens and resulting immune processes translates to individual and population disease dynamics.     

Mycobacterial ecology as a modulator of tuberculosis vaccine success

Natural infection with Mycobacterium tuberculosis, as well as cross-immune reactions with the constituent of standard vaccines, attenuated M. bovis, and other species of mycobacteria confer partial immunity to subsequent M. tuberculosis infection. It has been shown in the past that the immune response to mycobacteria found naturally in the environment reduces the benefit of vaccination as assessed by means of vaccine efficacy. In this paper we show that efficacy is a poor measure of the potential success of new anti-tuberculous vaccines due to its inability to account for the relative weight of reinfection in disease dynamics. We advocate instead the use of vaccine effectiveness when evaluating the impact of new control methods against infections that confer partial immunity. Through the study of a simple model that incorporates cross-reactive responses to environmental mycobacteria (EM) and reinfection, we show how the particulars of the relation between EM abundance and vaccine effectiveness depend on the degree of protection conferred respectively by natural infection, vaccination and EM. The relative importance of reinfection as a transmission mechanism comes up as the most important source of variability in vaccine effectiveness. Our results suggest that control efforts should be placed in reducing the importance of reinfection through diminishing transmission rates. Vaccines that overcome preexisting immunity to other mycobacteria will still have varying degrees of success depending on the underlying rate of TB transmission.     

A Covering-Graph Approach to Epidemics on SIS and SIS-Like Networks

In this paper, we introduce a new class of epidemics on networks which we call SI(S/I). SI(S/I) networks differ from SIS networks in allowing an infected individual to become reinfected without first passing to the susceptible state. We use a covering-graph construction to compare SIR, SIS, and SI(S/I) networks. Like the SIR networks that cover them, SI(S/I) networks exhibit infection probabilities that are monotone with respect to both transmission probabilities and the initial set of infectives. The same covering-graph construction allows us to characterize the recurrent states in an SIS or SI(S/I) network with reinfection.     

The evolution of costly acquired immune memory

A key feature of the vertebrate adaptive immune system is acquired immune memory, whereby hosts launch a faster and heightened response when challenged by previously encountered pathogens, preventing full infection. Here, we use a mathematical model to explore the role of ecological and epidemiological processes in shaping selection for costly acquired immune memory. Applying the framework of adaptive dynamics to the classic SIR (Susceptible‐Infected‐Recovered) epidemiological model, we focus on the conditions that may lead hosts to evolve high levels of immunity. Linking our work to previous theory, we show how investment in immune memory may be greatest at long or intermediate host lifespans depending on whether immunity is long lasting. High initial costs to gain immunity are also found to be essential for a highly effective immune memory. We also find that high disease infectivity and sterility, but intermediate virulence and immune period, increase selection for immunity. Diversity in host populations through evolutionary branching is found to be possible but only for a limited range of parameter space. Our model suggests that specific ecological and epidemiological conditions have to be met for acquired immune memory to evolve.     

Implications of vaccination and waning immunity

For infectious diseases where immunization can offer lifelong protection, a variety of simple models can be used to explain the utility of vaccination as a control method. However, for many diseases, immunity wanes over time and is subsequently enhanced (boosted) by asymptomatic encounters with the infection. The study of this type of epidemiological process requires a model formulation that can capture both the within-host dynamics of the pathogen and immune system as well as the associated population-level transmission dynamics. Here, we parametrize such a model for measles and show how vaccination can have a range of unexpected consequences as it reduces the natural boosting of immunity as well as reducing the number of naive susceptibles. In particular, we show that moderate waning times (40–80 years) and high levels of vaccination (greater than 70%) can induce large-scale oscillations with substantial numbers of symptomatic cases being generated at the peak. In addition, we predict that, after a long disease-free period, the introduction of infection will lead to far larger epidemics than that predicted by standard models. These results have clear implications for the long-term success of any vaccination campaign and highlight the need for a sound understanding of the immunological mechanisms of immunity and vaccination.     

Epidemics with partial immunity to reinfection

We obtain analytical results about epidemics generated by the partial immunity model of Gomes et al. [3], in which infection confers partial immunity to reinfection. When the demographic process is excluded, the behavior switches from epidemic to endemic as the basic reproduction number R₀ crosses the reinfection threshold . We derive formulas for two quantities characterizing the size of the epidemic below the reinfection threshold: the attack rate A, which is the fraction of the population infected at least once, and the final size Z, which is the average number of infections per individual. We also derive a system of differential equations which can be used to obtain more detailed information, such as the fraction of the population infected n times throughout the epidemic, for every n.     

An in-host model of acute infection: measles as a case study

The epidemiology of acute infections is strongly influenced by the immune status of individuals. In-host models can provide quantitative predictions of immune status and can thus offer valuable insights into the factors that influence transmission between individuals and the effectiveness of vaccination protocols with respect to individual immunity. Here we develop an in-host model of measles infection. The model explicitly considers the effects of immune system memory and CD8 T-cells, which are key to measles clearance. The model is used to determine the effects of waning immunity through vaccination and infection, the effects of booster exposures or vaccines on the level of immunity, and the immune system characteristics that result in measles transmission (R(0)>1) even if an individual has no apparent clinical symptoms. We find that the level of immune system CD8 T-cells at the time of exposure to measles determines whether an individual will experience a measles infection or simply a boost in immunity. We also find that the infected cell dynamics are a good indicator of measles transmission and the degree of symptoms that will be experienced. Our results indicate that the degree of immunity in adults is independent of the source of exposure in early childhood, be it vaccine or natural infection.     

The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection

In stochastic modelling of infectious spread, it is often assumed that infection confers permanent immunity, a susceptible-infective-removed (SIR) model. We show how results concerning long-term (endemic) behaviour may be extended to a susceptible-infective-removed-susceptible (SIRS) model, in which immunity is temporary. Since the full SIRS model with demography is rather intractable, we also consider two simpler models: the susceptible-infective-susceptible (SIS) model with demography, in which there is no immunity; and the SIRS model in a closed population. For each model, we first analyse a deterministic model, then approximate the quasi-stationary distribution (equilibrium distribution conditional upon non-extinction of infection) using a moment closure technique. We look in particular at the effect of the immune period upon infection prevalence and upon time to fade-out of infection. Our main findings are that a shorter average immune period leads to higher infection prevalence in quasi-stationarity, and to longer persistence of infection in the population.     

Mucosal immunity and vaccination

Abstract The gut mucosal immune system is a critical component of the body's defense against pathogenic organisms, especially those responsible for enteric infections associated with diarrhoeal disease. Attempts to vaccinate against infections of mucosal tissues have been less successful than vaccination against systematic infections, to a large extent reflecting a still incomplete knowledge about the most efficient means for inducing protective local immune responses at these sites. Secretory IgA (SIgA) is the predominating immunoglobulin along mucosal surfaces, and SIgA antibodies generated in gastrointestinal, respiratory or genito-urinary mucosal tissues can confer protection against infections affecting or originating in these sites. An efficacious intestinal SIgA immunity-inducing oral vaccine against cholera has been developed recently, and development of oral vaccines against other enteric infections such as those caused by enterotoxigenic Escherichia coli, Shigella and rotaviruses is in progress as well. Based on the concept of a common mucosal immune system through which activated lymphocytes from the gut can disseminate immunity to other mucosal and glandular tissues, there is currently also much interest in the possibility of developing oral vaccines against infections in the respiratory and urogenital tracts. However, the large and repeated antigen doses often required to achieve a protective immune response still makes this vaccination approach impractical for many purified antigens. There is, therefore, a great need to develop strategies for enhancing delivery of antigen to the mucosal immune system as well as to identify mucosa-active immunostimulating agents (adjuvants). These and other aspects of mucosal immunity in relation to immunization and vaccine development are discussed in this short review article.     

A model for tuberculosis with exogenous reinfection

Following primary tuberculosis (TB) infection, only approximately 10% of individuals develop active T.B. Most people are assumed to mount an effective immune response to the initial infection that limits proliferation of the bacilli and leads to long-lasting partial immunity both to further infection and to reactivation of latent bacilli remaining from the original infection. Infected individuals may develop active TB as a consequence of exogenous reinfection, i.e., acquiring a new infection from another infectious individual. Our results in this paper suggest that exogenous reinfection has a drastic effect on the qualitative dynamics of TB. The incorporation of exogenous reinfection into our TB model allows the possibility of a subcritical bifurcation at the critical value of the basic reproductive number R(0)=1, and hence the existence of multiple endemic equilibria for R(0)<1 and the exogenous reinfection rate larger than a threshold. Our results suggest that reducing R(0) to be smaller than one may not be sufficient to eradicate the disease. An additional reduction in reinfection rate may be required. These results may also partially explain the recently observed resurgence of TB.     

Microparasite population dynamics and continuous immunity.

A mathematical model is presented for the transmission of a microparasite where the hosts occupy one of two states, uninfected or infected. In each state, the hosts are distributed over a continuous range of immunity. The immune levels vary within hosts due to the processes of waning of immunity (when uninfected), and increasing immunity (when infected), eventually resulting in recovery. Immunity level also influences the host''s ability to infect or be infected. Thus the proposed model incorporates both inter- and intra-host dynamics. It is shown from equilibrium results that this model is a general form of the susceptible-infected-resistant (SIR) and susceptible-infected-susceptible (SIS) family of models, a development that is useful for exploring multistrain pathogen transmission and use of vaccines which confer temporary protection.     

Antibody dynamics in childhood diseases: waning and boosting of immunity and the impact of vaccination

The introduction of vaccination against acute diseases such as measles induced a dramatic decline in the prevalence of the disease, and a more gradual rise in the proportion of the population whose immunity is derived solely from vaccination. These two factors combine to constitute an important shift in the dynamics of immunity, especially in highly vaccinated populations. We develop a general model to describe both loss of immunity in the absence of disease, and boosting of immunity corresponding to subclinical infection in individuals whose immunity has waned. We consider the interaction between infection and immunity and identify the key parameters that determine the eradication threshold. We explore the dynamics in the years following the introduction of vaccination using a stochastic version of the model, and consider the effect of different assumptions concerning the nature of immunity. A comparison of the model results with recently published data suggests that heterogeneity in host immune response is an important feature of the antibody dynamics.     

Longitudinal studies on human schistosomiasis

A major difficulty in understanding the epidemiology of human schistosomiasis has been to distinguish between acquired immunity and reduced exposure as possible reasons for an observed decline, in older individuals, of levels of superinfection or of reinfection after chemotherapy. A series of studies of Schistosoma mansoni infections in Kenya has been undertaken to approach this problem, by investigation of intensities of reinfection after treatment of individuals whose levels of contact with contaminated water is subsequently observed. Intensities of reinfection are highest among younger children, thereafter declining sharply. This decline can be attributed only in part to age-related changes in the duration and nature of exposure; there is also evidence for the development of an acquired resistance to reinfection that is dependent both on age and on previous experience of infection, and that may be immunologically mediated. Evidence has been obtained that the slow development of this acquired immunity with age may be associated with the early development and subsequent slow decline of inappropriate immune responses that 'block' the effect of potentially protective responses. Implications of these findings for immunological intervention through vaccination are discussed.     

Evidence for frequent reinfection with human immunodeficiency virus type 1 of a different subtype

A major premise underlying current human immunodeficiency virus type 1 (HIV-1) vaccine approaches is that preexisting HIV-1-specific immunity will block or reduce infection. However, the recent identification of several cases of HIV-1 reinfection suggests that the specific immune response generated for chronic HIV-1 infection may not be adequate to protect against infection by a second HIV-1 strain. It has been unclear, though, whether these individuals are representative of the global epidemic or are rare cases. Here we show that in a population of high-risk women, HIV-1 reinfection occurs almost as commonly as first infections. The study was designed to detect cases of reinfection by HIV-1 of a different subtype and thus captured cases where there was considerable diversity between the first and second strain. In each case, the second virus emerged approximately 1 year after the first infection, and in two cases, it emerged when viral levels were high, suggesting that a well-established HIV-1 infection may provide little benefit in terms of immunizing against reinfection, at least by more-divergent HIV-1 variants. Our findings indicate an urgent need for studies of larger cohorts to determine the incidence and timing of both intersubtype and intrasubtype reinfection.     

A predominant role for antibody in acquired immunity to chlamydial genital tract reinfection

Acquired immunity to murine Chlamydia trachomatis genital tract reinfection has long been assumed to be solely dependent on cell-mediated immunity. However, in this study, we identify a previously unrecognized protective role for Ab. Immunity develops in Ab-deficient mice following the resolution of primary chlamydial genital infection. Subsequent depletion of CD4+ T cells, but not CD8+ T cells, in those immune Ab-deficient mice before secondary infectious challenge, resulted in an infection that did not resolve. Passive immunization with immune (convalescent) serum conferred a marked level of protective immunity to reinfection, which was characterized by a striking decrease in bacterial shedding, from >100,000 inclusion forming units to fewer than 10 inclusion forming units, and a shortened duration of infection. Furthermore, mAbs to the chlamydial major outer membrane protein and LPS conferred significant levels of immunity to reinfection and reduced chlamydial shedding by >100-fold. Anti-heat shock protein 60 mAb had no protective effect. In contrast to the marked protective efficacy of immune serum on reinfection, the course of primary infection was essentially unaltered by the passive transfer of immune serum. Our results convincingly demonstrate that Abs contribute importantly to immunity to chlamydial genital tract reinfection, and that Ab-mediated protection is highly dependent on CD4+ T cell-mediated adaptive changes that occur in the local genital tract tissues during primary infection. These results impact our understanding of immunity to chlamydial genital infection and may provide important insight into vaccine development.