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JANF designed the model, undertook the analyses and drafted the paper with input from ACG. EMR, CJD and CJS advised on model structure and analysis and contributed to the drafting of the paper. All authors have seen and approved the final manuscript.

¤a Current address: Department of Plant Sciences, University of Cambridge, United Kingdom

¤b Current address: Department of Infectious Disease Epidemiology, Imperial College London, London, United Kingdom

The authors have declared that no competing interests exist.

Acquisition of partially protective immunity is a dominant feature of the epidemiology of malaria among exposed individuals. The processes that determine the acquisition of immunity to clinical disease and to asymptomatic carriage of malaria parasites are poorly understood, in part because of a lack of validated immunological markers of protection. Using mathematical models, we seek to better understand the processes that determine observed epidemiological patterns. We have developed an age-structured mathematical model of malaria transmission in which acquired immunity can act in three ways (“immunity functions”): reducing the probability of clinical disease, speeding the clearance of parasites, and increasing tolerance to subpatent infections. Each immunity function was allowed to vary in efficacy depending on both age and malaria transmission intensity. The results were compared to age patterns of parasite prevalence and clinical disease in endemic settings in northeastern Tanzania and The Gambia. Two types of immune function were required to reproduce the epidemiological age-prevalence curves seen in the empirical data; a form of clinical immunity that reduces susceptibility to clinical disease and develops with age and exposure (with half-life of the order of five years or more) and a form of anti-parasite immunity which results in more rapid clearance of parasitaemia, is acquired later in life and is longer lasting (half-life of >20 y). The development of anti-parasite immunity better reproduced observed epidemiological patterns if it was dominated by age-dependent physiological processes rather than by the magnitude of exposure (provided some exposure occurs). Tolerance to subpatent infections was not required to explain the empirical data. The model comprising immunity to clinical disease which develops early in life and is exposure-dependent, and anti-parasite immunity which develops later in life and is not dependent on the magnitude of exposure, appears to best reproduce the pattern of parasite prevalence and clinical disease by age in different malaria transmission settings. Understanding the effector mechanisms underlying these two immune functions will assist in the design of transmission-reducing interventions against malaria.

While the processes that determine the acquisition of immunity to

Here we develop a mathematical model to better understand the impact of the development of immunity on observed epidemiological patterns, and also aspects of the immunology which might be inferred from the epidemiology such as time scales of acquisition and loss. Whilst a number of malaria transmission models have been developed in the past which incorporate immunity [

We first developed an age-structured transmission model for malaria in which acquired immunity acts at three different stages of a host's history of infection: 1) susceptibility to symptomatic disease (severe and clinical cases) upon infection or re-infection, assuming susceptibility decreases with cumulative exposure to infectious bites (e.g., as a result of antibody-mediated strain-specific immunity); 2) natural recovery from asymptomatic to undetectable infection (i.e., effective clearance of parasites), which increases with cumulative exposure to infectious bites after a delay during childhood representing maturation of the immune system, 3) natural clearance of undetectable subpatent infection, assuming increased tolerance and slower clearance of such infection.

Each response, which we call an

(A) Prevalence of parasitaemia by age, region, and altitude (<600 m, 600-1200 m, and >1200 m) from studies in Northern Tanzania.

(B) Clinical episodes by age and altitude for region 2 (Usambara mountains) from severe malaria admissions to district, regional, and referral hospitals.

(C,D) Prevalence of parasitaemia by age, year, and season (wet/dry) from North Bank (C) and South Bank (D) of River Gambia.

The corresponding patterns predicted by different versions of the model are shown in

(A,B) No immunity; (C,D) immunity acting on clearance of subpatent parasites (immunity function 3); (E,F) immunity acting on clearance of detectable parasites (immunity function 2); (G,H) immunity acting on susceptibility to clinical disease (immunity function 1); (I,J) immunity acting on clearance of detectable parasites and susceptibility to clinical disease (immunity functions 1 and 2). Parameters are as shown in

Summary of Model Parameters and Their Values

We next considered combining the different functions to identify which combination best reproduces the observed age-prevalence patterns in

The age-prevalence patterns in

(A) Patterns predicted by the model compared to those observed in region 2 in Northern Tanzania by altitude. EIRs for the model are 110 for low altitude (measured EIR 28–108), 18 for medium altitude (measured EIR 0.4–7.6), and 0.5 for high altitude (measured EIR 0.01–0.32), percentage treated

(B) Patterns predicted by the model compared to those observed on the north and south banks of the River Gambia. Model EIRs were 50 for the north bank and 15 for the south bank. Percentage treated

An alternative way of testing the immunity functions (conditional on the remaining model structure and assumptions being valid) is to compare the predicted mean infectivity by age, which may be regarded as the probability of carrying gametocytes (although not all gametocyte carriers will be infectious), with the observed age-prevalence of gametocytes. The patterns predicted by our best model (incorporating immunity functions 1 and 2) closely match the patterns observed in northern Tanzania and The Gambia (

(A) Predicted infectivity by age from the model with different immunity functions. If1= immunity function 1 (susceptibility to clinical disease); If2 = immunity function 2 (clearance of detectable parasites); If3 = immunity function 3 (clearance of subpatent infection), If2* denotes EIR-independent version of If2. Parasitaemia is calculated in the model as symptomatic cases plus asymptomatic infections (D_{H}+A_{H}). All runs assume an annual EIR = 40 ibppy and that parameters are as before (_{D} is adjusted (for If2 and If3) to make comparable the curves corresponding to different immunity function models.

(B–D) Observed gametocytaemia by age from (B) the low altitude area of region 2 in Tanzania, (C) The Gambia south of the river bank, and (D) The Gambia north of the river bank. Parameters for the model are annual EIR = 110 (B), 50 (C), 15 (D), infectivity C_{D} = 0.3 as before (B,D), 0.4 (C), percentage treated

Our determined half-lives of clinical and parasite immunity were 5 y and 20 y, respectively. By varying these parameters, we explored whether patterns of age-prevalence can inform possible bounds for these parameters.

Reducing the half-life for the duration of clinical immunity below 5 y results in a sharp increase in the proportion of all infections that are symptomatic cases and, in addition, results in less-pronounced age-prevalence peaks which begin to deviate from those observed in data. Increasing the duration of clinical immunity does not substantially change age-prevalence patterns but does have an impact on the proportion of infections that are symptomatic cases (

(A,B) Sensitivity to the duration of the immune response that reduces susceptibility to clinical disease where dS is the half-life; (A) shows the relationship between parasitaemia and age, and (B) shows the proportion of people predicted by the model to be symptomatic cases, have asymptomatic infections, and be parasitaemic (i.e., have patent infections) for different values of dS. Subpatent infections are not shown. For dS less than 5 y, the model predicts too high a proportion of all infections to be symptomatic cases rather than asymptomatic (B).

(C,D) Sensitivity to the duration of the immune response that increases clearance of detectable parasites where dA is the half-life; (C) shows the relationship between parasitaemia and age, and (D) shows the proportion of people predicted by the model to be symptomatic cases, asymptomatic infections, and parasitaemic for different values of dA. For dA less than approximately 20 y, the model predicts that high levels of parasitaemia will persist into adulthood (C). Results are presented for an annual EIR of 110 ibppy. Similar patterns are obtained for lower EIR values.

Reducing the half-life for the duration of parasite immunity below 20 y similarly has an impact on the age-prevalence curves and at very low values (<10 y) gives rise to curves that saturate rather than decline at older ages. The proportion of infections that are asymptomatic and parasitaemic is also increased. However, increasing the duration of parasite immunity has little impact on either outcome (

Our results demonstrate that, while distinct models can explain patterns of parasitaemia observed in individuals aged 0–5 y, in order to reproduce full age-prevalence patterns of parasitaemia and clinical disease observed in endemic malaria settings at least two distinct acquired immunity processes are required: 1) an early age (or early exposure) reduction in clinical susceptibility, and 2) a process of parasite immunity that increases the rate of natural recovery from infection and which develops substantially later in life (late childhood to adolescence). Adopting one of these processes in isolation does not reproduce observed patterns of age-prevalence of asexual parasitaemia, disease, and infectivity (gametocytaemia) across different endemicities (as measured by EIR). Moreover, while both clinical and parasite immunity were allowed to vary with age and EIR, the model in which natural recovery from infection (e.g., asymptomatic to subpatent) is determined solely by age better matches observed patterns than a model in which this is also determined by the intensity of exposure (EIR). This suggests parasite immunity in non-naïve individuals may be controlled by physiological processes rather than by amount of exposure (provided there is exposure). These findings agree with the current view that parasite immunity may require ageing to develop, but subsequently can persist without high antibody titres and therefore be maintained by occasional infrequent boosting [

Incorporating a prolonged duration of (subpatent) infections, i.e., continual reinfection that prolongs infection and boosts an immune response that allows parasitaemia to persist at subpatent levels, worsened the model predictions. However, we cannot exclude that an overall immune-modulated increase in duration of infections takes place, as suggested by recent hypotheses from within-host models [_{H}) is likely to rapidly become asymptomatic with subpatent parasitaemia upon reexposure (i.e., is immune to symptomatic and to patent asymptomatic infection), tantamount to frequent subpatent infection but with recovery and reinfection modelled explicitly. Other models [

Our model additionally allowed us to explore what age-prevalence patterns can tell us about the duration of clinical and parasite immunity. Our results suggest that clinical immunity has shorter memory (with a half-life of the order 5 y or more), while parasite immunity is effectively everlasting (with a half-life of 20 y or more after onset in adolescence). These durations are in line with evidence that migrating adults returning to endemic areas tend to become more sensitive to clinical attack but have lower parasite levels than children [

There are limitations in the epidemiological data that are available to inform model parameters. In particular, there are few and uncertain estimates of EIR by altitude range [

The model presented here clearly makes a number of simplifying assumptions. One of the main limitations is that the immunity functions, whilst generated based on current immunological understanding, could not be constrained by data. Further data on the way in which immunity develops and on the factors driving its development could help to refine these functions. The model also does not allow for partial immunity to reinfection, which would be relevant from the point of view of treating or vaccinating against pre-erythrocyte stages. While sterilising or partial pre-erythrocyte immunity are likely to be rare [

This age-structured malaria transmission model shares many features with existing models [

Clearly, it is never possible to determine whether the structural assumptions behind any model represent the true processes generating the observed data, and it is likely that more complex model structures could also generate similar patterns. One alternative method that could be employed is to track parasite density rather than infection alone. Such an approach explicitly acknowledges variation in parasite load between individuals, and this variation may influence the development of immunity. However, such an approach also has its limitations. In particular, the distinction between disease and asymptomatic and subpatent infection requires definition of arbitrary parasite density thresholds for becoming diseased once infected and for detection by microscopy. Our assumption that susceptibility and recovery vary continuously via dependence on cumulative exposure is, however, analogous to the effect of immunity in bringing parasite density below such thresholds.

A second alternative method for incorporating immunity into mathematical models is to explicitly model strains and hence incorporate long-lasting strain-specific immunity. As noted above, our assumption that immunity develops with exposure and has finite memory essentially reproduces the patterns that would be obtained from such a model. The model does not imply that parasite density or strain-specific immunity are unimportant; as indeed there is strong evidence to support both playing a role in the development of immunity. Rather, our simpler model structure which implicitly incorporates these processes through immunity functions allows us to explore the timescales over which clinical and parasite immunity develop and are lost as well as the role of ageing and exposure on these functions.

Few previous models have been consistent in checking that they can reproduce the patterns of infection observed across a range of endemicities. By validating output against such patterns, we have sought to develop a model that is both informative about the impact of immunity on

We model a human population with continuous age structure in which individuals of a given age can be in one of the following states: susceptible or not infected (S_{H}), latent infection (E_{H}), infected with symptomatic disease (including severe and clinical cases) (D_{H}), asymptomatic with detectable parasites (A_{H}), and asymptomatic infection with undetectable (subpatent) parasite density (U_{H}). The main distinction between states D_{H} and A_{H} is that individuals in state A_{H} do not prompt treatment that leads to a change in infection state. The state U_{H} is included to account for the fact that measured parasitemia often decays with age, while highly sensitive parasite detection techniques suggest parasitemia continues increasing with age nearing 100% in highly endemic areas [_{M}), exposed (latent) (E_{M}), or infectious (I_{M}). _{H}) receive effective drug treatment and recover at rate r_{T}, while the remaining cases recover naturally without treatment at rate r_{D}. If clinical treatment or natural recovery is fully successful at removing parasites (with probability φ), the host returns to the susceptible state and otherwise moves to the asymptomatic state. Asymptomatic infections become subpatent at rate r_{A}, and these subpatent infections are cleared at rate r_{U} with individuals returning to the susceptible state. Those in the asymptomatic state may additionally develop disease through superinfection at rate φΛ. Each human infection state, namely D_{H}, A_{H}, and U_{H}, has a specific level of infectivity (transmission of mature gametocytes) to biting mosquitoes. The full equations for this model and further parameter definitions are given in

States are shown in circles, and subscripts denote the population (H = humans, M = mosquitoes): susceptible S_{H}/S_{M,} latent infection E_{H}/E_{M}, infected with symptomatic disease (severe and clinical cases) D_{H}, asymptomatic patent infection A_{H}, infected with undetectable (subpatent) parasite density U_{H}, infectious mosquitoes I_{M}. Λ_{H} /Λ_{M} is the force of infection on the human and mosquito populations, respectively, 1/h is the mean latent period in humans, 1/g the mean latent period in mosquitoes, φ is the proportion of human infections that develop disease, _{T} the rate of recovery on treatment, r_{D} the rate of recovery without treatment, r_{A} the rate at which asymptomatic infections become subpatent, and r_{U} the rate at which subpatent infections are cleared. The coloured circles denote the stages at which acquired immunity can have an effect (modifying φ, r_{A}, and r_{U}). The parameters and their values are described in

_{H},A_{H},U_{H}) on age and EIR because this is currently less-well-understood [_{D}) in the absence of treatment is identical to that from asymptomatic infection (r_{A}), and that the rate of recovery of treated cases (r_{T}) is determined by treatment only.

Unknown parameters (

To explore the impact that acquired immunity can have on patterns of age prevalence in endemic settings, we extend the basic transmission model above to incorporate immunity acting at three different stages of a host's history of infection. Mathematical details of the functions, described in brief below, are given in

_{M}. Following birth, clinical immunity accumulates due to exposure at a rate dependent on the force of infection in the population, Λ. This acquired immunity decays with a half-life d_{S}. The schematic for this model is shown in

(A,C,E) Show schematically how each model assumes that immunity is developed (through exposure and/or age) and lost.

(B,D,F) Show the resulting effect of these immunity levels on (B) susceptibility to clinical disease, (D) the rate of clearance of detectable parasites, and (F) the clearance of subpatent infection as people age and for five different transmission settings (identified by the EIR in ibppy). Further mathematical details are given in

_{A} (immunity function 2)._{l}, and any maternal immunity is lost during this period. Parasite immunity then decays with half-life d_{A}. The schematic for this model is shown in _{A} is assumed to increase with levels of parasite immunity through a nonlinear function which saturates at higher levels of immunity. The overall dependence of recovery on age and EIR resulting from this model is shown in

As an alternative, we also consider a model in which parasite immunity is determined only by age (given some exposure to infection) and not by EIR.

_{U} (immunity function 3)._{U} as in previous models of superinfection [

We thank Hugh Reyburn and Paul Milligan for providing additional epidemiological data and Lucy Okell for helpful discussions.

entomological inoculation rate