Introduction

It is widely accepted that relationships exist between the regional epidemiology of malaria and the genetic diversity of local parasite populations. For example, the origin and spread of antimalarial drug resistance, the rate and directions of parasite migration, and the local intensity of transmission all have an impact on population genetic diversity (reviewed in [1–7]). In most cases, however, the precise nature of these relationships remains unclear. From a modelling perspective, exploring these relationships would require that both genetic processes (including mutation, drift and meiosis) and epidemiological ones (including the transmission dynamics and life cycle of malaria) are combined into a single framework. At present such integrated models are rare, yet without them, parasite genetic data will be under-utilized as a resource for malaria surveillance.

One epidemiological parameter of central importance to malaria surveillance is transmission intensity, as it is used by National Malaria Control Programs (NMCPs) to prescribe malaria control interventions and assess their efficacy [8]. NMCPs can attempt to measure transmission intensity with a variety of statistics. These include the basic reproduction number (R₀), defined as the number of secondary infections produced by a single infected individual in a naive population; the entomological inoculation rate (EIR), defined as the mean number of infectious bites received by an individual per annum; parasite prevalence or rate (PR or PfPR if the focus is P. falciparum), defined as the fraction of individuals in the population carrying detectable infection; or simply the rate of clinical incidence (reviewed in [9]). However, there are well-documented issues with all of these statistics. Though a theoretical gold-standard, R₀ is difficult to measure in practice, with estimation methods relying either on exploiting equilibrium relationships with other measures of transmission, or formulae involving several poorly-characterised parameters [9, 10]. The EIR suffers from small-scale variability in mosquito density, a lack of standardisation across mosquito catching methods, and difficulties associated with catching sufficient mosquitoes when transmission intensities are low [9, 11, 12]. Rates of clinical incidence are confounded by variation in acquired immunity and treatment seeking behaviour, as well as incomplete record keeping [12]. Parasite prevalence is the most widely collected measure and has been used as the basis of large-scale maps [13–15], yet it requires prohibitively extensive sampling at low transmission intensities [12], and must address biases in detection power that may arise from infection-course and age-dependent variation in parasitemia [9, 16]. Thus, a means to either estimate or improve existing estimates of transmission intensity with genetic data would be valuable.

The problem of estimating an epidemiological parameter like transmission intensity from genetic data is superficially similar to the demographic inference problems that are commonly encountered in population genetics. For example, a multitude of methods now exist for estimating effective population size (N_e) from genetic data [17–20], and it could be hypothesized that the regional transmission intensity of malaria is a function of the N_e of the local parasite population. However, there are at least two challenges unique to epidemiological inference from malaria genetic data that make it a distinctive and more difficult problem.

First, it is not clear that the classical models in population genetics (including the Wright-Fisher, Moran, and other models that converge to Kingman’s n−Coalescent in the infinite population limit [21]) that are often employed by demographic inference methods are suitable for malaria. The life cycle of P. falciparum involves oscillating between human host and mosquito vector populations, which may be of different sizes, and may also induce different rates of drift and mutation. Within both the host and the vector, parasite populations likely experience bottlenecks (for example, as the ookinetes penetrate the midgut wall of the vector), exponential growth phases (merozoites replicating in the blood of the host), and interactions with the host or vector immune system. Indeed, there has been work demonstrating that the malaria life cycle simultaneously intensifies drift and selection; a result contrary to what is expected under a Wright-Fisher model [22]. So, while classical models in population genetics have the advantage of being extensively studied and mathematically tractable, with a known set of relationships between equilibrium genetic diversity statistics and demographic parameters, they do not readily apply to P. falciparum. Finally, even if these models did apply, the relationships between demographic parameters (such as N_e) and epidemiological ones (such as transmission intensity) would need to be defined.

Conversely, the epidemiological models that have been designed to reflect malaria biology and transmission do not explicitly incorporate genetic processes. The most well known class of epidemiological models are the compartment-based models, where individual hosts and vectors transition between compartments which can represent a variety of disease states (such as susceptible, infected, or immune), and the overall population is represented by the total number of hosts and vectors occupying each state (reviewed in [23, 24]). A canonical compartment model for malaria is the Ross-Macdonald, where hosts and vectors can be either susceptible or infected [24]. Conveniently, in these models, the equilibrium prevalence of infected hosts and vectors is a function of the parameters specifying the transition rates between compartments. However, the absence of genetic processes (and indeed, individual parasites) means they offer no insight into how parasite genetic diversity relates to these transition rates or, as a corollary, any epidemiological parameters derived from them. Thus, at least with respect to malaria, the traditional modelling landscape cannot address questions that involve both genetic data and epidemiology.

A second challenge specific to epidemiological inference using genetic data is that the ultimate aim is often to inform disease control and, as a result, the time-dimension of the inference is of critical importance. Many demographic inference methods base their estimates of N_e on distributions of coalescent times between segments of DNA. As these coalescent events typically occur on the scale of N_e generations in past, the estimates are historical; reflecting the average population size over hundreds or thousands of generations. Such approaches are not suitable for disease control, where policy decisions need to be made on the basis of information about the near-present, or predictions about the future.

In view of these challenges, the development of integrated genetic-epidemiological models is a critical undertaking for the future of malaria genomic surveillance. To date, there have been two notable examples of such integrated models. The first, developed by Daniels et al. in 2015, was designed to support epidemiological inference from single-nucleotide polymorphism (SNP) data, collected during a period of intensified intervention in Thiès, Senegal [25]. Here, the model was used with an Approximate Bayesian Computation (ABC) algorithm to independently corroborate a decline and rebound in transmission intensity on the basis of 24-SNP barcodes [25]. More recently, Watson et al. have developed a model of substantially greater epidemiological complexity—including features such as individual vectors and a vector development cycle, six host infection states, host age and immunity—and used this model to estimate parasite prevalence in five locations from Uganda and Kenya after fitting a single parameter in the model to SNP data [26]. In both these cases, the focus on SNP barcodes reflected the data available. Yet, as P. falciparum whole-genome sequencing (WGS) data continues to increase [27], there is a need for modelling frameworks that can investigate the broader suite of genetic diversity statistics calculable from WGS data.

Here, we aim to address the lack of integrative genetic-epidemiological models by developing a new forward-time model called forward-dream. forward-dream merges the Ross-Macdonald model with a continuous-time intra-host and intra-vector Moran model, and further incorporates meiosis within the vector (allowing for multiple oocysts), multiple infection (by either super- or co-infection), and a representation of the transmission bottlenecks. Implemented as a stochastic simulation, we use the model to explore relationships between measures of genetic diversity and parasite prevalence, both at equilibrium and in response to malaria control interventions that perturb equilibrium. We confirm that a variety of genetic diversity statistics are correlated with parasite prevalence, although to varying degrees and over different time-scales. In addition, we find that interventions that affect the duration of infection in hosts have a greater influence on parasite genetic diversity than those that influence vector biting rate or density. Overall, our results suggest that statistics based on the complexity of infection (COI) are strongly, robustly, and rapidly responsive to changes in prevalence, highlighting their potential value for malaria surveillance.

Results

Developing a model of P.falciparum malaria transmission and evolution

We developed an agent-based simulation of P.falciparum malaria incorporating features of its transmission and life cycle, as well as explicitly modelling the genetic material of parasites. Our integrated genetic-epidemiological model is called forward-dream (forward-time drift, recolonisation, extinction, admixture and meiosis) and is comprised of three layers: (1) a stochastic epidemiological layer, which controls how malaria spreads through a population of hosts and vectors and reaches equilibrium; (2) a stochastic infection layer, which controls the behaviour of malaria parasites during individual transmission events and within individual hosts and vectors; and (3) a stochastic genetic layer, which controls how the genetic material of individual parasites is represented, mutated and recombined (Fig 1). We describe each layer below and provide additional information, including a discussion of parameterisation, in the S1 Appendix.

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Fig 1. Schematic of forward-dream.

(a) The epidemiological layer. Hosts and vectors oscillate between susceptible and infected compartments according to a Ross-Macdonald model. (b) The infection layer. The capacity of individual hosts and vectors to be infected is represented by a fixed number of sub-compartments (black boxes) each which can harbour a unique parasite genome (colored circles). In a susceptible host/vector, all sub-compartments are empty. Upon infection, all sub-compartments are populated. Drift and mutation occur among sub-compartments according to a continuous-time Moran model. Parasite genomes undergo meiosis during transmission from host to vector. Super-infection can occur, resulting in an average of half of all sub-compartments being replaced with newly transmitted parasite genomes. Note that the infection layer can be nested within the Ross-Macdonald model. (c) The genetic layer. The genome is represented by a fixed-length array of 0’s and 1’s. Mutation is reversible, converting 0 to 1 or 1 to 0. Recombination occurs during meiosis.

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Epidemiological layer.

For the epidemiological layer we implemented a stochastic, agent-based version of the Ross-Macdonald model (reviewed in [24]). In this model, a fixed number of hosts (N_h) and vectors (N_v) alternate between susceptible and infected based on four fixed rate parameters (Fig 1a). The model can be described by two coupled differential equations: where h₀ and h₁ are the number of susceptible and infected hosts, respectively, with N_h = h₀+ h₁; and v₀ and v₁ are the number of susceptible and infected vectors, respectively, with N_v = v₀+ v₁. Note that in this model λ and ϕ are compound parameters representing the rate at which hosts and vectors become infected. In particular, where b is the daily vector biting rate, (N_v/N_h) gives the vector density, and π_h is the probability that an infectious bite from a vector produces an infected host (the vector-to-host transmission efficiency). Similarly, we have, where b is again the daily vector biting rate and π_v is the probability that a vector that bites an infected host becomes infected (the host-to-vector transmission efficiency). We allow for mixed infections however, for the epidemiological layer, they have no consequence: both clonal and mixed infections are assigned to the infected compartments (h₁ or v₁).

Note that the epidemiological layer dictates the equilibrium prevalence of infection in hosts (X_h = h₁/N_h) and vectors (X_v = v₁/N_v). In particular, the equilibrium prevalence in hosts is given by: (1) and in vectors by: (2)

We have confirmed that our implementation of the Ross-Macdonald model in forward-dream converges to the expected equilibrium prevalence values (Fig 2a and S1 Fig). In total, the behaviour of the epidemiological layer is specified by seven parameters.

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Fig 2. Simultaneously monitoring parasite prevalence and genetic diversity using forward-dream.

(a) A forward-dream simulation seeded with ten infected hosts harbouring identical parasite genomes and run for 50 years. Prevalence of all infected hosts (‘Host All’, which corresponds to PfPR) and multiply-infected hosts (‘Host Mixed’) is indicated by red and pink lines, respectively. The same is shown for vectors in blues. The prevalence of infected hosts and vectors fluctuates around their Ross-Macdonald equilibrium values (X_h and X_v), indicated with red and blue horizontal lines, respectively. (b) The same simulation as in (a), but visualizing eight genetic diversity statistics that were computed by collecting parasite genomes from twenty randomly selected infected hosts every 30 days. The statistics are defined in S1 Table. For reference, the light and dark grey shaded areas show the host and vector prevalence (right y-axis), corresponding to the red and dark blue lines in panel (a).

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Infection layer.

The infection layer specifies the biology of individual infection and transmission events. This includes a representation of: (i) the capacity of hosts and vectors to harbour infection, (ii) the evolution of infections within hosts and vectors, and (iii) the transmission of infection between hosts and vectors.

We model an individual host’s capacity to be infected with n_h within-host sub-compartments (Fig 1b). When an individual host is in the susceptible state (i.e. is uninfected), all n_h sub-compartments are empty. When an individual host becomes infected, all n_h sub-compartments simultaneously become populated. Each sub-compartment can potentially harbour a unique parasite genome (such that the maximum complexity of infection k = n_h), although typically multiple sub-compartments will be occupied by identical, or near-identical, genomes. For example, if a host is infected by a vector carrying a single distinct parasite genome, all n_h sub-compartments will initially be occupied by that genome. Alternatively, if a host is co-infected by a vector carrying two distinct parasite genomes, its sub-compartments will be drawn from a mixture of the two genomes.

Moving forward in time, the P. falciparum infection of a host evolves according to a continuous-time Moran process for a population with n_h individuals, parameterised by a drift rate (d_h) and mutation rate (θ_h) [28]. We have confirmed that our implementation of the Moran process yields fixation times consistent with theoretical expectation (S2 Fig). The Moran process continues until the infection is cleared and the host returns to the susceptible state, with all sub-compartments becoming simultaneously empty. For hosts, clearance occurs at rate γ, as specified in the epidemiological layer. The infection of a vector evolves according to a comparable process as in the host, but with a set of parameters n_v, d_v, θ_v, and ϵ.

In forward-dream, a P. falciparum infection is transmitted from a vector to a host through a transmission bottleneck, such that not all of the parasites within the infectious vector establish themselves in the host. In particular, a random subset of all parasites (where is the set of all parasites in the vector) is transmitted. The number of transmitted parasites is drawn from a truncated binomial:

This results in an average of parasites passing through the transmission bottleneck; the size of the bottleneck is controlled by p_v. For all of the simulations presented here, p_v = 0.2 and n_v = 20, resulting in an average of ≃4 parasites passing through the bottleneck.

Note that in cases where n > 1 and contains unique parasites, co-infection may occur. If the vector is infecting a susceptible (i.e. uninfected) host, n_h parasites are drawn with replacement from with each having an equal probability (1/n) of being drawn. These n_h parasites then populate the n_h host sub-compartments. Alternatively, super-infection occurs if the host is already infected. Defining the parasites already within the host by the set , we first create the union . From this set n_h parasites are drawn, where each parasite has probability 1/2n_h or 1/2n of being drawn, if it is from h or , respectively. As a consequence, on average super-infection results in half of the within-host compartments being occupied by new parasites.

The transmission from host to vector is the same as above, but with the addition of meiosis. In brief, the n parasite strains selected at random from the infected host may undergo meiosis before populating the the n_v within-vector sub-compartments. The meiosis model is based on a simplified implementation of our previously published meiosis simulator, pf-meiosis [29]. The number of recombination events is drawn from a Poisson distribution scaled with respect to chromosome length, such that an average of one cross-over event occurs per bivalent during meiosis. Recombination breakpoints are sampled uniformly from along the chromosome. The model includes multiple oocysts, allowing for parallel rounds of meiosis to occur during a single transmission event, with the number of oocysts being drawn from a truncated geometric distribution: ∼min[10, Geo(p_oocysts)].

In total, the infection layer is specified by nine parameters.

Genetic layer.

The genetic layer of forward-dream describes the malaria genome model. We represent the genetic material of an individual parasite as a single fixed-length array of zeros and ones, defined by the parameter N_snps (Fig 1c). In effect, this array represents a single chromosome marked with N_snps single-nucleotide polymorphisms (SNPs). Mutation is symmetric and reversible. The only parameter specific to the genome evolution layer is N_snps.

Overall, forward-dream is specified by 17 parameters (Table 1). It is implemented in Python and available on GitHub at https://github.com/JasonAHendry/fwd-dream.

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Table 1. Complete list of simulation parameters for forward-dream.

Values given represent those of a simulation with a host (X_h) and vector (X_v) prevalence of 0.65 and 0.075, respectively. This corresponds to the Initialise epoch of all malaria control intervention simulations (see below). For details on how parameter values were selected see the S1 Appendix.

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Relationships between parasite prevalence and genetic diversity at equilibrium

Within the forward-dream framework, it is straightforward to monitor the fraction of hosts that are infected (h₁/N_h), which corresponds to the most ubiquitously collected measure of transmission intensity, parasite prevalence (PfPR) (Fig 2a). It is also possible to monitor the fraction of vectors infected (v₁/N_v), which corresponds closely to the sporozoite rate (SP). Finally, the genetic diversity of the parasite population can be monitored by collecting parasite genomes from a sample of infected hosts and simulating DNA sequencing (see Materials and methods) (Fig 2b).

We sought to use forward-dream to elucidate relationships between PfPR and the genetic diversity of the parasite population. To this end, we varied PfPR across simulations and observed the resulting differences in parasite genetic diversity. Within the Ross-Macdonald framework, PfPR is a function of the four rate parameters (see Eq 1). In nature, what underlies prevalence differences observed between two geographies or points in time is often unknown, and likely the outcome of a myriad of epidemiological and environmental factors. To achieve different PfPR values in forward-dream, we choose three parameters to vary separately: (i) the human infection clearance rate (γ), which may vary between sites if, at one site there is quicker recourse to treatment, or differing proportions of symptomatic and asymptomatic individuals; (ii) the vector biting rate (λ), which may vary as a consequence of differences in vector species, environmental conditions, or the presence of bednets; and (iii) the number of vectors (N_v)—which is influenced by climate and weather, local geography, and also by insecticide-based interventions (see [30]). We tuned each of these parameters to achieve equilibrium PfPR values that varied from 0.1 to 0.8 in a population of one thousand human hosts (S3 Fig). At each prevalence value, forward-dream was seeded with forty infected hosts carrying identical parasites and then run to equilibrium. After reaching equilibrium, simulations were continued for an additional 10 years, during which time parasite genomes were collected every thirty days by sampling from the infected host population. Since we also wanted to explore the noise distributions of different genetic diversity statistics and how they are influenced by sample size, at each sampling time we collected parasite genomes from five randomly-drawn samples ranging in size from 20 to 100 infected hosts.

The genetic diversity statistics calculated from these parasite genomes are described in S1 Table. The statistics can be divided into three broad categories: (i) those related to mixed infections, which includes the fraction of mixed samples and the mean complexity of infection (COI); (ii) those that are related to the size and shape of the genetic genealogy of the sample, which includes the number of segregating sites, the number of singletons, nucleotide diversity (π), Watterson’s Theta (θ_w), and Tajima’s D; and (iii) those that summarize the structure of identity-by-state (IBS) and identity-by-descent (IBD) within the population, including, between a pair of samples, the average fraction of the genome in IBD or IBS and the average length of an IBD or IBS segment. We note that these statistics are not independent, indeed many are co-linear (S4 Fig), however they reflect commonly-used measures of genetic diversity in population genetics.

We conducted a total of thirty replicate simulations at each prevalence level and aggregated all the samples of a given size to explore both relationships with PfPR and sampling noise. The relationships for all eleven genetic diversity statistics are shown (Fig 3a and S5–S7 Figs), partitioned by which epidemiological parameter was varied. Linear regression was used to determine the fraction of the variance (r²) in each genetic diversity statistic that could be explained by variation by PfPR, and how this changed with sample size (Fig 3b). We observed that nearly all genetic diversity statistics have a considerable proportion of their variance explained by prevalence. For many statistics—the fraction of mixed samples, mean COI, number of segregating sites, nucleotide diversity, Watterson’s θ, and fraction IBS—the r² exceeded 0.9 for at least one epidemiological driver. The two statistics with the lowest r² values were Tajima’s D and the mean IBD segment length. A stable value for Tajima’s D would suggest that the parasite population’s genealogical tree has a consistent shape across different prevalence values. One possible cause for the modest r² values we observe for Tajima’s D is recurrent mutation that results from a finite genome size. A lower r² for the mean IBD segment length may be in part related to a lack of power to detect small IBD segments at higher prevalence values.

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Fig 3. Equilibrium relationships between genetic diversity and parasite prevalence.

(a) Boxplots showing the equilibrium parasite prevalence in hosts (x-axis) versus genetic diversity of the parasite population, by different statistics. Each individual box summarizes 30 replicate simulations at the indicated prevalence; in each, 40 infected hosts were sampled every 30 days for ten years to collect parasite genomes from which diversity statistics were computed. Left, middle, and right columns show relationships when equilibrium parasite prevalence is varied as a function of the host clearance rate (γ, in blue), vector biting rate (b, in green) or number of vectors (N_v, in red). The variance explained in an ordinary linear regression (r²) is shown at the top left for each relationship, and in (b) the variance explained is shown across a larger panel of genetic diversity statistics and as a function of sample size. The variation in r² as a function of sample size is indicated by point size. Slope of the line of best fit is indicated at left (+, increasing; -, decreasing). Boxplots for all statistics in (b) can be found in S5–S7 Figs.

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A second major observation is that there are pronounced differences in the variance explained by PfPR when different epidemiological parameters drive variation in prevalence. In particular, varying host clearance rate (γ) results in a significantly higher r² for all of the genealogy statistics, as well as for all of the IBD and IBS statistics, excluding mean IBS segment length. For example, the variance in nucleotide diversity explained by PfPR drops from 94% to 60% (for samples of size 40) when parasite prevalence is modulated by the number of vectors (N_v), instead of the host clearance rate (γ). This reduction in explanatory power is associated with a plateau in diversity at higher prevalence values observed when either the number of vectors (N_v) or biting rate (b) are increased. In contrast, the two statistics related to mixed infections have consistently high r² values (>75%), regardless of which epidemiological parameter is driving prevalence variation.

The r² value is a reflection of the signal-to-noise ratio for different genetic diversity statistics, and may be influenced by sample size. Here, we find that for some statistics there is considerable benefit to collecting additional samples, whereas for others there is only little benefit (Fig 3b and S8 Fig). When prevalence change is driven by the number of vectors (N_v) or biting rate (b), the variance in the fraction of mixed samples explained by PfPR climbs from 0.81 to 0.94 as the sample size is increased from 20 to 100. For the same increase in sample size, the r² of nucleotide diversity increases only from 0.62 to 0.68. The r² of the mean IBD segment length also increases substantially with an increase in the number of samples collected (from 0.01 to 0.15). Here, a small sample size can result in a very high variance in mean estimates, due to the chance detection of rare, long IBD segments. We found that increasing from 20 to 100 samples reduced the variance in mean IBD segment length by up to 60% (S8 Fig).

There are multiple sources for the noise that remains even after increasing sample size, related to the stochastic nature of both the epidemiological and genetic processes. Epidemiological noise means that prevalence fluctuates around its equilibrium value, and this can impact some diversity statistics. Noise on the genetic level can exist deep within the structure of the genealogical tree (nearer to the root), but also also more recently. To assess these contributions, we performed a variance decomposition analysis, determining what fraction of the variation in different genetic diversity statistics was occurring within- versus between- replicate simulations (S9 Fig). Since we collect parasite genetic data for 10 years for each replicate simulation, variation on a timescale longer than this will manifest as between- rather than within-simulation. Overall, we found that the majority of variation occurred between replicate simulations for the genealogy related statistics, indicating that they vary over long timescales (S9 Fig). Conversely, more than 90% of the variation in COI statistics existed within individual simulations, indicative of a short timescale of variation.

Non-equilibrium relationships between parasite prevalence and genetic diversity

An important application of our model is in settings where malaria control interventions are actively occurring. In such cases, and also in cases with seasonal variation in parasite prevalence, it is unlikely that the parasite population will be in equilibrium. Thus, our second aim with forward-dream was to explore which measures of genetic variation are most predictive of instantaneous PfPR in non-equilibrium settings.

Malaria control interventions.

In order to understand how genetic diversity statistics relate to PfPR in scenarios where a malaria control intervention has been deployed, we developed a framework where individual forward-dream simulations pass through three distinct epochs: Initialise, Crash and Recovery (Fig 4). In the Initialise epoch, simulations are run until equilibrium at a parasite prevalence of 0.65, under the parameters listed in Table 1. At the start of the Crash epoch, one of either the host clearance rate (γ), the vector biting rate (b), or the number of vectors (N_v), is changed such that the new equilibrium PfPR is 0.2. The parameter change occurs incrementally over a period of thirty days following a logistic transition function, as to mimic the staged introduction of a malaria control intervention. As a consequence, the simulation leaves equilibrium, with host and vector prevalence declining. The Crash epoch is allowed to continue until the population has regained equilibrium. At the start of the Recovery epoch, we return the changed simulation parameter to its original value, again over thirty days following a logistic transition function. Again, this results in the simulation leaving equilibrium, with the PfPR increasing back to 0.65. As with the Crash epoch, the Recovery epoch continues until the population regains equilibrium. In summary, the three epoch model allows us to explore both a parasite population decline and rebound.

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Fig 4. Responses of genetic diversity statistics to a crash and recovery in parasite prevalence.

(a) An individual forward-dream simulation where the number of vectors (N_v) is reduced at time zero (x-axis, indicated by grey vertical bar), resulting in a decline of parasite prevalence in hosts from 0.65 to 0.2. (b) Left column, the same simulation as in (a), but showing the response of three genetic diversity statistics (colored lines) to the prevalence change. Light and dark grey areas show host and vector prevalence, as in Fig 2. Right column, mean value of each genetic diversity statistics across 100 independent replicate simulations, with shading showing the 95% confidence interval. (c) Same simulation as in (a) at a later time, where the number of vectors is returned to its original value. Parasite prevalence increases back to 0.65. (d) Same as (b), but corresponding to the recovery shown in (c).

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Throughout the three epochs parasite genomes are collected every five days from twenty hosts selected at random, allowing the same suite of genetic diversity statistics discussed above to be followed through time. An immediate observation was that in individual simulations, the trajectories of genetic diversity statistics exhibited considerable noise, tending to fluctuate through time even in the absence of any change to simulation parameters (Fig 4). To better discern the average behaviour of different statistics, we sought to smooth their trajectories by computing a rolling mean. However, consistent with our variance decomposition analysis, we found that the timescale of random fluctuations differs greatly among genetic diversity statistics. Statistics such as the mean COI and the fraction of mixed samples fluctuate rapidly, and variation around their true mean could be substantially reduced (to <20% the original variance) by averaging the genetic data collected over roughly a month (S10 Fig). Yet we found that other statistics, such as nucleotide diversity, are more inertial in nature; they can trend up or down over very long time periods without any change to the underlying simulation parameters. Reducing their initial variance to an equivalent degree required averaging over 10 years worth of genetic data (S10 Fig).

Thus, we took an alternate approach to extract underlying trends: we averaged the trajectories of each statistic across 100 independent, replicate forward-dream simulations (Fig 4b and 4d). Several observations emerged. First and consistent with population genetic theory, Tajima’s D increases in the period where PfPR is declining (indicative of a contracting N_e), and falls below zero in the period where PfPR is climbing (indicative of an expanding N_e) [31] (S11 Fig). Second, there are marked differences in the rate at which different genetic diversity statistics respond to changing prevalence. For example, the COI-related statistics respond faster than IBD or IBS related statistics, which in turn respond faster than nucleotide diversity. Finally, we noted that the rate at which a given genetic diversity statistics responds to a decline in PfPR may be different from the rate at which it responds to an increase in PfPR (Fig 4).

We developed two metrics to summarize the temporal responses of different genetic diversity statistics to changes in parasite prevalence in our simulations (Fig 5, see also Materials and methods). Within the three epoch simulation framework, we could construct equilibrium distributions for each statistic before and after each prevalence change (i.e. equilibrium distributions for Initialise, Crash, and Recovery). Using these distributions, we computed a “detection time” (t_d), which we define as the amount of time, following an intervention, until a given genetic diversity statistic takes on values outside of its pre-intervention equilibrium. In effect, t_d is an estimate of how long it takes to detect that a change in parasite prevalence has occurred by monitoring a given genetic diversity statistic through time. We also computed an “equilibrium time” (t_e), which we define as the amount of time until a given genetic diversity statistic reaches its new, post-intervention equilibrium. Note that these metrics were designed to be informative summaries of the simulations only; it is unlikely they could be deployed in real-world settings.

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Fig 5. Detection and equilibrium times of genetic diversity statistics following a crash in parasite prevalence.

(a) Each plot shows the behaviour of a genetic diversity statistic in an individual simulation through a crash in parasite prevalence, induced by: left column, increasing the host clearance rate (γ); middle column, reducing the vector biting rate (b); or right column, reducing the number of vectors. The intervention occurs at time zero (x-axis, grey vertical bar) in all cases. For each plot, the detection time (vertical dashed bar) and equilibrium time (vertical solid bar) of the genetic diversity statistic is indicated. Note that here a single simulation is shown for each intervention type. (b) Empirical cumulative density functions (ECDFs) of the detection and equilibrium times of diversity statistics, created from 100 independent replicate simulations for each intervention type. The y-axis gives the fraction of replicate simulations with a detection (dashed line) or equilibrium (solid line) less than the time indicated on the x-axis. Line color specifies the type of intervention. Open and closed circles give medians for the detection and equilibrium times, respectively. The first year is magnified for clarity.

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Fig 5b shows the empirical cumulative density functions for t_d and t_e across all considered genetic diversity statistics, following the host prevalence decline at the beginning of the Crash epoch. Consistent with our observations from the averaged trajectories, the mean COI and fraction of mixed samples had both the shortest detection times (median of 6–9 mo., depending on intervention) and equilibrium times ( 2–3 yrs). The statistic with the next shortest detection time was the number of segregating sites ( 1–2 yrs), however this statistic had a very long equilibrium time ( 40–75 yrs). The next two fastest statistics were the fraction IBD and mean IBS segment length. We note these statistics are highly co-linear within our simulations (S4 Fig, Pearson’s R ≥ 0.89), and had both fast detection and equilibrium times (detection 1–4 yrs, equilibrium 8–16 yrs). The least responsive statistic was nucleotide diversity, which had a median detection time of 3–6 years and a median equilibrium time of 45–75 years.

We hypothesised that the fast detection time for the number of segregating sites was a consequence of starting from a relatively high prevalence (65%). In this context, the parasite population carries a large and relatively stable number of segregating sites (CV <10%), and so a significant reduction can be quickly detected. At the same time, IBD related statistics are less sensitive to changes that occur at high prevalence. To explore this further, we repeated the above experiment but declined from an initial prevalence of 30% to a post-intervention prevalence of 10%. Consistent with this hypothesis, the detection time for IBD fraction was faster than number of segregating sites in this context (S12 Fig).

Finally, we sought to understand how the detection and equilibrium times we observed depended on the size of the population under consideration. To this end, we repeated the above experiment but with 400, 700, and 1000 hosts. For statistics related to the genetic genealogy, we found that the equilibrium times grew markedly with the host population size (S13 Fig). For example, the median equilibrium time for nucleotide grew from 28 years for a host population size of 400, to 75 years for a host population size of 1000. Equilibrium times for the mean IBS segment length and Fraction IBD statistics also increased with population size, but to a much lesser degree. Interestingly, the detection times exhibited little dependency on host population size.

In terms of the relative behaviour of the statistics, the t_d and t_e values for the Recovery epoch were similar to that of the Crash epoch (S14 Fig), with mean COI and the fraction of mixed samples being the fastest statistics, and nucleotide diversity being the slowest. Overall, The median times tended to be longer, in particular for t_e. This is likely a consequence of the rate of diversity being re-established (by mutation) being slower than the rate of it being eliminated (by an intervention).

Seasonality.

We next aimed to explore whether any measures of genetic diversity were responsive to changes in parasite prevalence driven by seasonality. To this end, we developed a simulation framework where the number of vectors oscillates between a peak reached in the wet season and trough reached in the dry season, with PfPR fluctuating between ∼0.6 and ∼0.2 (see Materials and methods). Vectors that die entering the dry season are selected at random, with the dry season lasting 170 days and the wet season lasting 195 days. Consistent with our results from the intervention analysis, we found that the fraction of mixed samples and mean COI showed clear correlations with seasonal change in prevalence (r² = 0.62 for mean COI, r² = 0.59 for fraction mixed samples; Fig 6). We found that the mean IBD and IBS segment lengths also exhibited weak correlations with seasonally varying prevalence (r² = 0.12 and r² = 0.06, respectively). However, compared to equilibrium patterns, the relationships are in the opposite direction—with an increase in mean segment length at higher prevalence values. This is likely driven by “epidemic expansion” in the early wet season—with the parasite population expanding faster than it acquires new mutations, resulting in increased IBD [32]. Consistent with this, we observed similar increases in mean IBD and IBS segment length in the first year of the Recovery epoch explored in the previous section (S15 Fig).

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Fig 6. Responses of genetic diversity statistics to seasonal change in parasite prevalence.

Annual variation in parasite prevalence was induced by varying the number of vectors (top row, see Materials and methods). The behaviour of genetic diversity statistics for an individual simulation is shown in the left column from second row onwards. The mean behaviour of 10 independent replicate simulations is shown at middle, with shaded areas giving the 95% confidence intervals. Scatterplots at right show the relationship between each genetic diversity at parasite prevalence across the six years of seasonal fluctuation. Each point represents a genetic diversity estimate (y-axis) computed from sampling parasite genomes from 20 infected hosts in an individual simulation; parasite prevalence (x-axis) is computed across the entire host population at the same time. Data from all 10 replicate simulations has been aggregated. Variance explained r² from an ordinary linear regression is indicated at top left.

https://doi.org/10.1371/journal.pcbi.1009287.g006

Notably, none of the other genetic diversity statistics we calculated exhibited correlations with seasonal fluctuation in prevalence.

Discussion

The collection of parasite genetic data may, over the next few years, become a routine part of malaria surveillance. Yet, deriving the maximum benefit from this data will require an understanding of the relationships between malaria genetic diversity and epidemiology. Such understanding can be guided by modelling approaches, but only if both evolutionary and epidemiological processes are integrated. At present this is rare, as classical models in population genetics are poor approximations of the malaria life cycle, and classical epidemiological models don’t incorporate the evolution of parasites. To address this issue, we have combined the Ross-Macdonald and Moran models into a single framework, which we have implemented as a stochastic simulation called forward-dream.

We have used forward-dream to investigate the relationships between parasite genetic diversity and parasite prevalence in equilibrium and non-equilibrium settings. We find that many measures of parasite genetic diversity correlate with parasite prevalence at equilibrium. Our findings align with existing empirical data [25, 32–35], and support the idea that the rate of mixed infection (and as a consequence the rate of recombination) is positively correlated with parasite prevalence [36]. Moreover, we find that, for a given human host population size, statistics that reflect the long-term effective population size (N_e) of the parasite, such as a nucleotide diversity and the number of segregating sites, also increase with equilibrium prevalence. We also explored the behaviour of these genetic diversity statistics in non-equilibrium settings, most importantly in response to changes in parasite prevalence that mimic malaria control interventions. Other authors have emphasised that the viability of a genomic approach to malaria surveillance will depend on how rapidly signals of epidemiological change become detectable in a reasonably sized sample of parasite genetic data [25]. We find that statistics related to the COI distribution respond most rapidly (on the order of months), whereas other statistics, such as nucleotide diversity, may take decades to respond to a change in parasite prevalence. These pronounced differences in response times are related to the type of events that must occur to change different statistics’ values. The COI distribution changes directly in response to individual infection events (on the timescale of the rate of transmission and clearance), whereas other statistics change in response to drift (on the timescale of N_e) and/or mutational events (on the timescale of μ).

In addition to understanding the relationships of different genetic diversity statistics with prevalence, it is also important to understand sources of noise and the extent to which they can be controlled. The noise we observe in forward-dream has contributions from across the layers of our simulation: for a given set of epidemiological parameters, there will be variation in host prevalence due to the stochastic nature of infection and clearance events; for a given host prevalence, there will be variation in genetic diversity driven by the stochastic nature of drift, mutation and recombination events; and for a given population-level genetic diversity, there will be variation in sequencing data introduced by sampling a subset of all infected hosts. Moreover, on the genetic level, noise can be generated deep within the genealogical tree, related to the timings of coalescent events nearer to the root; and also in the more recent past, driven by more recent coalescent events. The extent to which different statistics are influenced by these sources of noise dictates the value of collecting additional samples, either as part of a cross-sectional survey or longitudinally. For statistics that are influenced by epidemiological noise (such as those derived from the COI distribution) and recent coalescent events (in particular mean length of IBD and IBS segments), increasing sample size can act to average over temporal fluctuations, or reduce sampling variation in a cross-sectional survey. In contrast, collecting additional samples in the present day does little to reduce variation driven by historical events and manifest in statistics like nucleotide diversity or Watterson’s Theta (θ_w). For these statistics, relatively few samples are required to produce good estimates, and additional longitudinal sampling in the short-term has little added value. Though not explored here, one way to average over such deep genealogical noise is by assessing a larger proportion of the genome.

Our results also demonstrate how relationships between prevalence and genetic variation are sensitive to the underlying epidemiological process. Specifically, we find that changes to the host clearance rate had a more profound effect on several genetic diversity statistics than changes to either the vector biting rate or density. The statistics exhibiting this behaviour are all related to θ = N_e μ. As we did not alter the mutation rate across simulations, we expect this observation is being driven by effects on N_e. Where a population’s size fluctuates through time, the N_e can be approximated as the harmonic mean of those sizes, and thus is more influenced by periods where the population is small [37]. Similarly, a P. falciparum lineage alternates between a large vector population and a small host population, and so one explanation for our observation is that the amount of diversity is impacted more by changes influencing the smaller host population. This result complicates efforts to use genetic variation metrics to compare parasite prevalence across space or time, as it implies that only under certain conditions will changes in prevalence be reflected by changes in genetic variation. Furthermore, it implies that it will be important to assess which modes of prevalence variation are observed more frequently in nature. This assessment can be aided directly by epidemiological data. For example, the empirical observation that the relationship between EIR and prevalence is log-linear suggests that variation in the number of vectors or biting rate may be a dominant epidemiological driver of geographical prevalence variation [38].

forward-dream has several limitations, most obviously with respect to its simplicity and scale. With regards to simplicity for example, heterogeneous biting, acquired immunity, and migration are all phenomena that have been proposed to influence the rate of mixed infections [2], yet they are not included in forward-dream. We do not explore the effect of selection, though this may be relevant in many contexts, particularly in Southeast Asia where drug resistance is widespread [39]. With regards to scale, we simulate only one thousands hosts, and our genome model is limited to single chromosome of eight thousand sites. As a consequence, some genetic diversity statistics have absolute values that differ from existing empirical observations. Two examples include our estimate of Fraction IBD at low prevalence values, which is higher than observation [40] as a result of the inbreeding that may occur in the small populations we simulate; and our estimate of the number of segregating sites, which is low as a result of our much smaller genome.

Many of these limitations could be addressed with the continued development of forward-dream, however, there are at least two salient considerations. The first is that increasing either complexity or scale will likely increase computational costs. Merging the Moran and Ross-Macdonald models resulted in forward-dream being more computationally expensive than either, and for most of the individual simulations in this study, run-times were on the order of several hours (S16 Fig). The majority of this was associated with the time required to bring forward-dream to equilibrium; a general problem faced by forward-time genetic models (see [41]). Reverse-time simulations harnessing coalescent theory can have greatly accelerated computational times by omitting processes extraneous to the sample of genetic data collected (for example [42]), however the reverse-time formulation of the model described here is yet to be elucidated. In a similar vein, the development of models separating epidemiological and genetic processes are underway [43], and could result in significantly faster simulations. Second, it is important to consider that more complex models typically require more parameters. Most likely these models will be both analytically intractable and statistically non-identifiable, thus making inference about their values impossible without additional and complex field experiments. Indeed, many of the parameters used within the present model have substantial uncertainty and were hard to find in current literature (see S1 Appendix). Community efforts to collate existing knowledge and address key uncertainties through experimental work would greatly benefit the field.

The immediate value of forward-dream, as demonstrated here, is as a tool through which relationships between genetics and epidemiology can be explored and experimental and analytical strategies can be evaluated. Moreover, as methods for epidemiological inference from malaria genetic data are developed, forward-dream can provide a basis for assessing their expected performance and designing ideal sampling strategies under different epidemiological scenarios. We do not see forward-dream as likely powering such inference methods directly, for example through techniques such as Approximate Bayesian Computation (ABC) (reviewed in [44, 45]), because of the limitations described above. Rather, it will enable the identification of approaches to hypothesis testing and estimation that are insensitive to the specific values of a multitude of epidemiological parameters (for example, assessing relative changes in prevalence by monitoring COI). It is in these ways that forward-dream, and other simulations like it, can provide a platform for interpreting the signals within the projected tens of thousands of malaria genomes that will be collected over the next decade, and can help to leverage those signals for malaria surveillance.

Materials and methods

Supporting information

Acknowledgments

We thank Tim Anderson, Lisa White, Jerome Kelleher, Dan Bridges and Penny Hancock for helpful discussions.