Agent-based models of malaria transmission: a systematic review

A systematic literature review was performed, consistent with the PRISMA statement [11]. Database searches of OVID Medline, CINAHL Plus and OVID Embase were performed as outlined in Additional file 1.

The search strategy aimed to return publications referring to each of the following three concepts in their subject heading, keywords list, title or abstract:

No limits were placed on publication type, language, location, dates or publication status. Further to database searching, additional articles were found through expert knowledge, and backward and forward citation searching (the latter using Google Scholar) of included articles.

The abstracts of all returned studies were assessed for suitability, with those that mentioned an agent-based, individual-based, or microsimulation model of malaria transmission selected for full-text review. Full-text articles were excluded if no agent-based model of hosts or vectors was described or used, or if no components of malaria transmission (such as human disease, vector biting, or interventions) were explicitly modelled. Models of vector life cycles or ecology without malaria-specific elements were excluded. Articles that described comparison or ensemble modelling of pre-existing models were considered separately, and were not “included studies” for the purposes of this review.

Due to the anticipated variation within the literature, a data extraction form was not developed prospectively. Instead, for included publications, key characteristics of each model were identified, with common themes and unique properties of models characterized. After model themes were identified, a classification for key components of models was created (see Additional file 2). The properties of each model were collated as discussed in the “Results” section.

Figure 1 outlines the timeline of models in the included papers. 27 original model frameworks have been published, with 15 models reported once to date, and 12 ‘baseline’ models collectively resulting in 63 further publications. Of the 27 original ABMs, six models were developed by extending compartmental or within-host models to an individualized framework [12‐17]. Model progression was largely similar across the largest model frameworks, including OpenMalaria [18], EMOD [19] HYDREMATS [20], and work from Imperial College [14], such that the topic of each paper progressed logically from previous works. Extensions included simulating interventions [21], spatial mapping [22], or embedding “sub-modules” of key disease concepts into pre-existing works, such as host infectivity [23], mortality [24], and potential impacts of climate change [25]. When interventions were adequately simulated, cost-effectiveness analysis (CEA) was sometimes performed [26‐28].

Methods of modelling at the individual level are influenced by factors including the question at hand, characteristics of the agent, and interactions of interest. Certain ABM frameworks naturally arise from the above concepts; in malaria modelling, key considerations include the choice of agent and whether to focus on disease states or transmission. While these choices are not mutually exclusive, three broad methods of individual simulation were common in the literature, each with differing degrees of agent autonomy. First, models developed from a compartmental structure typically used probabilities in place of flow rates to determine whether an individual transitioned to a new disease state at a given time step (e.g. Gurarie and McKenzie [13], and McKenzie et al. [29]). Using this method, the success or failure of a Bernoulli trial generally dictated disease progression. Second, and particularly in models focused on host parasite densities (see the OpenMalaria simulations [18]), distributions of key variables were sampled to generate differences between agents, with temporal disease state changes governed by a set of equations. In the third method, the specific actions of individuals, for example blood meal searching, resting and oviposition, were simulated according to a process represented by a flow chart (e.g. Pizzitutti et al. [30], Zhu et al. [31] or the EMOD framework [19]). Figure 2 presents a hypothetical flow chart of vector actions, to illustrate a typical set of transitions available to a mosquito.

The diversity of methods led to a broad range of agent constructions, affecting the number and type of variables, the variation between individuals, and the type of agents required. Because methods to construct ABMs are so broad, two examples are provided below for comparison. These two models are examples of ABMs being tailored to the question at hand and highlights the variation in model construction.

Consider an early OpenMalaria simulation of the relationship between the entomological inoculation rate (EIR) and force of infection [18]. Human agents had individual ages, leading to an age-adjusted EIR. Individual mosquitoes were not explicitly modelled; instead, EIR was used to guide biting rates [32]. Two individuals of the same age may have a different number of infections on a particular day, but these numbers are drawn from the same distribution [33]. As OpenMalaria grew to answer new questions, these individuals also had different gametocyte densities [34], pyrogenic thresholds [35], and morbidity and mortality [24].

In contrast, Zhu et al. [21, 31, 36] developed a spatial ABM to investigate the impact of the location of food, hosts, and resting and breeding sites on malaria transmission. This required simulation of a physical landscape, with both mosquito and human locations guided by movement patterns. Mosquito agents had a sequence of actions to be undertaken, and therefore required variables capturing their current hunger, breeding state, and flight distance since last meal. The number of bites for each human was also recorded. The distance to the target of interest (sugar, host, resting site, breeding site) and time of day could dictate vector flight paths. Capping the number of eggs per larval habitat required the current number of eggs to be recorded.

There were also considerations relating the method of agent simulation and size of the time step. Where compartmental models inspired an ABM, daily time steps were most common [15]. To simulate host infections using distributions for variables, a temporal resolution of 1–5 days was sufficient to represent changing disease characteristics [32]. However, if agent actions were explicitly modelled, hourly status updates were most common, with some models tracking vectors as frequently as each second [21, 31].

Models with only humans as agents commonly focused on the impact of infection characteristics or medical interventions on transmission. For example, the OpenMalaria models from the Swiss Tropical and Public Health Institute had at their core individual humans with varying parasite densities [18]. These models were extended to incorporate individual effects of innate immunity [34], pyrogenic thresholds [35] and the impact of vaccination [43], particularly pre-erythrocytic vaccines [18, 33, 44]. The models from this group accounted for 20 of the 49 human-only studies.

Key simulated human factors included age and host immunity, with immunity either being maternal, [13, 14] acquired [18, 39, 45] or both [34]. Details about immunity varied from general descriptions of the impact of immunity on transmission, to individualized antibody concentrations for sexual-stage parasites [46]. Human behaviours regarding intervention use [30] and treatment decisions have been simulated for both patients [47] and carers [28]. One model simulated the impact of anti-malarial use on HIV-positive pregnant women [48], including disease severity and improvements in birth weight.

The most commonly simulated disease aspects were gametocyte density (incorporated in all OpenMalaria models [18]) and the infectiousness of hosts to mosquitoes [49], with other common elements including fever [35], disease morbidity and mortality [24] and disease severity [24, 50].

Of 81 papers with human agents, 50 stratified individuals by age (Additional file 2, column 13), either to vary disease profiles or report results. Commonly, age was used in the calculation of human biting rates, because of its strong correlation with body surface area [18]. Other age-varying factors were adaptive and maternal immunity and duration of infection [17]. If interventions were age-specific, for example vaccination programmes [33], stratification was used to assess the impact at different levels of the community [44].

Models with mosquitoes as agents most commonly assessed interventions impacting vector mortality, such as habitat removal [30], IRS [16] or ITN [51]. Detailed models of vector life cycle and ecology were common, with mosquito behaviours, such as feeding habits [31, 52], movement patterns [30, 53] and biting frequency [54], modelled in detail. Vector movement was often modelled as random motion, but CO₂ gradients were also considered [20, 55‐57]. Simulations such as the EMOD framework modelled the egg population as a cohort for their progression to adulthood [19], and two models simulated eggs as individual agents [58, 59].

In most publications, only female mosquitoes were simulated [60, 61], as the sex responsible for transmission. Male mosquitoes were generally modelled when a complete population was required, for example to assess vector control interventions (e.g. [21]). Males were specifically required to examine the effects of introducing gene drive mosquitoes into an environment, such as “driving-Y” modified male vectors [62]. Although dominated by An. gambiae representation, models also varied by Anopheles species, including An. vagus [63], An. stephensi [64], An. arabiensis [62], and An. darlingi [30, 37]. One An. gambiae model [65] was adapted to An. vagus vectors [63] to simulate transmission in Bangladesh, and several models incorporated multiple Anopheles populations [14, 56, 66‐68], were adaptable across species [51, 58, 59, 69], or were nonspecific in Anopheles species [60, 70, 71].

When both hosts and vectors were simulated, models focused on mosquito life cycles [19], population dynamics [58], physical environments [55] and interventions for vector control [21]. Human agents were often included to relate Anopheles populations and malaria transmission, in models with vector dynamics as a core component. When models also included a spatial component, malaria transmission generally required vector and host to be co-located. These simulations used a ‘decision tree’ to represent the timing of movements and other necessary actions [21, 30, 31] (e.g. Fig. 2). Simulation of both hosts and vectors at the individual level was regularly used to assess interventions directed at mosquitoes and their effects on malaria transmission [19, 72].

Of 54 models that specified a malaria parasite (see Additional file 2, column five), 51 modelled the dynamics of Plasmodium falciparum, two simulated both P. falciparum and Plasmodium vivax [30, 37], and one Plasmodium berghei [64]. The in-host modelling of P. falciparum dynamics by Molineaux et al. [12, 18, 73] was used in 21 papers. In some studies, Plasmodium infections within an individual were modelled as agents [30, 37]; when a Plasmodium agent reached specific points in its life-cycle, the vector or host agent would change infectious status.

One study characterized the infectious reservoir in humans according to parasite characteristics [68], with parasite densities known to be linked to host infectiousness. Studies investigated the impacts of infection not only on hosts, but also vector behaviours, such as altered biting rates [56, 57]. Parasite biology was also considered, including parasite strains [17, 74], PfHRP2 status [75, 76] and recrudescence of P. vivax malaria [30]; the latter model considered the disease to be a submodule of the overall simulation framework. Three models simulated multiple parasite clones [15, 17, 77], and four allowed antigenic parasite variation [15, 78‐82], particularly to capture antimalarial resistance.

The choice of parasite, much like the Anopheles species, was often based on the dominant species in the target location. The dominance of P. falciparum malaria simulation reflects the attention paid to it, which is largely due to the historic relative burden of disease. Additionally, a number of interventions simulated were specific to P. falciparum, in particular the pre-erythrocytic RTS,S vaccine [14, 44, 83].

Modelling different Plasmodium species may require changes to model structure, for example to account for the recrudescence seen in P. vivax but not P. falciparum malaria. Pizzitutti et al. [30, 37] incorporated these differences by adding parameters governing recurrence time and risk of P. falciparum-triggered P. vivax recurrence. The parameters representing infection vary between Plasmodium species, based on prior evidence. Beyond these changes, the decision-tree model structures dictating the actions of agents were not changed.

Core environmental aspects related to vector activity were included, such as water sources for oviposition (egg-laying) [20], houses for blood meal locations [72], and meteorological data to account for seasonal patterns in transmission [58, 84]. Rainfall and temperature data were regularly used; model questions included the impact of wet season lengths [42] and hysteresis [40] on vector populations.

Thirty of the 90 models explicitly incorporated a spatial framework into their model (Fig. 1 and Additional file 2, column 11; explored further in Additional file 4). Seventeen models investigated malaria transmission in Africa, by simulating a specific location (e.g. [20, 85]) or using a hypothetical landscape representing a typical African village (e.g. [31, 53]). Two models represented transmission in the Amazon region [30, 37] and one each in Papua New Guinea [17], France [71], Haiti [86] and Bangladesh [63]. Spatial models also simulated local vector activity in a generic physical environment [51, 57, 58].

While many spatial models had detailed representations of a known landscape [25], this was not always the case, particularly with early spatial models [22, 86]. Three methods of spatial construction predominated: grid-based systems, patches and continuous landscapes. Additionally, Depinay et al. [58] explicitly linked vector locations to houses, but did not assign spatial coordinates, whilst Karl et al. [17] generated a probability of transmission based on the distance between host and vector locations to incorporate spatial biting factors.

In a grid-based landscape, the modelled area was most commonly divided into squares of a constant size, with elements of the landscape residing within a square (e.g. [22]). The HYDREMATS hydrology/entomology models were the exception, with grid size varying depending on proximity to human habitats, and complexity of water sources in the area [20]. There was interplay between the physical properties of the study area and the construction of the model. For example, the size of grid squares were related to parameters such as typical distance moved by mosquitoes in one time step, or larval habitat size [20]. The overall simulation area, typically a square between 600 m and 3000 m in length, was commonly justified based on Anopheles dispersal behaviour and typical village sizes being modelled [31]. Six models did simulate much larger areas [16, 17, 37, 62, 67, 85]; the last of these models used 1 km grids to describe the environment, yet characterized movement and interventions on much finer spatial scales [67]. Distances from which mosquitoes can target hosts or oviposition sites were guided by previous field data where possible [30, 31].

The additional spatial methods addressed opposing limitations of grid-based landscapes, namely total area and spatial precision. A patch framework was used by Silal et al. [16] and Eckhoff et al. [62] to represent areas of differing transmission characteristics (e.g. changing EIRs within districts), without explicitly modelling the environment. This method allowed for simulation of broader spatial areas. Rateb et al. [86] achieved this by representing Haiti as a set of “microenvironments” that host agents were impacted by. Conversely, models with a continuous landscape were used to allow variation in the location of houses, larval habitats, sugar sources and resting sites, and to model the impact of different physical environments [21, 31]. Each object had a specified location, and mosquitoes could sense objects within a certain circular distance, as opposed to a particular grid square. However, even when using a village-sized continuous landscape, models from Zhu et al. [21, 31, 36] still mapped random mosquito flight using a grid-based system, albeit on a 1 m × 1 m scale.

Some models also impute spatial results from non-spatial models by alternate means. The Malaria Atlas Project [87] and the Markham Seasonality Index [88] have generated methods to predict EIRs by location, and researchers have combined these tools with model outputs to generate maps of estimated disease burden across sub-Saharan Africa (SSA) [27, 54, 69, 78, 89, 90]. Models also used SSA rainfall data to replicate seasonality patterns [69, 91] and population estimates for host locations over large areas [91, 92]. In one instance, a spatial model of hypothetical locations was simulated in conjunction with geographic information systems (GIS) technology, to represent transmission in specific locations [61].

Fifty-eight studies assessed the impact of at least one intervention on malaria transmission, mosquito prevalence or EIR (Additional file 2, column 6). The majority of papers assessed multiple interventions, 21 assessed interventions in combination, and one investigated the removal of current malaria strategies [52]. Interventions could broadly be divided into those targeted at the human host (e.g. pharmacological) or the vector. Model structure tended to support simulation of the intervention being tested. For example, larval habitat elimination [51] or toxic sugar baits [21] were modelled spatially, requiring vectors as agents; whereas vaccination [18, 33, 44] and drug administration [93] studies involved human agents, in order to capture individual immune responses.

Interventions were included in models for two reasons: for assessment of effectiveness [18], or to replicate the ‘baseline’ characteristics of regional transmission [26], to allow additional interventions to be explored. In general, baseline scenarios included ITN [54] and “case management” [26], as defined by the current interventions in the location being modelled.

Simulation of interventions usually took one of three approaches. First, a specific intervention was defined to have a certain effect when in use, for example a medication having a fixed efficacy, and the impact on transmission is measured. Second, where environments were explicitly modelled, interventions were usually defined by their physical impact. For example, from a baseline scenario, an intervention might alter individual larval habitats [51] or remove sugar sources to reduce vector feeding [31], and simulations highlighted the change in the output of interest, such as disease burden or vector populations. Third, hypothetical interventions were described by their impacts, to target a specific aspect of malaria transmission [65]. The hypothetical impact may approximate a pre-existing intervention, such as halving human biting rates for a fixed period [94], or evaluate novel targets that may guide future control techniques.

The ease with which ABMs allow for heterogeneity and model complexity enables some less often considered transmission factors to be simulated. Examples include anthropophily (i.e. vector preference for human hosts) [19], sugar sources (e.g. fruit trees) [21, 31], travellers [16, 95], the impacts of larval densities in oviposition sites [59, 65], and the pharmacokinetics and pharmacodynamics of anti-malarials [75, 80, 91, 96].

Most models used past literature or simulations to determine baseline parameter values (Additional file 2, column 8). Due to the nature of certain inputs, many parameters cannot or have not been estimated in field studies, and consequently authors used expert knowledge to select these values. Five papers provided no evidence that prior knowledge was used to estimate parameters. These models generally introduced a new modelling technique, or assessed hypothetical scenarios, as opposed to claiming accurate simulation of real-world situations.

Fifty-two papers used models that were either previously calibrated to data or presented calibration as a component of their work. Calibration techniques included the use of calibration vectors, least squares, maximum likelihood functions, and visual estimations. Twelve papers mentioned using Bayesian techniques for model fitting, with five providing credible intervals alongside point estimates of parameters [14, 23, 32, 50, 89, 91, 92, 97]. Of note, no papers calibrated all model parameters to data, which necessarily excludes certain parameter combinations that could produce accurate calibrations.

Validation was performed on models used in 31 papers. All major model frameworks were reported as validated. Validation techniques were rarely explained in detail. When described, validation was most commonly performed by running a calibrated simulation, and comparing model outputs to a dataset not used for calibration [98]. Methods of comparing models to data were rarely explained; use of a square distance function [37], log likelihoods [16] and docking techniques [99] were outlined. Successful model validation was often used to justify extending a model framework to include interventions or to assess their potential impact in the location of interest. A number of studies concluded that their model did not accurately fit the data used for validation [98], suggesting incomplete data may explain any discrepancies.

Forty-five papers either explicitly mentioned sensitivity analysis or described parameter variation and comparison of outputs. The techniques used were generally informal, with methods used rarely explained in detail, and reporting of results was uncommon. Where sensitivity analysis was explained, it involved altering calibrated baseline parameter values by a fixed percentage and assessing changes to outputs. Formal techniques employed included Latin Hypercube sampling [70, 72], regression tree analysis [72], and one-way and probabilistic sensitivity analysis [28].

A number of papers referenced optimization use, including in titles [21, 45, 85] and keywords [100], and seven papers performed analysis for the clear purpose of optimization (Additional file 2, column 16). Methods included changing the timing [94, 95], location [21] or implementation of interventions [45, 85]. Five studies assessed between nine and 144 scenarios each. In contrast, two studies described formal optimization methods [69, 97], whereby one or more objective functions are optimized using a defined search algorithm. By changing the combination and coverage of interventions in a variety of baseline settings, these studies compared 98,784 and 306,000 scenarios, respectively.

CEA was performed in 11 studies, with six of these using the OpenMalaria platform to assess various medical interventions [101‐103] or pre-erythrocytic vaccine RTS,S/SA02A [26, 27, 104] (Additional file 2, column 14). CEA studies used known local costs of the physical intervention and the costs of implementation [27, 104] to determine the financial impact of interventions. These values were compared to benchmarks, such as the World Health Organization (WHO) standard of $150 per life-year gained [28], to determine the potential value of interventions. No studies investigated the maximum impact on outputs for a fixed cost, instead assessing cost-effectiveness of interventions for a specific level of impact.

When determining how to scale interventions for CEA, one method employed was incremental cost-effectiveness ratios (ICERs) [27, 103], in which a baseline scenario was compared to the introduction of a number of interventions, to determine the net cost per case averted of each method up to a certain threshold. Interventions are added iteratively in a manner that does not perfectly optimize their delivery but has the lowest additional cost for each new intervention block.

Where multiple models analyse the same transmission scenario, ensemble modelling can be used to highlight robustness of results, or to assess the appropriateness of model structure. Similar to the role of sensitivity analysis in evaluating the sensitivity of model outputs to parameter values, ensemble modelling can assess aspects of both parametric and structural uncertainty. A precursor to this was performed by Ross et al., extending a baseline model to five variants to assess hypotheses regarding treatment and illness patterns [105]. In addition to the individual studies in this review, 12 publications included ensemble modelling of OpenMalaria model variants [106‐117], and three studies performed “consensus modelling” across different frameworks [118‐120] (see Additional file 5).

The OpenMalaria ensembles used from six to 14 model variants, selected from 30 tested alternatives. The 16 rejected models were either very similar to an accepted choice or could not be parameterized to the dataset used. For the three articles that compared outputs across the ensemble, the selected models were in agreement for most outputs [106, 107, 113]. The most common divergence was differing intervention efficacies predicted by models with in-built transmission heterogeneity, compared to those with uniform transmission levels between individuals. Interestingly, even though the fourteen models often reached the same conclusion, the estimated parameter values from calibration varied significantly across the ensemble [107].

The nine additional OpenMalaria ensembles aggregated the outputs from model variants, to investigate interventions such as vaccination [108, 116], seasonal malaria chemoprevention [115], mass test-and-treat strategies [109], and long-lasting insecticidal nets (LLINs) and IRS [110‐112, 114]. One study predicted changes to disease outcomes and transmission upon the removal of vector control interventions from elimination programmes [117]. The aggregation of model outputs for each parameter set was performed to reduce the uncertainty arising from structural differences in models. In this way, the ensemble is considered as one model and its outputs interpreted as such.

Unlike the ensemble modelling above, when comparing inter-model variation, simulation outputs could not be aggregated. Despite the varied approaches, the consensus modelling largely drew consistent findings on the relationship between available malaria prevalence data and clinical incidence [118], and on the impacts of vaccination [119] and mass drug administration (MDA) [120].