The development of drug resistance is a two-step process, the de novo or the initial emergence of resistance, and its subsequent spread. Resistance spreads because of the higher reproductive rate of resistance infections in the presence of anti-malarial drugs. In this paper, the focus is on modeling the spread of resistance assuming that drug resistance has already emerged among the human population. Combinations prevent resistance by preventing de novo emergence. Modeling the de novo emergence of drug resistance is discussed elsewhere 910.

Transmission and anti-malarial drug resistance are modeled on an age-structured population-based model, where the size and age-structure of the human population were assumed to be constant over time and based on an average African age-structure http://esa.un.org/unpp. A stochastic component of new imported infections (migrant) is added to the model making it more realistic.

The dynamic of mosquito population was summarized into vectorial capacities (VC) and estimated from several epidemiological studies. It is assumed that all humans are equally attractive to biting mosquitoes. Malaria is transmitted when a susceptible human is bitten by an infected Anopheline mosquito. The rate at which a susceptible person becomes infected or the inoculation rate is a function of contact rate with infective mosquitoes and level of host susceptibility, based on the formula given by Dietz 11 (see equation 1, Additional File 1). By assuming that all mosquito biting vectors are equally susceptible, human infectiousness to mosquitoes is determined solely by the gametocyte density in humans with a non-linear relationship shown from the malariatherapy study by Jeffery and Eyles 12 (see equation 5, Additional File 1). Human infectiousness can be estimated as the model handles malaria like a "macro" parasite by quantifying the density of infection in the human host. The in-vivo effect of drugs on parasite density can be measured, allowing quantification of the pharmacodynamic properties of anti-malarial drugs 13. The average gametocytaemia depends directly on the average parasite biomass by age group and the estimated gametocyte switch rate (i.e. probability that an asexual parasite develops to a sexual parasite). For simplicity, the gametocyte switching rate (GSR) was assumed to be uniform among asexual parasites and infections, but vary among anti-malarial drugs, and between primary and recrudescent infections.

The dynamic in human mainly consists of the development of immunity by age and transmission intensity. Making the model data-driven, the parameters for immunity functions and malaria infection in the human host (asymptomatic, symptomatic and recrudescent infections) were obtained from clinical, laboratory and epidemiological studies. Appropriate curve fitting including multiple linear and non-linear regressions was fitted to different data from age-stratified epidemiological studies in areas with different transmission intensities to derive immunity functions of age and EIR (Additional File 2). Stepwise selection using the Akaike Information Criterion (AIC) was used to identify the best fit in the case of non-nested functions and the maximum-likelihood ratio test was used for nested functions. These relationships represent the acquisition of immunity to malaria by age and frequency of malaria exposures. Different facets of malaria immunity are incorporated into this model i.e. reducing host susceptibility to infection 1415, reducing the level of (largely asymptomatic) parasitaemia in infected people 1617, reducing the likelihood of fever and other symptoms in infected patients 181920, reducing the treatment failure rate for a particular level of anti-malarial drug resistance 212223242526, and increasing the recovery rate of an established infection i.e. shortening the duration of infection 272829 (Table S7, Additional File 2). Some immunity functions were not measured directly from epidemiological studies (i.e. host susceptibility, duration of infections and treatment failure), so a normalized function of age-stratified parasite density was used to estimate the relationship between age and host susceptibility and between age and duration of infections by specifying initially the "maximum host susceptibility" in a non-immune person and the "maximum duration" of treated and untreated infections. The host susceptibility and the duration of infection for any given age group in any transmission intensity setting are then determined by the shape of the immunity curve. As the shape of the age effect on treatment failure is similar to the relationship between age and severe malaria 21, the same technique to the normalized function of age-stratified risk of severe malaria is applied to adjust the treatment failure rates for any given age group in any transmission intensity setting. The maximum value of duration of infection and the maximum value of treatment failure are dependent on treatment type, drug resistance, and likelihood of patient adherence to therapy. Adherence is incorporated in the model by adjusting down the expected failure rates of treated infections. Multiple recrudescences are treated as one continuous recrudescence, and overall infectiousness is calculated from the area under the gametocyte-time curve (AUCgam). Multiple infections in an individual or "superinfection" is not considered in this model.

The population with selective residual anti-malarial drug concentrations focused on this model is the proportion with concentrations in the blood which prevent establishment of new drug sensitive infections but allow establishment of resistant infections. This proportion of population was based on published literature (equation 6 – 7, Additional File 1).

At the outset, a human population is assumed to have little or no exposure to malaria and, therefore, the population has no immunity to malaria. Infected mosquitoes bite randomly and infect humans with drug susceptible infections i.e. malaria infections initially are all drug-sensitive and symptomatic humans receive only monotherapy (drug A). As the population becomes exposed to malaria and gains some level of immunity, the model updates the age-stratified immunity of the population according to the Entomological Inoculation Rate (EIR) (which varies with VC and the infective human population (see equation 8, Additional File 1)), and is allowed to run until a steady state is reached (Figure 1). A steady state point is defined as the point at which the number of new malaria cases (i.e. excluding imported cases) has varied day to day by less than 1% over a year. Any changes in malaria prevalence and levels of drug resistance thereafter can reasonably be assumed to result from introduction of drug resistance and the impact of the different treatment strategies. Once the steady state is obtained, the resistance to drug A is introduced, either as importation of a small number of resistant infections or by the de novo emergence of resistance based on available clinical and laboratory data. At a specified threshold of resistance to drug A (defined as a model parameter) artemisinin or its derivatives, or a completely new drug can also be introduced and used in a combination with drug A, or it can be used as a monotherapy. Resistance can then be tracked for a specified length of 10 years after steady state to gauge the impact on model outcomes over time. The model provides a number of outputs including estimated EIR, proportion of symptomatic infections, proportion of treatment failure, malaria prevalence and percentage of resistant infections. Those relevant to policy, the proportion of infections with resistant parasites, the malaria prevalence, and the incidence of malaria are presented here. The sensitivity of the model was tested in the four baseline scenarios (scenario A – D). The details of fixed and variable parameters with their respective distributions are given in Tables S1 – S6, Additional File 2. Each scenario is repeatedly run for 5,000 simulations with a unique set of parameters selected using the Latin Hypercube Sampling technique (LHS). The Coefficients of Variation (CVs), which determine the uncertainties of the model outcomes (Table S9, Additional File 3), and the Partial Rank Correlation Coefficients (PRCCs), which identify influential factors, were calculated (Table S10, Additional File 3). Full technical details can be found in 30.

Model consistency and model sensitivity are explored in four baseline scenarios (low transmission setting with low ACT coverage (scenario A), low transmission setting with high ACT coverage (scenario B), high transmission setting with low ACT coverage (scenario C) and high transmission with high ACT coverage (scenario D)). The model consistency is expressed as the Coefficient of Variation (CV) i.e. the variations in the model outputs given that there are uncertainties in the input parameter estimated (Tables S9, Additional File 3). For sensitivity, the Partial Rank Correlation Coefficient (PRCC) was used to describe the correlation between the model output to each input parameter (Tables S10, Additional File 3). ACT coverage is defined as the proportion of ACT treatments among all symptomatic treated infections.

In the low transmission setting (EIR < 1), with low ACT coverage, variations in the estimated prevalence of malaria stay consistently high over time (CV ~ 90%) after the steady state is obtained, while the variation in the estimated resistance falls substantially till year 8 (CV ~ 9%) with resistance approaching fixation at 100% in our 10-year time horizon. In low transmission settings with high ACT coverage, the mean prevalence of malaria stays below 1% over 10 years. Migration plays an important role in sustaining malaria. Without imported cases, malaria is readily eradicated. Mean resistance increases at a slower rate than in a low coverage setting, reaching 80% in year 10. Compared to scenario A, variation in the estimated prevalence is slightly lower (CV ~ 75%) while variation in the rate of resistance is higher (CV > 30%). This indicates some levels of uncertainty in the consequences of deploying a high coverage of ACT on the malaria prevalence and the rate of resistance. Once resistance to the slowly eliminated partner drug has emerged the spread of drug resistance and the malaria prevalence could be slowed down only by deploying ACT at a high coverage rate while the resistance prevalence is still reasonably low.

In the low transmission setting, VC was the most influential parameter affecting malaria prevalence (PRCC ~ 0.6), while VC (PRCC ~ 0.7) and the proportion of treated infections (PRCC ~ 0.4) were the most influential parameters affecting the spread of drug resistance.

In the high transmission setting (EIR > 100), malaria cannot be eradicated by anti-malarial treatment of symptomatic cases alone because the major transmission reservoir is in asymptomatic persons who do not take anti-malarial drugs. In the high transmission setting with low ACT coverage, malaria prevalence increases from 36% to 44% within 10 years due to the spread of resistance. Resistance spreads more slowly compared to the low transmission setting, reaching 80% in year 10. Variation in estimated prevalence is small (CV < 30%) while variation in the rate of resistance is consistently high (CV > 45%) compared to estimates in the low transmission setting. Both outcomes and their variations are unaltered by deploying a high coverage of ACT. Treatment of symptomatic infections in the high transmission setting has much less effect than in the low transmission setting. Consequently, the spread of drug resistance is driven by the fraction of the population with some residual anti-malarial drug in their blood and not by treatment failure.

In the high transmission setting, the parameters influencing malaria prevalence are the characteristics of the infection in immune subjects with untreated infections. These are the parasite biomass (PRCC ~ 0.4), the gametocyte switch rate (PRCC ~ 0.4) and the duration of infection (PRCC ~ 0.3). The proportion of residual drug in the population (PRCC ~ 0.9) is the dominant factor driving the spread of drug resistance in this setting. The strong correlation between the fraction of population with residual drug concentrations and the levels of resistance from this model suggests that controlling the use of presumptive treatment and encouraging the use of combination therapy with matching half-lives to reduce the selective window would slow the spread of resistance down within this setting.

In the first scenario, the model is used to predict the impact of varying the timing of switch to a high coverage (i.e. 85% of all symptomatic treated infections) by combining an artemisinin derivative with a failing partner drug (e.g. mefloquine), where the timing of the switch is governed by an observed prevalence of resistance to the partner drug in low and high transmission settings (Figure 2 and 3).

In the low transmission setting, treatment dependent parameter values are given in Table S8, Additional File 2. The force driving resistance comes from two sources; the first is from symptomatic malaria infections failing treatment and the second is exposure of infections to residual drug taken for presumptive or previous malaria treatment. More than 85% of all infections are symptomatic and thus treated. The rate of spread of resistance is faster relatively to the high transmission setting where treatment failure was identified to be the main force driving the spread of resistance. Deploying a high coverage of effective treatment, such as an ACT, when the level of resistance is still low delays the spread of drug resistance, a result consistent with previous results from simple epidemiological models 31.

In the high transmission settings, immunity prevents most acquired infections transmitting (and synergizes with anti-malarial drugs). Approximately 94% of all infections are asymptomatic and untreated. In the absence of anti-malarial treatment, the resistant infections have no survival advantage over the sensitive ones, and may have a fitness disadvantage. The main driving force for resistance is created from the selective filter provided by people carrying low residual concentrations of drugs, which protect against the establishment of new sensitive infections but not the resistant ones 3233. Without this residual drug effect, the rate of resistance would be much lower than shown in Figure 2B. Residual drug levels come mainly from previous 'presumptive' treatments (normally for other febrile illness), and are largely unrelated to the peaks of parasitaemia 9 and are, thereby, assumed to have little or no influence on the de novo resistance selection probability.

Combining an artemisinin with a drug to which resistance has arisen delays the spread of resistance in the low transmission setting (Figure 2A), but this delay decreases exponentially the later the switch is made to the ACT (Figure 3A). By contrast, in the high transmission setting, varying the time to switch to the ACT has only a small impact on delaying the spread of resistance (Figure 2B and 3B).

In this scenario, it is assumed that the monotherapy is the artemisinin and that drug resistance emerges to the artemisinin compound, rather than to the partner drug in an ACT (e.g. either piperaquine or lumefantrine), which is assumed to remain effective (see Table S8, Additional File 2). This simulates the current scenario in places such as Cambodia, where artesunate monotherapy use is widespread 34. It is assumed that the switch to the combination therapy is made when the resistance to artemisinin is as low as 1% (Figure 4). If the switch is made very early, when there are still very few cases of drug resistance, then the higher the coverage with the ACT, the greater is the delay in the spread of resistance. At coverage rates of > 80%, the level of resistance to the artemisinin does not reach 50% within the time span of 10 years. In general, the impact of ACT deployment on malaria incidence and prevalence is as expected. By deploying ACTs at a high coverage, the prevalence of malaria can be kept at a very low level over time (0.5%) and incidence is less than 50 cases per year, indeed in the model eradication is only prevented by the influx of malaria in immigrants (Figure 5). Similar to the first scenario, the impact of ACT in the high transmission setting is much less.

The model shows that deploying ACT in a high transmission setting has a small impact on the spread of resistance and malaria prevalence. However, as one of the model outcomes, treatment failure in both low and high transmission settings can be sustained to a low level by deploying high ACT coverage (25% of recrudescence compared with 10% when deploying high ACT coverage at year 10). In order to make an impact on malaria transmission and resistance, vector control strategies need to be applied to reduce the VC. Reduction in the transmission intensity results in fewer infections, which, as host immunity declines, are more likely to be symptomatic and eventually this makes malaria control by drugs more effective.