Antiretroviral therapy (ART) has drastically reduced mortality and morbidity of HIV/AIDS in resource-rich countries; however there are now fairly high levels of drug resistance in many of these countries 
. Therefore there is concern that the rollout of ART in Africa could quickly lead to high levels of transmitted resistance arising which would reduce the effectiveness of epidemic control efforts. The World Health Organization (WHO) has developed a surveillance system for monitoring the emergence of transmitted resistance in resource-poor countries 
. The WHO detection methodology is based upon a binomial sequential lot quality assurance (BSLQA) sampling method with an arbitrary detection threshold of 5% 
. This threshold enables determination of whether the incidence of transmitted resistance lies in one of three categories: less than 5%, 5% to 15% or over 15%. The range of the minimal number of newly infected treatment-naïve adults that need to be sampled to detect a 5% threshold ranges from 50 to 70 newly infected treatment-naïve adults 
. Hence the BSLQA methodology is cost-effective and easy to implement, but will not be able to detect transmitted resistance until it has exceeded 5% of the incidence rate. It is unknown when the 5% threshold will be reached in any country, because the WHO has not made any country-specific predictions to determine how quickly transmitted resistance is likely to evolve. Here we develop a mathematical model that can be used to make short-term predictions for the emergence and stochastic evolution of transmitted resistance in a resource-poor country. We use our model to determine when the WHO's 5% threshold is likely to be exceeded in Botswana.
Botswana is one of the world's worst hit countries by the AIDS pandemic. An estimated 39% of Botswana's 730,000 adults between the ages of 15 and 49 are HIV-infected 
. In 2002 HIV/AIDS was declared the most serious threat to the country, and their ART program was initiated. Currently the program treats 42,000 patients 
. The goal is to treat 85,000 patients (or equivalently 30% of their HIV-infected individuals) by 2009 
. Botswana has by far the highest treatment rate in Africa, and has already slightly exceeded their scheduled annual treatment targets. Botswana is now considered a test nation by foreign investors for ART programs in sub-Saharan Africa. Recent substantial financial investments have enabled the opening of 31 treatment centers, ensuring that Botswana is quite likely to reach the government treatment goal by 2009. We use our stochastic model to predict the expected magnitude, and the temporal dynamics, of transmitted resistance in Botswana by modeling their government treatment plan. Specifically, we predict the likely evolutionary trajectory of transmitted resistance (due to the Botswana government's treatment plan) by varying: (i) the rate of acquired resistance, and (ii) the relative transmissibility (i.e., fitness) of resistant strains. We conclude by discussing the implications of our results in the context of the WHO's surveillance system for transmitted resistance.
The first-line drug regimen used in Botswana is a fixed combination of Zidovudine, Lamivudine, and Nevirapine or Efavirenz 
. The evolution of transmitted resistance is driven by three key parameters: the treatment rate, the rate of development of acquired resistance and the fitness of the drug-resistant strains that evolve 
. We modeled the specific treatment rate for Botswana (see Materials & Methods and Mathematical Details S1 section 1
). Thus, we investigated the effect of the other two key parameters on the evolution of transmitted resistance in Botswana. The rate of acquired resistance depends upon adherence, viral replication rate, and the specific mutations that arise 
. Acquisition of one mutation will not necessarily reduce the clinical efficacy of a regimen; thus we did not model the accumulation of mutations, we modeled the virological failure rate due to acquired resistance. Under perfect adherence, only 5–10% of patients are likely to develop drug resistance in the first year 
. However, if adherence is less than 100%, a greater proportion of patients will develop drug resistance within a year 
. Recent studies on adherence in sub-Saharan Africa indicate that very favorable adherence rates can be reached and that these adherence rates can often be higher than those reported from North America: 77% adherence rates in Africa compared to 55% adherence rates in North America 
. However patients in Botswana may suffer from poor treatment accessibility and interruption of drug supply which could compromise their adherence to treatment 
; hence it is possible that rates of acquired resistance in Botswana could be high. In order to make worst case predictions, and to maximize the speed at which transmitted resistance could arise, we modeled high rates of acquired resistance. To make our short-term predictions we assumed that (on average) either 20% or 33% of treated patients could develop acquired resistance per year 
. By considering the worst case scenarios for the rates of acquiring resistance we were able to determine the earliest times that transmitted resistance could reach the 5% detection threshold.
Patients who acquire resistance have the potential to transmit resistant strains via sexual transmission. Relatively little is known about the fitness of resistant strains of HIV in vivo
. However, in vitro
experiments have shown that the replication rates of resistant strains are generally less than that of wild-type strains 
. Thus, it has often been assumed that resistant strains will always be less transmissible than wild-type strains. However, certain mutations may substantially reduce fitness whereas other mutations may have little effect. Hence it is possible that some drug-resistant strains could be almost as transmissible as wild-type strains. To consider all possibilities we examined the potential impact on transmitted resistance of a wide variety of mutations by varying the fitness of drug-resistant strains relative to wild-type strains from 0% to 75%. Specifically, we modeled the potential impact on transmitted resistance of drug-resistant strains with a relative fitness of: (i) 100%, (ii) 50%, or (iii) 25% 
. The parameter values of the transmissibility/fitness of the wild-type strains and of the drug-resistant strains were calculated based upon the viral load of the treatment regimen (i.e., Zidovudine, Lamivudine, and Nevirapine or Efavirenz 
) that is used in Botswana (see Mathematical Details S1 section 1.2
for further discussion of calculation of these parameters). It should be noted that resistant strains that have low fitness may acquire compensatory mutations and hence increase their fitness over the long-term. Since our predictions are only short-term (i.e., only to 2009) we ignored the evolution of fitness.
There are relatively few mathematical models of ART and drug resistance; notably the first was in 2000 by Blower et al.
. Since then other studies have presented similar models 
. All three of these models have been based on deterministic formulations of the dynamics of HIV resistance. However, the early stages of the dynamics of an ART program are subject to large variability and therefore necessitate a stochastic treatment. In the analysis presented in this manuscript we are the first to develop a stochastic model of the evolution of drug resistance. We made projections for transmitted resistance in Botswana by first formulating a continuous-time Markov chain model; see Materials & Methods
for mathematical details. To construct this model we developed a combination of deterministic and stochastic equations (see Mathematical Details S1 sections 1 & 2
for detailed description and discussion). Deterministic equations were used to predict the dynamics of the number of adults in Botswana who are: (i) susceptible or (ii) infected with wild-type strains of HIV and treatment-naïve or (iii) infected with wild-type strains of HIV and on treatment. Their values are relatively large and currently are approximately 460,000 and 280,000 and 34,000 respectively. Thus, relative stochastic fluctuations in these three groups could be neglected and the mean-field dynamics from the deterministic version of our stochastic model used. However ART has only been available in Botswana since 2002 therefore the number of adults infected with drug-resistant strains is relatively small and quantifying the stochastic fluctuations in transmitted incidence is important. Essentially we are predicting the initial stage of an epidemic; specifically, the early stage of a drug-resistant HIV epidemic. At this early stage the population dynamics of transmitted resistance will be subject to a high degree of random (i.e., stochastic) fluctuations1
. Capturing the effect of these stochastic fluctuations is an important part of any predictions made of the early stages of an epidemic. Therefore, in addition to mean-field predictions of transmitted resistance we also predicted the dynamics of the variance and skewness of these stochastic fluctuations by formulating and using a stochastic dynamic version of the deterministic model (see Mathematical Details S1 section 2.1
). We derived a birth-death-immigration Master equation from our stochastic dynamic model, solved the Master equation analytically and used this analytical solution to determine explicit expressions for the mean, variance and skewness of the evolutionary dynamics 
(see Mathematical Details S1 section 2.3
). We then used these explicit expressions to predict the stochastic evolutionary trajectory, to 2009, of transmitted drug-resistant strains in Botswana. This approach is valid over the short-term dynamics. We chose not to simulate the process numerically via Monte Carlo simulations as our aim was to obtain expressions from which one can directly obtain the magnitude of the fluctuation