Parameters for a stochastic model were estimated from longitudinal data from approximately 5000 people within 16 separate Ethiopian villages during the first 6 months after a mass distribution. The MLE estimates for our model gave β = 0.044 and γ = 0.017. Gamma can be interpreted as the reciprocal of the duration of infection, and here would be approximately 1 year.
With estimated parameters, our mathematical models did indeed show that elimination is possible with repeated treatments. The stochastic element of the model recreates what we see happening in real life. In some villages we see infection returning, while in others it disappears. As more treatments are administered, we see more and more villages achieving elimination. Observing results from a 1000 simulations using the stochastic model, the average
prevalence of infection is progressively reduced with each periodic treatment as in the deterministic model (Figure and , red curve). However, this reduction does not necessarily occur in all of the villages. In some simulated villages infection is eliminated with a single mass treatment. In others, it is reduced to a low level and fades out by chance (Figure , red curve). In still others, infection may return quite rapidly (Figure , blue curve). Simulations of 1000 communities demonstrate that infection is eliminated in more villages with each subsequent treatment. If not eliminated, it returns on average to the same level before the next scheduled treatment (green curve, Figure ). In fact, if we set aside the villages that have achieved elimination, the distribution of the prevalence of infection after the 3rd
treatment is similar to that after the 4th
treatment, and prevalence after the 4th
even more similar to that after the 5th
(Figure ). After two treatments, simulated communities quickly approach a distribution which, if conditioned on non-elimination, is stationary (termed a quasi-stationary distribution) [21
]. This quasi-stationary distribution can be determined analytically (violet curve in Figure , see methods).
Figure 2 a. Simulation data after one treatment: In identical simulated communities, infection responds to identical treatment in different ways. It may return after a single mass antibiotic treatment relatively rapidly (blue curve) or fade out (red curve) due (more ...)
Subsequent data from the same sixteen Ethiopian villages at 12, 18, and 24 months appear to confirm that elimination is possible. In one village, 50% of the children were infected at baseline, but no infection was identified in any of the 5 visits after the first treatment. In another, 45% of children were infected at baseline, but no infection could be found after the second treatment. If infection was not eliminated in a village then the distribution to which it returned 6 months after the 2nd treatment approaches that to which it returns after the 3rd treatment (Figure ). There is no statistical difference between distributions 6 months after the 3rd and 4th treatments (Kolmogorov-Smirnov test, P = 0.95 comparing 18 and 24 month data).
This stochastic model does not incorporate re-introduction of infection from neighbouring villages. The rate that infection is re-introduced is not known and is difficult to estimate. Re-introduction appears not to be frequent, as communities do well in the short term, whether or not neighbouring villages were treated (unpublished data). On the other hand, it may well be occurring [13
]. Some villages in which no infection could be found in children for several visits had infection identified at a subsequent visit. Even occasional transmission from other communities would prevent sustained elimination. Interestingly, the quasi-stationary distribution can be estimated from models by only considering villages where infection has occurred (the limiting case of the curves in Figure ), by preventing the last infectious case from disappearing (violet curve in Figure ), or by artificially introducing a single infection into a community whenever infection has been eliminated, much as might be expected with re-introduction[21
]. Since each of these methods mimics re-introduction, then the quasi-stationary distribution may be a reasonable estimate of the distribution of prevalence of infection.
Investigators are struggling to determine why infection returns in some villages and not others. This search is important, but may not be fruitful, since a great deal of variation is expected even in otherwise similar villages. Simulations allow us to construct communities that are absolutely identical, and variation is still observed due to the vagaries of who infects whom. Infection will be eliminated in some fortunate villages. In other, essentially identical villages, it will continue to return to the same average level even after multiple treatments (Figure ). In practice, we have also observed a varied response to treatment in similar villages (Figure ) [12
Often overall prevalence is reported, but we believe there needs to be a shift in the way we evaluate progress. The average prevalence of infection across several villages can be misleading. A regional prevalence of 5% does not mean that 1 in 20 children in each village are infected. More likely, infection has been eliminated in most villages but has returned in a few villages to a level that may far exceed 5%. A better measure of success of a trachoma program may be the proportion of villages in which infection has been eliminated. While the power of sampling allows the mean prevalence to be obtained relatively easily, it should be interpreted in terms of what it reveals about the underlying process; after several rounds of treatment, the mean represents some villages where infection has been completely eliminated and others where infection has returned to a level chosen from the quasi-stationary distribution. Thus the decision for stopping distributions may need to depend on whether infection has been eliminated locally.
Periodic mass antibiotic treatments can locally eliminate the ocular chlamydia that cause blinding trachoma, if given frequently and to a large portion of the population [7
]. Models suggest it will return to the same distribution before the next scheduled treatment. If infection is not eliminated from a community after several mass treatments, then stochastic Ethiopian data demonstrates that this quasi-stationary distribution is closely approached after only two or three treatments. The WHO's strategy requires only for antibiotics to reduce the level of infection and expects that other measures will be necessary for a permanent solution. Their current recommendation of three treatments before re-evaluation is reasonable. If constant re-infection of communities prevents elimination, then subsequent treatments may only serve to maintain the prevalence in a stationary distribution, not lower it. However, if infection between neighbouring communities is rare or can be reduced by treating large areas within a narrow time frame, then elimination over larger areas than a village may be achieved. There is some concern that communities from which trachoma has been partially eliminated will lose much of their immunity to chlamydia, only to have it return with a vengeance after treatments have been discontinued [24
]. This has yet to be demonstrated convincingly in practice [26
], but potential loss of immunity could easily be included in future models.
Partners in the WHO's trachoma program have distributed over 30 million doses of oral azithromycin so far, and over 1 million in Ethiopia. These programs are expanding rapidly. There has been evidence of subsequent re-emergence of infection in some recent trials of azithromycin, suggesting coverage and dosing intervals in mass therapy need careful consideration [7
]. Biannual treatments are costly, require more resources, and may contribute to antibiotic resistance in chlamydia and other pathogens such as Streptococcus pneumoniae
]. The benefits of mass therapy thus need to be weighed against the potential emergence of antibiotic resistance which will require long term surveillance. But if necessary for local elimination biannual distributions may be more cost effective in the long run in severely affected areas. The elimination of a bacterial disease from a large area with mass antibiotics would be major medical breakthrough.
Simulations were run to estimate the number of rounds of treatment necessary to achieve elimination in 50%, 75%, and 95% of the villages. Elimination depends on the efficacy and coverage of treatment, as well as the effective population size in a community. It should be noted that these models were developed using data from a hyper-endemic region, and they are not generalizable to all areas.
The stochastic model suggests that treatment given to 90% of the population biannually would eliminate infection in 95% of the villages after 5 years. Lower coverage or less frequent treatments would not be as successful (Figures and ). For example, the WHO's recommended strategy of annual treatment of 80% of the population would eliminate infection in an estimated 95% of hyper-endemic villages in 12 years. Not surprisingly, areas with a lower baseline prevalence of infection should achieve elimination more rapidly (Figure ). If the rate that an individual recovers from an infection is more rapid than estimated here, then it will be more difficult to eliminate infection with repeated treatments (Figure ). Infection can be eliminated from smaller communities more rapidly than larger ones, in part because stochastic fade out is more likely with fewer cases (Figure ). Other factors, such as the cost of medication and of distributions, were not analyzed in this report, but clearly play a role in determining the optimal strategy for a given program.
Figure 4 Coverage vs. Years until elimination: Here we vary coverage in the stochastic model, while keeping other parameters the same. Other parameters are biannual treatment an effective population size of 100 children, γ = 0.017, and β = 0.044. (more ...)
Figure 5 Number of months between treatments vs. Years until elimination: Here we vary treatment frequency in the stochastic model, while keeping other parameters the same. Other parameters are 90% effective coverage, an effective population size of 100 children, (more ...)
Figure 8 Population vs. Years until elimination: Here we vary the effective population size of children in the stochastic model, while keeping other parameters the same. Other parameters are biannual treatment with 90% effective coverage, γ = 0.017, and (more ...)