Francis I. Proctor Foundation, University of California, San Francisco, USA

Mathematics Department, San Francisco State University, USA

Department of Public Health, San Francisco, USA

Orbis International, Addis Ababa, Ethiopia

Department of Ophthalmology, University of California, San Francisco, USA

Department of Epidemiology & Biostatistics, University of California, San Francisco, USA

Institute for Global Health, University of California, San Francisco, USA

Abstract

Background

The World Health Organization recommends periodic mass antibiotic distributions to reduce the ocular strains of chlamydia that cause trachoma, the world's leading cause of infectious blindness. Their stated goal is to control infection, not to completely eliminate it. A single mass distribution can dramatically reduce the prevalence of infection. However, if infection is not eliminated in every individual in the community, it may gradually return back into the community, so often repeated treatments are necessary. Since public health groups are reluctant to distribute antibiotics indefinitely, we are still in need of a proven long-term rationale. Here we use mathematical models to demonstrate that repeated antibiotic distributions can eliminate infection in a reasonable time period.

Methods

We fit parameters of a stochastic epidemiological transmission model to data collected before and 6 months after a mass antibiotic distribution in a region of Ethiopia that is one of the most severely affected areas in the world. We validate the model by comparing our predicted results to Ethiopian data which was collected biannually for two years past the initial mass antibiotic distribution. We use the model to simulate the effect of different treatment programs in terms of local elimination of infection.

Results

Simulations show that the average prevalence of infection across all villages progressively decreases after each treatment, as long as the frequency and coverage of antibiotics are high enough. Infection can be eliminated in more villages with each round of treatment. However, in the communities where infection is not eliminated, it returns to the same average level, forming the same stationary distribution. This phenomenon is also seen in subsequent epidemiological data from Ethiopia. Simulations suggest that a biannual treatment plan implemented for 5 years will lead to elimination in 95% of all villages.

Conclusion

Local elimination from a community is theoretically possible, even in the most severely infected communities. However, elimination from larger areas may require repeated biannual treatments and prevention of re-introduction from outside to treated areas.

Background

Trachoma remains the major cause of infectious blindness in the world

A single dose of oral azithromycin is clearly effective in eliminating ocular chlamydial infection in an individual

In previous studies, we have used deterministic models of difference

a. Deterministic Model of Time vs. Prevalence with biannual treatments: Results from a differential equation based model demonstrating that biannual coverage of 80% of the population should progressively reduce ocular chlamydial infection (blue curve)

a. Deterministic Model of Time vs. Prevalence with biannual treatments: Results from a differential equation based model demonstrating that biannual coverage of 80% of the population should progressively reduce ocular chlamydial infection (blue curve). The deterministic model is an excellent approximation for the expectation of the stochastic model (mean of 1000 simulations, red curve). b. Stochastic Model of Time vs. Prevalence with biannual treatments: The mean of 1000 simulations of a stochastic model, assuming biannual treatments with 80% coverage (again, red curve) vs. the average prevalence of only those villages which still harbor infection (green curve). After the third treatment, the average prevalence of infection in these villages returns to approximately the same level with each subsequent treatment (green curve).

Methods

Clinical data

Sixteen villages in the Gurage region of southern Ethiopia were enrolled in a mass antibiotic treatment program for trachoma as previously described

Models

Previously, we constructed a simple SIS (susceptible, infected, susceptible) model of ocular chlamydial infection in a core group of children using differential equations

where

Transitions used in stochastic model

**Transition**

**Description of transition**

**Hazard/Risk**

S→I

susceptible becomes infected

I→S

infected recovers naturally

I→S

periodic antibiotic treatment

c*f

Parameters used in stochastic model

**Parameter**

**Definition**

**Value**

**Units**

**Estimated or Known**

N

Number of children in a village

100

people

Known

I

Number of children infected

varies by time

people

Known

Weeks between a mass treatment

26

weeks

Known

c*f

Effective Coverage

90%

percentage

Known

Rate of recovery

0.017

1/weeks

Estimated using MLE

Rate of infection transmission

0.047

1/weeks

Estimated using MLE

Note that treatment is not given at a constant rate, but is given periodically to a certain proportion of the population. Each member of the community has a chance of being treated based on the coverage level and the efficacy of antibiotic in an individual.

We constructed the analogous Markov model by letting _{i}(

between the _{ij }is the Kronecker delta, which equals 1 if

Parameters for this stochastic model were fit to the 6 month Ethiopian data using maximum likelihood estimation. We started out simulations at the average prevalence, and ran for 24 months to allow the distribution of prevalence to approximate the pre-treatment distribution at the time point zero We approximated the joint probability distribution function of the 3 points (baseline, 2, and 6 month post-treatment prevalence) by fitting a standard kernel density estimator (a 3-dimensional product-Epanechnikov kernel), to the results of 1000 simulations. The bandwidth of the kernel in each dimension was set using standard techniques

For the sensitivity analysis, we varied

Baseline prevalence vs. Years until elimination: Here we vary

Baseline prevalence vs. Years until elimination: Here we vary

Recovery rate vs. Years until elimination: Here we vary the recovery rate in the stochastic model, while keeping other parameters the same

Recovery rate vs. Years until elimination: Here we vary the recovery rate in the stochastic model, while keeping other parameters the same. Other parameters are biannual treatment with 90% effective coverage, an effective population size of 100 children, and

To incorporate the effects of treatment, we used the analogous discrete time, Markov process where time is incremented every period of mass antibiotic distribution (e.g. 26 weeks). The quasi-stationary distribution (Figure ** q**, with

a. Probability Density of infection prevalence found in biannually treated villages where "village level" elimination has not yet occurred: Probability distribution of 1000 simulations at baseline, 6, 12, 18, and 24 months

a. Probability Density of infection prevalence found in biannually treated villages where "village level" elimination has not yet occurred: Probability distribution of 1000 simulations at baseline, 6, 12, 18, and 24 months. The prevalence of infection in a simulated community pre-treatment varies in a normal distribution [21]. Each mass treatment eliminates infection in some villages, but in those that it does not, the distribution is shifted to the left, rapidly approaching a quasi-stationary distribution 3b. Probability density graphs using Ethiopian data: Distribution of the prevalence of infection in pre-school children in 16 Ethiopian villages. Baseline, 2, and 6 month data were used to fit the parameters of the stochastic model. Subsequent data from 12, 18, and 24 months confirm that the distribution of infection in the villages also approaches the quasi-stationary distribution.

**M**·q^{t }= q^{t+treatment period}

Note that _{0, }the state that would represent elimination, is specifically not included in this formulation, so only communities where infection has not been eliminated are followed. The largest eigenvalue of

Results and Discussion

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

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, ^{rd }treatment is similar to that after the 4^{th }treatment, and prevalence after the 4^{th }even more similar to that after the 5^{th }(Figure

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 to the effects of chance

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 to the effects of chance. b. Real Ethiopian villages after one treatment: In Ethiopian communities with similar pre-treatment prevalence of infection and similar antibiotic coverage levels, infection may return relatively rapidly (blue curve), or fade out (red curve).

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 2^{nd }treatment approaches that to which it returns after the 3rd treatment (Figure ^{rd }and 4^{th }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

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

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

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

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

Coverage vs. Years until elimination: Here we vary coverage in the stochastic model, while keeping other parameters the same

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,

Number of months between treatments vs. Years until elimination: Here we vary treatment frequency in the stochastic model, while keeping other parameters the same

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,

Population vs. Years until elimination: Here we vary the effective population size of children in the stochastic model, while keeping other parameters the same

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,

Conclusion

The local elimination of infectious trachoma is possible with repeated mass antibiotic distributions. A stochastic mathematical model suggests that elimination (reduction of the prevalence of infection to zero in a community) can occur within a reasonable time period even in the most severely affected areas, as long as the coverage and frequency of distributions are high enough. For example, in hyper-endemic areas of Ethiopia, we estimate that biannual treatment with 90% effective coverage would result in elimination in more than half of communities in 3 years, and in 95% within 5 years. Most areas have far less trachoma than the 16 villages in Ethiopia presented here, so elimination should prove far easier. The WHO's current recommendation of three annual treatments with at least 80% coverage will likely eliminate infection in a large number of communities worldwide. The time to elimination is also dependent on whether transmission of infection can be effectively reduced by any other means, such as hygiene (clean faces are associated with less clinically active trachoma) and fly control (Musca sorbens is thought to play a role in the transmission of ocular chlamydia), both of which are major parts of the WHO's overall trachoma strategy

Competing interests

The author(s) declare that they have no competing interests.

Authors' contributions

KR acquisition of data, mathematical modelling, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis.

TP study concept and design, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis.

KH acquisition of data, analysis and interpretation of data, critical revision of the manuscript for important intellectual content, administrative, technical, or material support, study supervision.

DL mathematical modeling, analysis and interpretation of data, critical revision of the manuscript for important intellectual content, statistical analysis.

WA study concept and design, acquisition of data, critical revision of the manuscript for important intellectual content, obtained funding.

MM study concept and design, acquisition of data, critical revision of the manuscript for important intellectual content, obtained funding.

TL study concept and design, acquisition of data, critical revision of the manuscript for important intellectual content, obtained funding.

EY acquisition of data, critical revision of the manuscript for important intellectual content, administrative, technical, or material support, study supervision.

JH acquisition of data, critical revision of the manuscript for important intellectual content, administrative, technical, or material support, study supervision.

JC acquisition of data, analysis and interpretation of data, critical revision of the manuscript for important intellectual content.

JW study concept and design, acquisition of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, obtained funding.

BG study concept and design, acquisition of data, critical revision of the manuscript for important intellectual content.

TL conceived of the study concept and design, acquisition of data, mathematical modelling, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, study supervision.

Acknowledgements

We would like to thank the Bernard Osher Foundation, the International Trachoma Initiative, the Bodri Foundation, the South Asia Research Fund, the Harper Inglis Trust, Research to Prevent Blindness, That Man May See, and the National Institutes of Health (Grant U10-EY016214) for financial support, and Dr. Arek Goetz and Dr. Mohammad Kafai for guidance.

Pre-publication history

The pre-publication history for this paper can be accessed here: