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The authors have declared that no competing interests exist.

Conceived and designed the experiments: LB LM BL IS. Performed the experiments: LB LM BL IS. Analyzed the data: LB LM BL IS. Contributed reagents/materials/analysis tools: LB LM BL IS. Wrote the paper: LB LM BL IS.

Disease control is of paramount importance in public health, with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Intervention controls are typically defined on a deterministic schedule. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.

Understanding the processes underlying disease extinction is an important problem in epidemic prediction and control. Currently, total eradication of infectious disease is quite rare, but continues to be a major theme in public health. Temporary eradication, sometimes called fade out, tends to happen in local spatial regions, and may be followed by the reintroduction of the disease from other regions

One main reason that diseases go extinct is due to the stochasticity that is inherent to populations of finite size

To properly model the random interactions occurring in populations, the study of disease extinction requires a stochastic modeling approach. There are numerous studies from time series analysis and epidemic modeling supporting stochastic fluctuations due to random interactions

We remark here that although escape has been considered for systems of Langevin type, the theory we present in this paper is for discrete finite populations modeled as a master equation. In continuous systems, a rigorous theory of escape rates for systems driven by white Gaussian noise was developed by Freidlin and Wentzell

Treatment programs are common methods used to speed up the extinction of a disease in a population

In general, little work has been done in analyzing stochastic models with random treatment intervention. In this context, we assume that treatments would be applied to infected individuals, removing them from that group. Most intervention schedules are designed as periodic, especially for childhood and seasonal diseases

Thus, one of the main problems in understanding treatment scheduling is that deterministic schedule models are

In this paper, we use the stochastic SIS compartmental model as a basic example to clearly demonstrate our mathematical methods analytically and numerically. The methods can be extended for use in more complex models, as necessary for a disease of interest. The SIS model tracks the number of individuals in a population of size

Associated with the parameters for a particular disease is the basic reproduction number,

For disease control, the stochastic model assumes a treatment schedule that occurs at randomly chosen times with a frequency

We use the master equation approach to describe the time evolution of the stochastic system. The general theory of applying the WKB method to finite populations begins by assuming that the population of

When the probability current at the extinct state is sufficiently small, there will exist a quasi-stationary probability distribution with a non-zero number of infected individuals that decays into the stationary solution over exponentially long times. The rate at which the extinction of infected individuals occurs may be calculated from the tail of the quasi-stationary distribution. It has been shown that a WKB approximation to the quasi-stationary distribution allows one to approximate the mean-time to extinction with high accuracy for a sufficiently large population

Approximating the probability by

In the first model, we approximate the SIS dynamics by reducing the dimension of the problem. Assume the average population size is

Since the population variable in the master equation is integer-valued, we choose to keep the integer part

Note that for any particular realization of the master equation, the treatment ceases to have an effect whenever

Next, we rescale the state variable by the population by using the normalized variable

While stochastic die out state is similar to the disease free equilibrium having

Note that the endemic state exists only if

The second model is the unconstrained SIS treatment model in two-dimensions. We calculate

For the two-dimensional model, let the state vector be

Here, as in Model 1, the non-integer quantity

We now use these Hamiltonian models to approximate the mean time to extinction. Topologically, the solution that describes an extinction event in the Hamiltonian system will connect the endemic state (

From the definition of the momentum,

Using this quantity, we approximate the mean time to extinction by evaluating

It is usually not a trivial task to identify the set of points that describe the optimal path. In some cases, it can be found analytically. One example is Model 1 with

Because the Hamiltonian system for the constrained model is in two dimensions, the first approximation to the action path simplifies to

For this model, an asymptotic approach can be used to approximate the action along the optimal path to extinction. We assume

The first term in the expansion ^{−1} and recovery rate ^{−1}. These generic parameters are chosen to represent a slowly spreading disease, but with a large

For Model 1, plot of the numerical approximation of the action (dashed curve) and the asymptotic approximation (solid curve) as a function of the treatment, ^{−1} and

Note that while the action does not depend on the size of the population, to first order, the mean time to extinction does. The population size must be large enough for the system to be quasi-stationary. Our model assumes that disease extinction is a rare event, which occurs in the tail of the distribution described by

To quantify where the system is quasi-stationary, we evaluate evaluate ^{−1} and a population of 8,000. The parameters region to the right of

A contour plot of ^{−1} and

The final step in finding the mean time to extinction is approximating the prefactor in Eq. (9). Following the approach in

(We use Eq. (49) of

To quantify the accuracy of the approximation to the mean time to extinction in Eq. (9), with

A plot of the mean time to disease extinction, ^{−1}. The results for the Monte Carlo simulations are averaged over 2,000 realizations and plotted as symbols. The curves of the same color show the approximation of the mean time to extinction by finding the action. The parameters are ^{−1}. Note the exponential decrease in the mean time to extinction as the treatment fraction is increased.

The full SIS model has a Hamiltonian system in four dimensions and asymptotic approximations of the optimal path and action are not tractable. Therefore, we rely on numerical approximations. For example, we show the probability density of extinction prehistory and the optimal path to extinction in

The red point denotes the endemic state. A simulation starts with the population at the endemic state and stops when the number of infected individuals is zero. The probability density uses the last five years of data from 200,000 Monte Carlo extinction realizations.The parameters are ^{−1}, ^{−1},

A plot of the mean time to extinction, ^{−1}. The averages of 2,000 Monte Carlo simulations are shown with symbols. The curves of the same color show the numerical approximation of ^{−1}.

We also comment on the differences in the constrained and unconstrained SIS models with treatment. In

This plot shows the quantitative difference in the action approximation for Model 1 (blue) and Model 2 (red) as we vary ^{−1}, ^{−1}, and

In this paper, we quantified how treatment enhances the extinction of epidemics using a stochastic, discrete-population framework. Specifically, we based our study on a general formulation of an SIS model with treatment that is applied randomly in a Poisson fashion, accounting for the limited amount of resources. We used a WKB approximation to the master equation of the stochastic process to calculate the average time to extinction starting from the endemic state, as a function of the transmissibility of the disease and the strength and frequency of the treatment. We compared the extinction times obtained analytically and numerically from the WKB approximation with the values obtained from Monte Carlo simulations.

In addition, we explored the significance of the quasi-stationarity assumption that is fundamental to the WKB approximation. The existence of a quasi-stationary distribution peaked at the endemic point produces a meta-stable state in which the population fluctuates in a neighborhood around the same endemic point. In contrast, the extinct state lies in the exponentially small tail of the distribution. When a quasi-stationary distribution exists, the extinction of a disease is a rare event, i.e. the mean time to extinction is exponentially long. As we show in

Deterministic models of treatment are not accurate representations of the process in practice when applied to finite population realizations. A more realistic description is that, on average, treatment scheduling has a mean period or cycle, but is itself a random process. To quantify the difference between the deterministic and the stochastic descriptions, we compared the mean time to extinction for a strictly periodic and a Poisson-distributed treatment schedule obtained by averaging the Monte Carlo simulation results of many extinction events starting from the endemic state. We assume that a fraction

For Model 1, a plot of the Monte Carlo simulated mean time to disease extinction for random (points connected by dotted lines) and periodic (symbols) treatment schedules vs. the fraction of infected vaccinated during each treatment. Results are shown for treatment frequencies, ^{−1} averaged over 2,000 realizations. The parameters are ^{−1}. Note that the random treatment schedule has average extinction times consistently lower than the periodic treatment schedule.

The treatment program that we implement in our model has two degrees of freedom: the frequency

Using Model 1 with a fixed treatment supply ^{−1}. Note that the mean time to extinction is a decreasing function of

In conclusion, we have described a method to quantify the effectiveness of a random treatment program. We find that increasing the magnitude and frequency of randomly scheduled treatments provide an exponential decrease in average extinction times. We have presented evidence that supports how larger campaigns applied less frequently are the most effective in facilitating disease eradication. Several assumptions in the model clarify the accuracy of the analytic approximation to the mean time to extinction, but its exponential rate of decrease as we increase the intervention is consistent with simulations throughout our analysis as populations get very large. The techniques considered here can be easily generalized to other diseases, such as those that include seasonality or population structure. Future work in this area could provide a more targeted control strategy that would be robust in fluctuating environments as well as more efficient and economical disease eradication.

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