Modeling Shocks in COVID 19 with Stochastic Differential Equations¶

Contents¶

• Modeling Shocks in COVID 19 with Stochastic Differential Equations
  • Overview
  • The Basic SIR/SIRD Model
  • Introduction to SDEs
  • Ending Lockdown
  • Reinfection

Overview¶

Coauthored with Chris Rackauckas

This lecture continues the analyzing of the COVID-19 pandemic established in this lecture.

As before, the model is inspired by

• Notes from Andrew Atkeson and NBER Working Paper No. 26867
• Estimating and Forecasting Disease Scenarios for COVID-19 with an SIR Model by Andrew Atkeson, Karen Kopecky and Tao Zha
• Estimating and Simulating a SIRD Model of COVID-19 for Many Countries, States, and Cities by Jesús Fernández-Villaverde and Charles I. Jones
• Further variations on the classic SIR model in Julia here.

Here we extend the model to include policy-relevant aggregate shocks.

Continuous-Time Stochastic Processes¶

In continuous-time, there is an important distinction between randomness that leads to continuous paths vs. those which have (almost surely right-continuous) jumps in their paths. The most tractable of these includes the theory of Levy Processes.

Among the appealing features of Levy Processes is that they fit well into the sorts of Markov modeling techniques that economists tend to use in discrete time, and usually fulfill the measurability required for calculating expected present discounted values.

Unlike in discrete-time, where a modeller has license to be creative, the rules of continuous-time stochastic processes are much stricter. In practice, there are only two types of Levy Processes that can be used without careful measure theory.

1. Weiner Processes (as known as Brownian Motion) which leads to a diffusion equations, and is the only continuous-time Levy process with continuous paths
2. Poisson Processes with an arrival rate of jumps in the variable.

Every other Levy Process can be represented by these building blocks (e.g. a Diffusion Process such as Geometric Brownian Motion is a transformation of a Weiner process, a jump diffusion is a diffusion process with a Poisson arrival of jumps, and a continuous-time markov chain (CMTC) is a Poisson process jumping between a finite number of states).

In this lecture, we will examine shocks driven by transformations of Brownian motion, as the prototypical Stochastic Differential Equation (SDE).

Setup¶

In addition, we will be exploring packages within the SciML ecosystem and others covered in previous lectures

The Basic SIR/SIRD Model¶

To demonstrate another common compartmentalized model we will change the previous SEIR model to remove the exposed state, and more carefully manage the death state, D.

The states are are now: susceptible (S), infected (I), resistant (R), or dead (D).

Comments:

• Unlike the previous SEIR model, the R state is only for those recovered, alive, and currently resistant.
• As before, we start by assuming those have recovered have acquired immunity.
• Later, we could consider transitions from R to S if resistance is not permanent due to virus mutation, etc.

Transition Rates¶

See the previous lecture, for a more detailed development of the model.

• $ \beta(t) $ is called the transmission rate or effective contact rate (the rate at which individuals bump into others and expose them to the virus)
• $ \gamma $ is called the resolution rate (the rate at which infected people recover or die)
• $ \delta(t) \in [0, 1] $ is the death probability
• As before, we re-parameterize as $ R_0(t) := \beta(t) / \gamma $, where $ R_0 $ has previous interpretation

Jumping directly to the equations in $ s, i, r, d $ already normalized by $ N $,

$$ \begin{aligned} d s & = - \gamma \, R_0 \, s \, i \, dt \\ d i & = \left(\gamma \, R_0 \, s \, i - \gamma \, i \right) dt \\ d r & = (1-\delta) \gamma \, i \, dt \\ d d & = \delta \, \gamma \, i \, dt \\ \end{aligned} \tag{1} $$

Note that the notation has changed to heuristically put the $ dt $ on the right hand side, which will be used when adding the stochastic shocks.

Introduction to SDEs¶

We start by extending our model to include randomness in $ R_0(t) $ and then the mortality rate $ \delta(t) $.

The result is a system of Stochastic Differential Equations (SDEs).

Shocks to Transmission Rates¶

As before, we assume that the basic reproduction number, $ R_0(t) $, follows a process with a reversion to a value $ \bar{R}_0(t) $ which could conceivably be influenced by policy. The intuition is that even if the targeted $ \bar{R}_0(t) $ was changed through social distancing/etc., lags in behavior and implementation would smooth out the transition, where $ \eta $ governs the speed of $ R_0(t) $ moves towards $ \bar{R}_0(t) $.

Beyond changes in policy, randomness in $ R_0(t) $ may come from shocks to the $ \beta(t) $ process. For example,

• Misinformation on Facebook spreading non-uniformly.
• Large political rallies, elections, or protests.
• Deviations in the implementation and timing of lockdown policy between demographics, locations, or businesses within the system.
• Aggregate shocks in opening/closing industries.

To implement these sorts of randomness, we will add on a diffusion term with an instantaneous volatility of $ \sigma \sqrt{R_0} $.

• This equation is used in the Cox-Ingersoll-Ross and Heston models of interest rates and stochastic volatility.
• The scaling by the $ \sqrt{R_0} $ ensure that the process stays weakly positive. The heuristic explanation is that the variance of the shocks converges to zero as R₀ goes to zero, enabling the upwards drift to dominate.
• See here for a heuristic description of when the process is weakly and strictly positive.

The notation for this SDE is then

$$ \begin{aligned} d R_{0t} &= \eta (\bar{R}_{0t} - R_{0t}) dt + \sigma \sqrt{R_{0t}} dW_t\\ \end{aligned} \tag{2} $$

where $ W $ is standard Brownian motion (i.e a Weiner Process.

Heuristically, if $ \sigma = 0 $, divide this equation by $ dt $ and it nests the original ODE used in the previous lecture.

While we do not consider any calibration for the $ \sigma $ parameter, empirical studies such as Estimating and Simulating a SIRD Model of COVID-19 for Many Countries, States, and Cities (Figure 6) show highly volatile $ R_0(t) $ estimates over time.

Even after lockdowns are first implemented, we see variation between 0.5 and 1.5. Since countries are made of interconnecting cities with such variable contact rates, a high $ \sigma $ seems reasonable both intuitively and empirically.

Mortality Rates¶

Unlike the previous lecture, we will build up towards mortality rates which change over time.

Imperfect mixing of different demographic groups could lead to aggregate shocks in mortality (e.g. if a retirement home is afflicted vs. an elementary school). These sorts of relatively small changes might be best modeled as a continuous path process.

Let $ \delta(t) $ be the mortality rate and in addition,

• Assume that the base mortality rate is $ \bar{\delta} $, which acts as the mean of the process, reverting at rate $ \theta $. In more elaborate models, this could be time-varying.
• The diffusion term has a volatility $ \xi\sqrt{\delta (1 - \delta)} $.
• As the process gets closer to either $ \delta = 1 $ or $ \delta = 0 $, the volatility goes to 0, which acts as a force to allow the mean reversion to keep the process within the bounds
• Unlike the well-studied Cox-Ingersoll-Ross model, we make no claims on the long-run behavior of this process, but will be examining the behavior on a small timescale so this is not an issue.

Given this, the stochastic process for the mortality rate is,

$$ \begin{aligned} d \delta_t & = \theta (\bar{\delta} - \delta_t) dt + \xi \sqrt{(\delta_t (1-\delta_t)} d W_t\\ \end{aligned} \tag{3} $$

Where the $ W_t $ Brownian motion is independent from the previous process.

System of SDEs¶

The system (1) can be written in vector form $ x := [s, i, r, d, R₀, \delta] $ with parameter tuple parameter tuple $ p := (\gamma, \eta, \sigma, \theta, \xi, \bar{R}_0(\cdot), \bar{ \delta}) $

The general form of the SDE is.

With the drift,

$$ F(x,t;p) := \begin{bmatrix} -\gamma \, R_0 \, s \, i \\ \gamma \, R_0 \, s \, i - \gamma i \\ (1-\delta) \gamma i \\ \delta \gamma i \\ \eta (\bar{R}_0(t) - R_0) \\ \theta (\bar{\delta} - \delta) \\ \end{bmatrix} \tag{4} $$

Here, it is convenient but not necessary for $ d W $ to have the same dimension as $ x $. If so, then we can use a square matrix $ G(x,t;p) $ to associate the shocks with the appropriate $ x $ (e.g. diagonal noise, or using a covariance matrix).

As the two independent sources of Brownian motion only affect the $ d R_0 $ and $ d \delta $ terms (i.e. the 5th and 6th equations), define the covariance matrix as

$$ \begin{aligned} G(x, t) &:= diag\left(\begin{bmatrix} 0 & 0 & 0 & 0 & \sigma \sqrt{R_0} & \xi \sqrt{\delta (1-\delta)} \end{bmatrix}\right) \end{aligned} \tag{5} $$

Implementation¶

First, construct our $ F $ from (4) and $ G $ from (5)

Next create a settings generator, and then define a SDEProblem with Diagonal Noise.

We solve the problem with the SOSRI algorithm (Adaptive strong order 1.5 methods for diagonal noise Ito and Stratonovich SDEs).

As in the deterministic case of the previous lecture, we are using an adaptive time-stepping method. However, since this is an SDE, (1) you will tend to see more timesteps required due to the greater curvature; and (2) the number of timesteps will change with different shock realizations.

With stochastic differential equations, a “solution” is akin to a simulation for a particular realization of the noise process.

If we take two solutions and plot the number of infections, we will see differences over time:

The same holds for other variables such as the cumulative deaths, mortality, and $ R_0 $:

See here for comments on finding the appropriate SDE algorithm given the structure of $ F(x, t) $ and $ G(x, t) $

• If $ G $ has diagonal noise (i.e. $ G(x, t) $ is a diagonal, and possibly a function of the state), then SOSRI is the typical choice.
• If $ G $ has additive and diagonal noise (i.e. $ G(t) $ is a diagonal and independent from the state), then SOSRA is usually the best algorithm for even mildly stiff $ F $.
• If adaptivity is not required, then EM (i.e. Euler-Maruyama method typically used by economists) is flexible in its ability to handle different noise processes.

Ensembles¶

While individual simulations are useful, you often want to look at an ensemble of trajectories of the SDE in order to get an accurate picture of how the system evolves.

To do this, use the EnsembleProblem in order to have the solution compute multiple trajectories at once. The returned EnsembleSolution acts like an array of solutions but is imbued to plot recipes to showcase aggregate quantities.

For example:

Or, more frequently, you may want to run many trajectories and plot quantiles, which can be automatically run in parallel using multiple threads, processes, or GPUs. Here we showcase EnsembleSummary which calculates summary information from an ensemble and plots the mean of the solution along with calculated quantiles of the simulation:

In addition, you can calculate more quantiles and stack graphs

Some additional features of the ensemble and SDE infrastructure are

• Plotting
• Noise Processes, Non-diagonal noise, and Correlated Noise
• Parallel Ensemble features
• Transforming the ensemble calculations with an output_func or reduction
• Auto-GPU accelerated by using EnsembleGPUArray() from DiffEqGPU

Changing Mitigation¶

Consider a policy maker who wants to consider the impact of relaxing lockdown at various speeds.

We will shut down the shocks to the mortality rate (i.e. $ \xi = 0 $) to focus on the variation caused by the volatility in $ R_0(t) $.

Consider $ \eta = 1/50 $ and $ \eta = 1/20 $, where we start at the same initial condition of $ R_0(0) = 0.5 $.

While the the mean of the $ d(t) $ increases, unsurprisingly, we see that the 95% quantile for later time periods is also much larger - even after the $ R_0 $ has converged.

That is, volatile contact rates (and hence $ R_0 $) can interact to make catastrophic worst-case scenarios due to the nonlinear dynamics of the system.

Ending Lockdown¶

As in the deterministic lecture, we can consider two mitigation scenarios

1. choose $ \bar{R}_0(t) $ to target $ R_0 = 0.5 $ for 30 days and then $ R_0 = 2 $ for the remaining 17 months. This corresponds to lifting lockdown in 30 days.
2. target $ R_0 = 0.5 $ for 120 days and then $ R_0 = 2 $ for the remaining 14 months. This corresponds to lifting lockdown in 4 months.

Since empirical estimates of $ R_0(t) $ discussed in [FVJ20] and other papers show it to have wide variation, we will maintain a fairly larger $ \sigma $.

We start the model with 100,000 active infections.

Simulating for a single realization of the shocks, we see the results are qualitatively similar to what we had before

However, note that this masks highly volatile values induced by the in $ R_0 $ variation, as seen in the ensemble

Finally, rather than looking at the ensemble summary, we can use data directly from the ensemble to do our own analysis.

For example, evaluating at an intermediate (t = 350) and final time step.

This shows that there are significant differences after a year, but by 550 days the graphs largely coincide.

In the above code given the return from solve on an EnsembleProblem , e.g. ensemble_sol = solve(...)

• You can access the i’th simulation as ensemble_sol[i], which then has all of the standard solution handling features
• You can evaluate at a real time period, t, with ensemble_sol[i](t). Or access the 4th element with ensemble_sol[i](t)[4]
• If the t was not exactly one of the saveat values (if specified) or the adaptive timesteps (if it was not), then it will use interpolation
• Alternatively, to access the results of the ODE as a grid exactly at the timesteps, where j is timestep index, use ensemble_sol[i][j] or the 4th element with ensemble_sol[i][4, j]
• Warning: unless you have chosen a saveat grid, the timesteps will not be aligned between simulations. That is, ensemble_sol[i_1].t wouldn’t match ensemble_sol[i_2].t. In that case, use interpolation with ensemble_sol[i_1](t) etc.

Reinfection¶

As a final experiment, consider a model where the immunity is only temporary, and individuals become susceptible again.

In particular, assume that at rate $ \nu $ immunity is lost. For illustration, we will examine the case if the average immunity lasts 12 months (i.e. $ 1/\nu = 360 $)

The transition modifies the differential equation (1) to become

$$ \begin{aligned} d s & = \left(- \gamma \, R_0 \, s \, i + \nu \, r \right) dt \\ d i & = \left(\gamma \, R_0 \, s \, i - \gamma i \right) dt \\ d r & = \left((1-\delta) \gamma i - \nu \, r\right) dt \\ d d & = \delta \gamma i \, dt \\ \end{aligned} \tag{6} $$

This change modifies the underlying F function and adds a parameter, but otherwise the model remains the same.

We will redo the “Ending Lockdown” simulation from above, where the only difference is the new transition.

The ensemble simulations for the $ \nu = 0 $ and $ \nu > 0 $ can be compared to see the impact in the absence of medical innovations.

Finally, we can examine the same early vs. late lockdown histogram