Stochastic And Deterministic Methods

The simulator contains several numerical methods. A model name ending in -stochastic includes random draws within each node. A model ending in -deterministic moves expected fractions of the population instead.

The travel model is separate from the disease model. The available binomial travel model is stochastic, so a deterministic disease model can still produce different network-level results across realizations when travel is enabled. Vaccine and antiviral stockpile releases are deterministic scheduled counts: they transfer fixed numbers of doses or people where eligible, and do not add another random draw to the simulation.

Method Summary

Method	Models	Population representation	Main runtime driver
Gillespie-style event queue	seaitrd-stochastic	Individual people and scheduled events	Total infections and events in the queue
Daily Poisson tau-leaping	seir-stochastic, seirs-stochastic, seitrs-stochastic, seihrd-stochastic, and SEITHRD configurations	Integer compartment counts	Nodes, demographic groups, and simulation days
Daily Euler update	seirs-deterministic, including SEIR and SEITRS compartment configurations; seaitrd-deterministic	Fractional compartment counts	Nodes, demographic groups, and simulation days
Binomial travel	binomial travel with any disease model	Integer new exposures	Origin-destination node pairs

Gillespie-Style SEAITRD

seaitrd-stochastic assigns a queued infection trajectory when a person enters E. Waiting times are sampled with rand_exp_min1, which enforces at least one day in each sampled stage. The queue then processes one-person progression events such as:

\[E \rightarrow A,\quad A \rightarrow I,\quad I,T \rightarrow R \text{ or } D.\]

The T compartment is entered only through stockpile-allocated antiviral treatment. The standard SEAITRD antiviral route is I -> T; natural untreated progression bypasses T.

Because each infected person creates several scheduled events, runtime grows with population size and epidemic size. This model can become impractical for real county populations, especially with a high \(R_0\) or many realizations. Use a small population or short validation scenario before committing to a large run.

The event queue preserves population: every valid progression event removes one person from its source compartment and adds one person to its destination. Events invalidated by earlier antiviral treatment are discarded.

Daily Poisson Tau-Leaping

The other stochastic models update compartment counts once per simulation day. For a transition with rate \(r\) and source count \(X\), the proposed movement is:

\[\Delta \sim \operatorname{Pois}(rX).\]

Each draw is capped by the people remaining in the source compartment. For competing exits, the implementation draws from the remaining population so compartments cannot become negative.

Tau-leaping is much faster than the individual SEAITRD queue because it tracks counts rather than a separate trajectory for every infected person. Runtime is usually more sensitive to the number of counties, demographic groups, and realizations than to the absolute population count.

Daily Euler Updates

Deterministic models use the same one-day transition structure but move the expected amount directly instead of drawing from a Poisson distribution. This allows fractional people and produces smooth within-node trajectories.

Euler updates can approach zero without reaching exactly zero because fractions continue moving between compartments. The simulator therefore uses a stopping tolerance rather than requiring every infectious compartment to be exactly zero.

Stochastic and deterministic versions of the same compartment structure use the same biological parameter meanings. Their difference is whether daily movements are random integer draws or expected fractional movements.

Binomial Travel

At the end of each simulation day, the travel model evaluates origin- destination node pairs. Infectious travelers can seed exposure at their destination, and susceptible travelers can be exposed to infection at the destination.

For each susceptible age, risk, and vaccine subgroup, new travel-associated exposures are drawn from a binomial distribution:

\[X_{\mathrm{travel}} \sim \operatorname{Binom}(S, p_{\mathrm{travel}}).\]

The probability combines the mobility flow, contact matrix, disease transmission coefficient, relative susceptibility, vaccination, and the configured transmitting and traveling compartments.

Consequently, -deterministic describes disease progression within nodes; it does not make the complete network simulation deterministic while binomial travel is active (more than one node in the network).

See Binomial Travel Model for the complete parameter reference and the distinction between traveling and locally transmitting compartments.

Choosing Realizations

One realization is useful for checking that an input runs, but not for estimating a stochastic outcome. Begin with roughly 5 to 100 realizations, depending on runtime and variability, then increase the count when rare epidemic emergence or extinction dominates the result. Small populations, \(R_0\) close to 1, or few initial exposures will drive-up stochasticity.

For example, an \(R_0\) below 1 should naturally decline without intervention. Conversely, seeding very few exposures in a rural county may require hundreds or thousands of realizations to distinguish a failed introduction from a scenario where an epidemic can emerge.