#Epidemic Model Comparison

#Scope Of The Comparison

The GoLT preset has the same named flow structure as an SIRSD model— $S\rightarrow I\rightarrow R\rightarrow S$ , with $I\rightarrow D$ —but implements it as a local stochastic cellular automaton. This chapter compares structures, not outputs, as no external model was run and the GoLT probabilities are not calibrated to a particular disease or unit of time.

#Compartmental SIR And SIRSD

The classical Kermack-McKendrick tradition models aggregate Susceptible, Infectious, and Removed populations with continuous equations and homogeneous or otherwise specified mixing. Hethcote's review describes the threshold concepts and model families built from that foundation. Wolff's SIRSD construction adds a separate disease-death compartment and loss of immunity from Recovered back to Susceptible.

Feature	Compartmental SIRSD	GoLT SIRSD Epidemic
State variables	Continuous population totals or fractions	One discrete state per lattice cell
Time	Continuous rate equations	Synchronous discrete generations
Contact	Population-level mixing term	Eight fixed Moore neighbors
Infection pressure	Usually scales with $S$ , $I$ , and a transmission rate	Fixed $50\%$ if at least $1$ `Infectious` neighbor exists
Recovery/death	Competing rates from $I$	Ordered $5\%$ recovery roll, then nominal $0.1\%$ conditional death roll
Immunity loss	Rate from $R$ to $S$	$1\%$ roll per `Recovered` cell per generation
Empty space	Usually not a compartment	Empty/`Dead` lattice sites break local contact paths
Spatial history	Not present in a homogeneous model	Clusters, fronts, local depletion, and toroidal paths emerge

Changing occupied density therefore has a different meaning in GoLT. It changes the connectivity of the contact substrate and the probability that infection can find a local path. In a homogeneous compartmental model, changing total population while keeping the same fractions and frequency-dependent rates need not create the same geometric transition.

#Spatial SIRS References

Joo and Lebowitz studied a stochastic SIRS process on $1$ - and $2$ -dimensional lattices and showed that spatial correlations can make mean-field approximations inaccurate. Van Ballegooijen and Boerlijst used an $8$ -neighbor grid-structured SIRS model in which local outbreaks, turbulent waves, and recurring waves arise for different infection and resistance parameters.

Those studies are structurally closer to GoLT SIRSD Epidemic preset because local contact and spatial organization matter. Important differences remain:

Their infection hazards can depend on the number of Infectious neighbors, whereas GoLT's preset saturates after $1$ .
Some models use continuous-time or small-step dynamics instead of $1$ synchronous generation.
Infectious and resistant periods may be fixed durations rather than geometric waiting times from per-step rolls.
The cited SIRS lattice models do not necessarily include GoLT's absorbing Dead state.
GoLT's experiment varies occupied-site density while keeping its transition probabilities fixed.

The local analysis's occasional resurgence events are qualitatively compatible with the possibility of recurring spatial waves under waning immunity.

#What A Direct Validation Would Require

A direct numerical comparison with Wolff's SIRSD equations or another SIRSD implementation would require:

A common interpretation of $1$ generation as physical time.
Matched infection, recovery, death, and immunity-loss hazards.
Agreement on whether infection hazard grows with the number of infectious contacts.
Matched initial S, I, R, and D fractions and population normalization.
Matched topology, population mobility, and contact network.
Identical endpoints and replicated stochastic seeds.

Comparing a spatial CA against a compartmental model would then be useful precisely because their differences are interpretable: divergence would quantify the effect of local correlations and empty-space connectivity rather than being attributed to unmatched protocols.

Primary sources for these comparisons are collected in References.