Mathematical Reference

This page summarizes the core equations. The stochastic implementation uses integer-valued daily transitions, typically drawn from Poisson processes and capped by compartment counts. The deterministic implementation uses the same daily transition structure but evaluates the transition amounts directly from rates (fractions of people) instead of drawing Poisson counts. The deterministic model is Euler’s method, so it will not converge without non-integer people being allowed to progress forward in time.

Intervention allocation and effectiveness equations are documented separately in NPI And Vaccine Mathematics and Antiviral Stockpile Model.

Force Of Infection

For susceptible age group \(i\), the force of infection is:

\[\lambda_i(t) = \beta_i(t) \sum_j C_{ij} \frac{I_j^\ast(t)}{N}\]

where:

• \(C_{ij}\) is the average number of daily contacts that a person in group \(i\) (contact initiator) has with people in group \(j\) (contactee)

• \(N\) is node population (e.g. county)

• \(I_j^\ast\) is the weighted infectious population in age group \(j\)

• \(\beta_i(t)\) is the baseline beta after NPI modification

For SEITRS and deterministic SEITRS:

\[I_j^\ast = I_j + \rho_T T_j\]

where \(\rho_T =\) rel_inf_T_to_I.

For SEITHRD:

\[I_j^\ast = \rho_{IP} IP_j + \rho_{IA} IA_j + IS_j + \rho_T T_j\]

where \(\rho_{IP} =\) rel_inf_IP_to_IS, \(\rho_{IA} =\) rel_inf_IA_to_IS, and \(\rho_T =\) rel_inf_T_to_IS.

Model Equation Systems

The systems below show the mean compartment flows without age, risk, vaccine, or node subscripts. Each equation applies separately to those groups, while \(\lambda\) includes the infectious pressure summed across the interacting groups. We’ll let the uncapped mean-field incidence be:

\[F = \lambda S.\]

The simulator advances these systems in one-day steps rather than using a continuous ODE solver. Deterministic models use capped mean flows; stochastic daily models replace eligible flows with capped Poisson draws.

SEIR And SEIRS

Let \(\sigma\) be the E -> I rate, \(\gamma\) the I -> R rate, and \(\omega\) the R -> S rate. For SEIR, set \(\omega=0\).

\[\begin{split}\begin{aligned} \dot S &= -F + \omega R, \\ \dot E &= F - \sigma E, \\ \dot I &= \sigma E - \gamma I, \\ \dot R &= \gamma I - \omega R. \end{aligned}\end{split}\]

SEITRS

Let \(d_T\) denote the configured duration T_to_R_days, so \(\tau_T = 1/d_T\). At the start of a simulation day, the antiviral stockpile model may move eligible people into T. Let \(E_{\mathrm{before}}\), \(I_{\mathrm{before}}\), and \(T_{\mathrm{before}}\) be the compartment counts immediately before allocation. If \(u_E\) exposed people and \(u_I\) infectious people receive treatment, the counts immediately after allocation are:

\[\begin{split}\begin{aligned} E_{\mathrm{after}} &= E_{\mathrm{before}} - u_E, \\ I_{\mathrm{after}} &= I_{\mathrm{before}} - u_I, \\ T_{\mathrm{after}} &= T_{\mathrm{before}} + u_E + u_I. \end{aligned}\end{split}\]

where \(u_E\) and \(u_I\) are allocated dose counts, not continuous disease rates. “After” means after antiviral allocation on the same simulation day, not the next day. The disease-progression step then uses these updated counts:

\[\begin{split}\begin{aligned} \dot S &= -F + \omega R, \\ \dot E &= F - \sigma E, \\ \dot I &= \sigma E - \gamma I, \\ \dot T &= -\tau_T T, \\ \dot R &= \gamma I + \tau_T T - \omega R. \end{aligned}\end{split}\]

Here \(F\) uses \(I^\ast=I+\rho_TT\). With no allocated doses and no initialized treated population, \(u_E=u_I=0\) and \(T\) remains zero.

SEIHRD

Let \(\alpha\) be the E -> IA/IP rate, \(p_A\) the asymptomatic branch proportion, \(\kappa\) the IP -> IS rate, \(\gamma_A\) the IA -> R rate, \(h\) the effective IS -> H rate, \(\gamma_S\) the IS -> R rate, \(\mu_H\) the H -> D rate, and \(\gamma_H\) the H -> R rate. The effective hospitalization rate \(h\) includes the configured age, risk, and vaccine modifiers.

\[\begin{split}\begin{aligned} \dot S &= -F, \\ \dot E &= F - \alpha E, \\ \dot{IA} &= p_A\alpha E - \gamma_A IA, \\ \dot{IP} &= (1-p_A)\alpha E - \kappa IP, \\ \dot{IS} &= \kappa IP - (h+\gamma_S)IS, \\ \dot H &= hIS - (\mu_H+\gamma_H)H, \\ \dot R &= \gamma_A IA + \gamma_S IS + \gamma_H H, \\ \dot D &= \mu_H H. \end{aligned}\end{split}\]

The stochastic implementation draws and caps the daily transitions, then resolves competing exits from the remaining source population.

SEITHRD

SEITHRD uses the SEIHRD progression rates above and adds treated hospitalization or recovery. At the start of a simulation day, antiviral allocation may transfer people from E, IA, IP, and IS into T. For each compartment, “before” means immediately before that day’s allocation and “after” means immediately after allocation, before disease progression:

\[\begin{split}\begin{aligned} E_{\mathrm{after}} &= E_{\mathrm{before}} - u_E, \\ IA_{\mathrm{after}} &= IA_{\mathrm{before}} - u_{IA}, \\ IP_{\mathrm{after}} &= IP_{\mathrm{before}} - u_{IP}, \\ IS_{\mathrm{after}} &= IS_{\mathrm{before}} - u_{IS}, \\ T_{\mathrm{after}} &= T_{\mathrm{before}} + u_E + u_{IA} + u_{IP} + u_{IS}. \end{aligned}\end{split}\]

The \(u\) values are whole-person daily allocations from the stockpile model. They are not rates in the ODE system, and these equations do not describe a change from one day to the next.

\[\begin{split}\begin{aligned} \dot S &= -F, \\ \dot E &= F - \alpha E, \\ \dot{IA} &= p_A\alpha E - \gamma_A IA, \\ \dot{IP} &= (1-p_A)\alpha E - \kappa IP, \\ \dot{IS} &= \kappa IP - (h+\gamma_S)IS, \\ \dot H &= hIS + h_TT - (\mu_H+\gamma_H)H, \\ \dot T &= -(h_T+\tau_T)T, \\ \dot R &= \gamma_A IA + \gamma_S IS + \gamma_H H + \tau_T T, \\ \dot D &= \mu_H H. \end{aligned}\end{split}\]

Here \(F\) uses the weighted infectious population defined in the Force Of Infection section above, including treated infectiousness. The treated hospitalization rate \(h_T\) uses the SEIHRD hospitalization split reduced by antiviral_effectiveness_hosp; for example, 0.25 means 25% lower hospitalization risk, or 75% relative risk, among treated people.

The T -> H branch uses the same target clock as untreated symptomatic hospitalization. Let \(d_{IS,H}\) be IS_to_H_days:

\[\eta_H = \frac{1}{d_{IS,H}}.\]

Treatment changes the desired realized hospitalization proportion. Let \(e_{AV,H}\) be antiviral_effectiveness_hosp:

\[p_{T \to H} = p_{IS \to H} \left(1 - e_{AV,H}\right),\]

then the competing-clock adjustment solves for the T -> H branch multiplier using target rate \(\eta_H\) and competing recovery rate \(\tau_T = 1 / d_T\), where \(d_T\) is T_to_R_days.

For a concrete low-risk example, take prop_IS_to_H_lowrisk = 0.10, IS_to_H_days = 3, IS_to_R_days = 7, T_to_R_days = 5, and antiviral_effectiveness_hosp = 0.25. Untreated IS has a realized hospitalization proportion of 0.10, so about 100 of 1,000 untreated symptomatic people eventually enter H. Treated T has:

\[p_{T \to H} = 0.10(1 - 0.25) = 0.075,\]

so about 75 of 1,000 treated people eventually enter H. Because T_to_R_days differs from IS_to_R_days, the branch multiplier changes, but the T -> H target clock remains \(\eta_H = 1/3\) per day.

For the untreated pathway, the competing-clock adjustment combines \(\eta_H = 1/3\) with \(\gamma_S = 1/7\). The adjusted daily rates are approximately:

\[h = 0.0152,\quad \gamma_S^\ast = 0.1364,\quad \frac{h}{h+\gamma_S^\ast} = 0.10.\]

For the treated pathway, the target clock remains \(\eta_H = 1/3\), but the desired realized hospitalization proportion is 0.075 and the competing recovery clock is \(\tau_T = 1/5\). The adjusted daily rates are approximately:

\[h_T = 0.0155,\quad \tau_T' = 0.1907,\quad \frac{h_T}{h_T+\tau_T'} = 0.075.\]

Thus the antiviral parameter changes the realized proportion from 10% to 7.5%, while the hospitalization target rate used in the adjustment is still the original IS_to_H_days rate. The recovery clock is also shortened from IS_to_R_days = 7 to T_to_R_days = 5, representing a 2-day antiviral reduction in symptomatic duration for people treated after entering IS.

SEAITRD

The deterministic SEAITRD mean-flow system uses transition-style input names: E_to_A_days, A_to_I_days, I_to_R_days, T_to_R_days, and I_to_D_invdays. Inputs ending in _days are durations that become rates; I_to_D_invdays is already a per-day low-risk mortality rate. highrisk_death_multiplier multiplies that rate for high-risk groups. SEAITRD requires T, but has no disease-rate flow into T; antiviral stockpile allocation is the only source of newly treated people.

Let \(r_{EA}=1/\text{E_to_A_days}\), \(r_{AI}=1/\text{A_to_I_days}\), \(r_{IR}=1/\text{I_to_R_days}\), \(r_{TR}=1/\text{T_to_R_days}\), \(r_{ID,g}=\text{I_to_D_invdays}\) for low-risk groups and \(r_{ID,g}=\text{I_to_D_invdays}\times\text{highrisk_death_multiplier}\) for high-risk groups, and \(r_{TD,g}=r_{ID,g}(1-e_{AV,D})\), where \(e_{AV,D}\) is antiviral_effectiveness_death.

\[\begin{split}\begin{aligned} \dot S &= -F, \\ \dot E &= F - r_{EA} E, \\ \dot A &= r_{EA} E - r_{AI} A, \\ \dot I &= r_{AI} A - (r_{IR}+r_{ID,g})I, \\ \dot T &= -(r_{TR}+r_{TD,g})T, \\ \dot R &= r_{IR}I+r_{TR}T, \\ \dot D &= r_{ID,g}I+r_{TD,g}T. \end{aligned}\end{split}\]

The stochastic SEAITRD model uses a Gillespie-style queue of individual events rather than numerically solving this ODE system. Its untreated trajectory bypasses T with A -> I; optional stockpile treatment replaces an eligible source compartment’s queued trajectory with a newly drawn trajectory beginning in T.

SEIRS Daily Transitions

For SEIRS:

\[S \rightarrow E,\quad E \rightarrow I,\quad I \rightarrow R,\quad R \rightarrow S\]

with rates:

\[\sigma = \frac{1}{\text{E_to_I_days}}, \quad \gamma = \frac{1}{\text{I_to_R_days}}, \quad \omega = \frac{1}{\text{R_to_S_days}}.\]

The stochastic transition counts are:

\[\Delta_{S \to E} = \min(\operatorname{Pois}(\lambda S), S)\]

\[\Delta_{E \to I} = \min(\operatorname{Pois}(\sigma E), E),\quad \Delta_{I \to R} = \min(\operatorname{Pois}(\gamma I), I),\quad \Delta_{R \to S} = \min(\operatorname{Pois}(\omega R), R).\]

The deterministic model uses the same capped transitions, replacing each Poisson draw with its mean, for example \(\Delta_{E \to I} = \min(\sigma E, E)\).

SEITRS Daily Transitions

SEITRS disease progression is:

\[S \rightarrow E,\quad E \rightarrow I,\quad I \rightarrow R,\quad T \rightarrow R,\quad R \rightarrow S.\]

The disease model does not contain an E -> T or I -> T rate. Treatment entry is handled by the antiviral stockpile model before disease progression for that day. It is analogous to vaccine allocation: doses are released, distributed to nodes, and then applied to eligible people until the dose supply or eligible population is exhausted.

The antiviral stockpile model creates treatment movements:

\[\Delta_{E \to T}^{AV},\quad \Delta_{I \to T}^{AV}\]

where each \(\Delta^{AV}\) is an allocation count from available doses, not a disease-rate draw. If no antiviral stockpile is released, these movement counts are zero forever and T remains zero unless initialized externally.

After antiviral allocation, the SEITRS disease step includes:

\[\Delta_{T \to R} = \min(\operatorname{Pois}(\tau_T T), T), \quad \tau_T = \frac{1}{d_T}\]

where \(d_T\) is T_to_R_days. For stochastic SEITRS, the transition is drawn from the Poisson distribution above. Deterministic SEITRS uses \(\Delta_{T \to R} = \min(\tau_T T, T)\).

SEITHRD Daily Transitions

SEITHRD is the antiviral-enabled SEIHRD variant with compartment order:

\[S, E, IA, IP, IS, H, T, R, D.\]

The disease progression transitions are:

\[\begin{split}\begin{aligned} S &\rightarrow E, & E &\rightarrow IA \text{ or } IP, & IP &\rightarrow IS, \\ IS &\rightarrow H \text{ or } R, & H &\rightarrow D \text{ or } R, & IA &\rightarrow R, \\ T &\rightarrow H \text{ or } R. \end{aligned}\end{split}\]

As with SEITRS, the disease model has no rate into T. With the SEITHRD antiviral input template’s routine-treatment compartment_priority of ["IS"], antiviral allocation creates:

\[\begin{aligned} \Delta_{IS \to T}^{AV}. \end{aligned}\]

This is governed by compartment_priority, node allocation, daily capacity, and stockpile availability. If a scenario explicitly includes E, IA, or IP in compartment_priority, those existing SEITHRD compartments may also be allocated to T; that represents prophylaxis or early treatment rather than the routine treatment template. Disease progression then moves treated people out of T:

\[\begin{split}\begin{aligned} \Delta_{T \to H} &= \min(\operatorname{Pois}(h_T T), T), \\ \Delta_{T \to R} &= \min( \operatorname{Pois}(\tau_T (T-\Delta_{T \to H})), T-\Delta_{T \to H} ), \\ \tau_T &= \frac{1}{d_T}. \end{aligned}\end{split}\]

Here \(d_T\) is T_to_R_days. The T -> H split is reduced by antiviral_effectiveness_hosp relative to the untreated IS -> H realized proportion, while the T -> H target rate remains 1 / IS_to_H_days. Treatment reduces severe outcomes without forcing all treated people to recover directly. No stockpile release means no new T entries.

Gillespie SEAITRD Events

SEAITRD is event-driven rather than a daily Poisson compartment update. At infection time, each person receives an event schedule:

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

Here T means Treated, and anyone in T is assumed to be receiving an antiviral. T is a required SEAITRD compartment, but untreated event times still bypass it. A person enters T only when an antiviral stockpile dose is allocated.

The optional stockpile model adds external queue edits:

\[I \xrightarrow{\text{dose}} T.\]

For each stockpile-allocated dose, the old I trajectory is marked stale and a new T trajectory is drawn. The treated death clock uses:

\[\nu_T = \nu(1 - e_{AV,D}),\]

where \(\nu\) is the configured I_to_D_invdays rate and \(e_{AV,D}\) is antiviral_effectiveness_death. For high-risk groups, \(\nu\) first includes highrisk_death_multiplier. There is no daily antiviral transition rate. Every event moves one person out of one compartment and into another. Therefore, if \(N_{c,\mathrm{before}}\) and \(N_{c,\mathrm{after}}\) are the counts in compartment \(c\) immediately before and after one event:

\[\sum_c N_{c,\mathrm{after}} = \sum_c N_{c,\mathrm{before}}.\]

This is an immediate event-level comparison, not a comparison between simulation days. Population is preserved because each event removes and adds the same one person. An event is discarded if its source person was already moved by treatment.

Progression events are reconciled after treatment. Contact events remain in the existing group-level contact queue because SEAITRD currently applies the same contact-generation process across A, T, and I.

Competing Clocks

When a compartment has competing exits and the desired realized fraction is \(p_i\), the simulator adjusts branch multipliers \(\pi_i\) so:

\[\frac{\pi_i r_i}{\sum_j \pi_j r_j} = p_i.\]

The implemented solution is:

\[\pi_i = \frac{p_i / r_i}{\sum_j p_j / r_j}.\]

For two branches, this reduces to the helper DiseaseModel.adjust_two_way_split_proportion.