Network SEIRV · Chapter 02

Compartment dynamics coupled through infection pressure

The construction has three layers: local reaction hazards, a contact-composition operator, and stochastic or deterministic time stepping.

The local reaction structure

First isolate the familiar five-state subsystem. The following equations explain the infection, vaccination, and recovery mechanisms; they are not a substitute for the full eight-compartment demographic model.

Model

Reduced reaction subsystem

Indices $i,a$ are suppressed. $\lambda$ is infection pressure, $\alpha$ is the exposed-to-infectious rate, $\gamma$ is recovery, $\nu$ is the vaccination rate, $q$ is primary vaccine failure probability, and $\theta$ is waning of vaccine protection. This reduction omits births, aging, mortality, maternal/infant states, and transfers. The active simulator uses finite-step hazard updates rather than directly integrating these five ODEs.

\begin{aligned} \dot S&=-\lambda S-\nu(1-q)S+\theta V,\\ \dot V&=\nu(1-q)S-\theta V,\\ \dot E&=\lambda S-\alpha E,\\ \dot I&=\alpha E-\gamma I,\\ \dot R&=\gamma I. \end{aligned}

Reaction hazards in the active stochastic and deterministic node updates.

Model

Two stages of maternal protection

Newborns enter $MI_1$ . Each maternal stage has a mean residence time of 4.5 model months before competing hazards are included. Susceptibility multipliers $\ell_1,\ell_2$ range from full protection (0) to full susceptibility (1). The model also permits infant vaccination into $VI$ and subsequent waning into $S$ ; that vaccination pathway is inactive under the current defaults.

MI_1\xrightarrow{\omega}MI_2\xrightarrow{\omega}S, \qquad \omega=\frac{1}{4.5\times30}\;\mathrm{day}^{-1}, \qquad MI_k\xrightarrow{\ell_k\lambda}E\quad(k=1,2)

Maternal-immunity transition rates in the active node updates.

A contact matrix is not a migration matrix

Ordinary mixing first averages age-specific prevalence across urban, rural, and nomadic sources. It then applies the recipient's age-contact matrix. School-year contact changes and seasonal transmission can vary this operator over time.

Definition

Contact composition for ordinary mixing

All source sums are over ordinary (non-IDP) communities, and $\kappa^+=\max(\kappa,0)$ . The capacity $c_i$ is the age-distribution-weighted mean contact count; $e_i$ is the total external contact input. These expressions apply when $c_i>0$ . If $c_i=0$ (including an empty recipient), the implementation sets the self-weight to 1 and external weights to 0. If $e_i>c_i$ , the weights renormalize to an entirely external mixture instead of becoming negative.

\begin{aligned} c_i&=\sum_a\frac{N_{ia}}{N_i}\sum_b C^{(i)}_{ab}, &e_i&=\sum_{j\ne i}\kappa_{ij}^{+},\\ w_{ij}&=\frac{\kappa_{ij}^{+}}{\max(c_i,e_i)}\quad(j\ne i), &w_{ii}&=\max\!\left(0,1-\frac{e_i}{c_i}\right). \end{aligned}

Contact-capacity and composition-weight routines in the active mixing module.

Identity

A valid prevalence mixture

For valid nonnegative compartment populations, the mixed prevalence stays a convex combination of source prevalences. This identity is a direct consequence of the contact-composition rule.

w_{ij}\ge0,\qquad\sum_jw_{ij}=1, \qquad 0\le\sum_jw_{ij}p_{jb}\le1

Proof / derivation

For $c_i>0$ , when $e_i\le c_i$ , external weights sum to $e_i/c_i$ and the self-weight is its complement. When $e_i>c_i$ , external weights sum to one and the self-weight is zero. The zero-capacity fallback is also a unit-sum vector. Nonnegative weights summing to one preserve the interval $[0,1]$ .

Model

Baseline force of infection

Age $b$ is the source and age $a$ the recipient. Here $p$ is infectious prevalence $I/N$ , set to zero for an empty age group. This baseline excludes main-IDP prevalence. The main displaced population has its own internal channel; its external contribution is added only to urban exposure. Temporary displaced groups add exposure only to their assigned rural recipients.

\lambda^{(0)}_{ia}(t) =\beta_i(t)\sum_b C^{(i)}_{ab}(t) \sum_{j\in\mathcal G_0}w_{ij}(t)\,p_{jb}(t)

Active baseline-plus-IDP force-of-infection assembly.

Competing risks, not independent outflows

If infection, vaccination, and death were sampled independently from the same susceptible population, their combined draws could overdraw that population. The stochastic step instead samples how many people leave, then allocates those departures among competing causes.

Model

One susceptible-compartment update

Here $\mu$ is natural mortality, $\Delta t$ is the time step, and $K$ counts transitions into exposure, vaccine protection, or death. Rates are frozen during the step. If $h=0$ , all departures are zero. The floor matches the integer count used for the binomial draw. Other compartments follow analogous competing-risk updates; births and aging are handled separately.

\begin{aligned} h&=\lambda+\nu(1-q)+\mu,\\ L&\sim\operatorname{Binomial}\!\left(\lfloor S\rfloor,1-e^{-h\Delta t}\right),\\ (K_E,K_V,K_D)\mid L &\sim\operatorname{Multinomial}\!\left(L, \frac{(\lambda,\nu(1-q),\mu)}{h}\right). \end{aligned}

Binomial departures and multinomial allocation in the active tau-leap implementation.

Rural and nomadic nodes use stochastic updates.
Urban and main-IDP nodes use stochastic updates below 50 infectious individuals, and deterministic expected-hazard updates otherwise.
A stochastic population can reach exact extinction; a smooth trajectory can retain arbitrarily small fractional infections.

Separate the scientific question from the random seed

Scientific configurations specify contact, immunity, seasonality, and displacement assumptions. Runtime configurations specify seeds and run counts. Saved configurations and seed manifests make a numerical comparison traceable; they do not make its parameters empirically validated.

Last updated September 26, 2026