Network SEIRV · Chapter 01

One population, several interacting scales

The mathematical question is how local disease transitions compose with age-dependent contact and a network of communities.

What is a node, and what is its state?

A node is a population group, not an individual person. The production topology used in the recurrence diagnostics has one urban, 18 rural, 500 nomadic, and one main displaced-population node. Each carries a separate state for every age group.

Definition

Age-structured community state

$S,E,I,R$ denote susceptible, exposed, infectious, and recovered populations; $V$ denotes vaccine-acquired protection. $MI_1$ and $MI_2$ are successive maternal-immunity stages, and $VI$ tracks vaccinated infants. Indices $i$ and $a$ select a community and one of $m$ age groups. Births, aging, mortality, and vaccination act on this eight-compartment state.

\begin{aligned} x_{ia}&=(MI_{1,ia},MI_{2,ia},VI_{ia},S_{ia},V_{ia},E_{ia},I_{ia},R_{ia})^\top,\\ N_{ia}&=\mathbf 1^\top x_{ia}, \qquad N_i=\sum_{a=1}^{m}N_{ia}. \end{aligned}

Active eight-compartment state definition and demographic update rules.

Contact and relocation are different operators

A contact changes exposure without moving residents. A transfer changes the population state itself. Keeping these mechanisms separate matters: an increase in infection burden at a destination need not mean that transmission intensified at the origin.

How the network enters the model
Mechanism	Mathematical role	Implemented scope
Ordinary contact	Mix source prevalence before applying age contacts	Urban, rural, and nomadic communities
Main displaced population	Separate internal exposure; added external pressure to urban nodes	Main IDP node and urban recipients
Temporary displacement	Stopover exposure followed by transfer to the main IDP population	Assigned rural recipient and temporary group
Ordinary travel	Transfer compartment populations between nodes	Optional rural/nomadic ↔ main IDP flows

IDP means internally displaced population. Ordinary travel is disabled by default; temporary-displacement shocks are a distinct mechanism.

What should a comparison measure?

Peak height alone misses duration. The diagnostics also measure infectious person-days, recurrence patterns, and stochastic fadeout. Initial immunity is part of the experiment, not a harmless implementation detail.

Definition

Infectious person-days

For a selected set of communities $\mathcal A$ , integrate the number currently infectious over the observation window. One individual infectious for seven days contributes seven person-days. This is neither the number of distinct cases nor the number of people displaced.

\operatorname{IPD}_{\mathcal A}(T) =\int_0^T\sum_{i\in\mathcal A}\sum_{a=1}^{m}I_{ia}(t)\,dt

Burden metric used in the displacement sensitivity diagnostics.

Last updated September 26, 2026