Lecture 04 - Group models

5 April 2022

Julien Arino

Department of Mathematics & Data Science Nexus
University of Manitoba*

Canadian Centre for Disease Modelling
Canadian COVID-19 Mathematical Modelling Task Force
NSERC-PHAC EID Modelling Consortium (CANMOD, MfPH, OMNI/RÉUNIS)

* The University of Manitoba campuses are located on original lands of Anishinaabeg, Cree, Oji-Cree, Dakota and Dene peoples, and on the homeland of the Métis Nation.

Outline

  • Formulating group models
    • Age-structured models
    • Models incorporating social structure
    • Models with pathogen heterogeneity
    • Models with immunological component
  • Analysing group models
  • Simulating group models

Formulating group models

  • Age-structured models
  • Models incorporating social structure
  • Models with pathogen heterogeneity
  • Models with immunological component

Limitations of single population ODE models

  • As discussed in Lecture 03, basic ODE assume that all individuals in a compartment are roughly the same
  • Individuals can spend differing times in a compartment (see Lecture 09), but they are all the same
  • As we have seen with COVID-19, different age groups are impacted differently

Groups can be used for many things

Groups allow to introduce structure in a population without using PDEs

  • Age structure
  • Social structure
  • Pathogen heterogeneity
  • Host heterogeneity (e.g., super spreaders)

In this lecture, we do not consider spatial heterogeity; this is done in Lecture 05

We start by considering a few examples

Age-structured models

First, a remark

In terms of modelling, ODEs are not the best way to incorporate structure such as age. We will come back to this in Lecture 09, but give one example here

A multi-group SIS model with age structure, by Feng, Huang & C

For different subgroups

where

with boundary and initial conditions, for

( fraction of newborns that is infected)

Basic reproduction number in group

  • Authors obtain some results in terms of global stability
  • Need simplifications to move forward
  • No numerics, because numerics for such models are hard

Going the ODE route

  • ODEs are way less satisfactory but can be used as-is and are much easier numerically
  • Caveat - ODE models with age structure are intrinsically wrong, since sojourn times in an age group is exponentially distributed instead of Dirac distributed! (See Lecture 09)

Models incorporating social structure

TB in foreign-born population of Canada

Preventing tuberculosis in the foreign-born population of Canada: a mathematical modelling study. Varughese, Langlois-Klassen, Long, & Li. International Journal of Tuberculosis and Lung Disease 18 (2014)

  • New immigrants from Canada come predominantly from countries in which TB is very active
  • It has been noticed that people develop TB in the first few years of their presence in CAN
  • Want to investigate this, together with effect of various screening measures

Models with pathogen heterogeneity

Importation of a new SARS-CoV-2 variant

Risk of COVID-19 variant importation – How useful are travel control measures? Arino, Boëlle, Milliken & Portet. Infectious Disease Modelling 6 (2021)

  • Consider what happens when a new variant N arrives in a situation where another variant O is already circulating

Coupling is through the force of infection

  • For now, we have discussed incidence functions
  • Here, we use a force of infection , for
  • Force of infection uses "outside" of function: it is the pressure that applies to individuals to make them infected
  • Of course, the two are equivalent, but in some contexts, it makes sense to use this
  • Here, for

Adding more groups - "Importation layer"

  • How can we evaluate how much "importations" contribute to propagation within a location?
  • If an individual arrives in a new location while bearing the disease, we put them in a special group, the importation layer
  • In importation layer, individuals make contacts with others in the population, but they remain in the importation layer until recovery or death

Force of infection with importation layer

For

where, for and

Models with immunological component

Global dynamics of a general class of multistage models for infectious diseases. Guo, Li & Shuai. SIAM Journal on Applied Mathematics 72 (2012)

  • Viruses such as HIV reside in the body for a very long time, potentially for life
  • Throughout the course of this residence, virus loads change and with it symptoms and infectiousness

Analysing group models

Model of Guo, Li & Shuai

  • Use Kirchhoff’s matrix tree theorem to show that is negative definite

Simulating group models

  • This is very similar to metapopulation models which are described in Practicum 02
  • The variant importation model simulations will be discussed in the stochastic lectures