Lecture 05 - Metapopulation epidemic 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 metapopulation models
  • Basic mathematical analysis
  • is not the panacea - An urban centre and satellite cities
  • Problems specific to metapopulations
  • Global stability considerations

Formulating metapopulation models

Disease spread process in a jurisdiction-based world

General principles (1)

  • geographical locations (patches) in a set (city, region, country..)
  • Patches are vertices in a graph
  • Each patch contains compartments
    • individuals susceptible to the disease
      • individuals infected by the disease
      • different species affected by the disease
      • etc.

General principles (2)

  • Compartments may move between patches, with rate of movement of individuals from compartment from patch to patch
  • Movement instantaneous and no death during movement
  • , defines a digraph with arcs
  • Arc from to if , absent otherwise
  • compartments, so each can have at most arrows multi-digraph

The underlying mobility model

population of compartment in patch

Assume no birth or death. Balance inflow and outflow

when we write

The toy SLIRS model in patches

center

is the birth rate (typically or )

= latently infected ( exposed, although the latter term is ambiguous)

-SLIRS model

with incidence

-SLIRS (multiple species)

a set of species

with incidence

-SLIRS (residency patch/movers-stayers)

with incidence

General metapopulation epidemic models

uninfected and infected compartments, and

For , and ,

where and

Basic mathematical analysis

Analysis - Toy system

For simplicity, consider -SLIRS with

with incidence

System of equations

Size is not that bad..

System of equations !!!

However, a lot of structure:

  • copies of individual units, each comprising 4 equations
  • Dynamics of individual units well understood
  • Coupling is linear

Good case of large-scale system (matrix analysis is your friend)

Notation in what follows

  • a square matrix with entries denoted

  • if for all (could be the zero matrix); if and with ; if . Same notation for vectors

  • spectrum of

  • spectral radius

  • spectral abscissa (or stability modulus)

  • is an M-matrix if it is a Z-matrix ( for ) and , with and

Behaviour of the total population

Consider behaviour of . We have

So

Vector / matrix form of the equation

We have

Write this in vector form

where -matrices with

The movement matrix

Consider a compartment . Then the following hold true:

  1. and corresponds to left eigenvector
  2. singular M-matrix
  4. irreducible has multiplicity 1

The nice case

Recall that

Suppose movement rates equal for all compartments, i.e.,

Then

Equilibria

given, of course, that (or, equivalently, ) is invertible.. Is it?

Perturbations of movement matrices

a movement matrix and a diagonal matrix. The following hold true:

  1. for all
  2. and is associated with an eigenvector . If, additionally, irreducible, then has multiplicity 1 and is associated with
  3. nonsingular M-matrix and
  4. irreducible and irreducible nonsingular M-matrix and

Nonsingularity of

Using a spectrum shift,

This gives a constraint: for total population to behave well (in general, we want this), we must assume all death rates are positive

Assume they are (in other words, assume nonsingular). Then is nonsingular and unique

Behaviour of the total population

Equal movement case

attracts solutions of

Indeed, we have

Since we now assume that is nonsingular, we have (spectral shift & properties of )

irreducible (provided , of course)

Behaviour of total population with reducible movement

Assume reducible. Let be the number of minimal absorbing sets in the corresponding connection graph . Then

  1. The spectral abscissa has multiplicity
  2. Associated to is a nonnegative eigenvector s.t.
    • if is a vertex in a minimal absorbing set
    • if is a transient vertex

From Foster and Jacquez, Multiple zeros for eigenvalues and the multiplicity of traps of a linear compartmental system, Mathematical Biosciences (1975)

The not-so-nice case

Recall that

Suppose movement rates similar for all compartments, i.e., the zero/nonzero patterns in all matrices are the same but not the entries

Let

and

Cool, no? No!

Then we have

Me, roughly every 6 months: Oooh, coooool, a linear differential inclusion!

Me, roughly 10 minutes after that previous statement: Quel con!

Indeed and are are not movement matrices (in particular, their column sums are not all zero)

So no luck there..

However, non lasciate ogne speranza, we can still do stuff!

Disease free equilibrium (DFE)

Assume system at equilibrium and for . Then and

Want to solve for . Here, it is best (crucial in fact) to remember some linear algebra. Write system in vector form:

where , -matrices ( diagonal)

at DFE

Recall second equation:

So unique solution if invertible.
Is it?

We have been here before!

From spectrum shift,

So, given , is the unique equilibrium and

DFE has

at the DFE

DFE has and , i.e.,

Recall: singular M-matrix. From previous reasoning, has instability modulus shifted right by . So:

  • invertible
  • nonsingular M-matrix

Second point (would have if irreducible)

So DFE makes sense with

Computing the basic reproduction number

Use next generation method with ,

Differentiate w.r.t. :

Note that

whenever , so

Evaluate at DFE

If , then

If , then

  • at DFE
  • at DFE

In both cases, block is zero so

Compute and evaluate at DFE

where . Inverse of easy ( block lower triangular):

where

as

Next generation matrix

where is block in . So

i.e.,

Local asymptotic stability of the DFE

Define for the -SLIRS as

Then the DFE

is locally asymptotically stable if and unstable if

From PvdD & Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Bulletin of Mathematical Biology 180(1-2): 29-48 (2002)

Global stability considerations

  • GAS is much harder
  • It has been done many times (look at my papers, but also those of Li, Shuai, Thieme, van den Driessche, Wang, Zhao..)
  • I am not aware of a way to do this generically

is not the panacea

An urban centre and satellite cities

JA & S Portet. Epidemiological implications of mobility between a large urban centre and smaller satellite cities. Journal of Mathematical Biology 71(5):1243-1265 (2015)

Context of the study

Winnipeg as urban centre and 3 smaller satellite cities: Portage la Prairie, Selkirk and Steinbach

  • population density low to very low outside of Winnipeg
  • MB road network well studied by MB Infrastructure Traffic Engineering Branch

Known and estimated quantities

City Pop. (2014) Pop. (now) Dist. Avg. trips/day
Winnipeg (W) 663,617 749,607 - -
Portage la Prairie (1) 12,996 13,270 88 4,115
Selkirk (2) 9,834 10,504 34 7,983
Steinbach (3) 13,524 17,806 66 7,505

Estimating movement rates

Assume movement rate from city to city . Ceteris paribus, , so . Therefore, after one day, , i.e.,

Now, , where number of individuals going from to / day. So

Computed for all pairs and of cities

Sensitivity of to variations of


with disease: ; without disease: . Each box and corresponding whiskers are 10,000 simulations

Lower connectivity can drive

PLP and Steinbach have comparable populations but with parameters used, only PLP can cause the general to take values larger than 1 when

This is due to the movement rate: if , then

since is then block diagonal

Movement rates to and from PLP are lower situation closer to uncoupled case and has more impact on the general

does not tell the whole story!


Plots as functions of in PLP and the reduction of movement between Winnipeg and PLP. Left: general . Right: Attack rate in Winnipeg

Problems specific to metapopulations

Inherited dynamical properties (a.k.a. I am lazy)

Given

with known properties, what is known of

  • Existence and uniqueness
  • Invariance of under the flow
  • Boundedness
  • Location of individual and general ?
  • GAS?

An inheritance problem - Backward bifurcations

  • Suppose a model that, isolated in a single patch, undergoes so-called backward bifurcations
  • This means the model admits subthreshold endemic equilibria
  • What happens when you couple many such consistuting units?

YES, coupling together backward bifurcating units can lead to a system-level backward bifurcation

JA, Ducrot & Zongo. A metapopulation model for malaria with transmission-blocking partial immunity in hosts. Journal of Mathematical Biology 64(3):423-448 (2012)

Metapopulation-induced behaviours ?

"Converse" problem to inheritance problem. Given

with known properties, does

exhibit some behaviours not observed in the uncoupled system?

E.g.: units have DFE GAS, 1 GAS EEP behaviour, metapopulation has periodic solutions

Mixed equilibria

Can there be situations where some patches are at the DFE and others at an EEP?

This is the problem of mixed equilibria

This is a metapopulation-specific problem, not one of inheritance of dynamical properties!

Types of equilibria

[Patch level] Patch at equilibrium is empty if , at the disease-free equilibrium if , where are some indices with and are positive, and at an endemic equilibrium if

[Metapopulation level] A population-free equilibrium has all patches empty. A metapopulation disease-free equilibrium has all patches at the disease-free equilibrium for the same compartments. A metapopulation endemic equilibrium has all patches at an endemic equilibrium

Mixed equilibria

A mixed equilibrium is an equilibrium such that

  • all patches are at a disease-free equilibrium but the system is not at a metapopulation disease-free equilibrium
  • or, there are at least two patches that have different types of patch-level equilibrium (empty, disease-free or endemic)

E.g.,

is mixed, so is

Suppose that movement is similar for all compartments (MSAC) and that the system is at equilibrium

  • If patch is empty, then all patches in are empty
  • If patch is at a disease free equilibrium, then the subsystem consisting of all patches in is at a metapopulation disease free equilibrium
  • If patch is at an endemic equilibrium, then all patches in are also at an endemic equilibrium
  • If is strongly connected for some compartment , then there does not exist mixed equilibria

Note that MSAC and for all

Interesting (IMHO) problems

More is needed on inheritance problem, in particular GAS part (Li, Shuai, Kamgang, Sallet, and older stuff: Michel & Miller, Šiljak)

Incorporate travel time (delay) and events (infection, recovery, death ..) during travel

What is the minimum complexity of the movement functions below

required to observe a metapopulation-induced behaviour?

Global stability considerations

-SLIRS model

The linear stability result for can be strengthened to a global result

Let be computed as explained earlier. If then the DFE of -SLIRS - is globally asymptotically stable

Proof

Since

and the equation for gives the inequality

Define a linear system given by the equation above and

  • This system linear has coefficient matrix , and so (by some argument in the proof of local stability based on ) satisfies and for
  • Using a comparison theorem, it follows that these limits also hold for the nonlinear system in and
  • That and follow from the equations for and

Thus for the DFE is GAS and the disease dies out

-SLIRS (multiple species)

a set of species

For the -SLIRS system - with equal movement rates for all states, define using the method described earlier and use proportional incidence. If , then the DFE is globally asymptotically stable

(Movement equal for all states: for )

Proof of the result

To establish the global stability of the DFE, consider the nonautonomous system consisting of , and , with written in the form

in which has been replaced by , and is a solution of the equation for the total population

To continue, we need this

Suppose movement is identical for all epidemiological states, that disease is not lethal (), and that in each patch, birth compensates natural death, that is, . Then the movement model is given, for all and all by

and there holds

Write the system , and as

where is the dimensional vector consisting of the , and

The DFE of the original system corresponds to the equilibrium in the the nonautonomous system

System for can be solved for independently of the epidemic variables, and result earlier implies that the time dependent functions as

Substituting this large time limit value for in the nonautonomous equation for gives

Therefore, the nonautonomous system is asymptotically autonomous, with limit equation

To show that 0 is a globally asymptotically stable equilibrium for the limit system, consider the linear system

where is the dimensional vector consisting of the , and . In , we replace with . Equations and are not affected by this transformation, whereas takes the form

Comparing and , we note that for all

In system , the equations for and do not involve . Let be the part of the vector corresponding to the variables and , and be the corresponding submatrix of .

The method of used to prove local stability can also be applied to study the stability of the equilibrium of the subsystem , with

Therefore, if , then the equilibrium of the subsystem is stable. When , takes the form

with and

We know that is a singular M-matrix. It follows that is a nonsingular M-matrix for each

Thus the equilibrium of the linear system in is stable

As a consequence, the equilibrium of is stable when

Using a standard comparison theorem, it follows that 0 is a globally asymptotically stable equilibrium of the asymptotically autonomous system , and

For , the linear system - has a unique equilibrium (the DFE) since its coefficient matrix is nonsingular

The proof of global stability is completed using results on asymptotically autonomous equations

Epilogue / Postlude

In conclusion

  • Space is a fundamental component of the epidemic spread process and cannot be ignored, both in modelling and in public health decision making

  • One way to model space is to use metapopulation models

  • Metapopulation models are easy to analyse locally, give interesting problems at the global level

  • We will see in Practicum 02 that simulation (deterministic and stochastic) can be costly in RAM and cycles but is easy

  • Metapopulation models are not the only solution!