Outline

• Steps of the analysis
• The basic stuff (well-posedness, DFE)
• Epidemic models
• Endemic models

Steps of the analysis

Step 0 - Well-posedness

• Do solutions exist?
• Are they unique?
• Are they bounded?
• Invariance of the nonnegative cone under the flow..?

For "classic" models, all of these properties are more or less a given, so good to bear in mind, worth mentioning in a paper, but not necessarily worth showing unless this is a MSc or PhD manuscript

When you start considering nonstandard models, or PDE/DDE, then often required

Step 1 - Epidemic model or endemic model ?

• Often a source of confusion: analysis of epidemic models differs from analysis of endemic models!!
• Important to determine what you are dealing with
• Easy first test (can be wrong): is there demography?
  • Demography can lead to constant population, but if there is "flow" through the system (with, e.g., births = deaths), then there is demography
• Other (more complex) test: what is the nature of the DFE?

Step 1 and a half - Computing the DFE

• If you are not yet sure whether you have an epidemic or endemic model, you need to compute the DFE (you will need it/them anyway)
• Usually: set all infected variables to 0 (I, L and I, etc.)
  • If you find a single or denumerable number of equilibria for the remaining variables, this is an endemic model
  • If you get something of the form "any value of works", this is an epidemic model

Step 2 - Epidemic case

• Compute
• Usually: do not consider LAS properties of DFE, they are given
• Compute a final size (if feasible)

Step 2 - Endemic case

• Compute and deduce LAS properties of DFE
• (Optional) Determine direction of bifurcation at
• (Sometimes impossible) Determine GAS properties of DFE or EEP

Why considering LAS properties of epidemic model is wrong

Consider the IVP

and denote its solution at time through the initial condition

is an equilibrium point if

is locally asymptotically stable (LAS) if open in the domain of s.t. for all , for all and furthermore,

If there is a continuum of equilibria, then , where is some curve in the domain of , s.t. for all . We say is not isolated. But then any open neighbourhood of contains elements of and taking , , implies that . is locally stable but not locally asymptotically stable!

The basic stuff (well-posedness, DFE)

Existence and uniqueness

• Is your vector field ?
  • If so, you are done
  • If not, might be worth checking. Some of the models in particular have issues if the total population is variable and under circumstances
• Probably not worth more than "solutions exist and are unique" in most instances...

Invariance of the nonnegative cone under the flow

• Study of this can be warranted
• What can be important is invariance of some subsets of the nonnegative cone under the flow of the system.. this can really help in some cases

Example: SIS system

First, remark that - is , giving existence and uniqueness of solutions

Invariance of under the flow of -

If , then becomes

cannot ever become zero: if , then for all . If , then for small and by the preceding argument, this is also true for all

For , remark that if , then is positively invariant: if , then for all

In practice, values of for any solution in are "carried" by one of the following 3 solutions:

1. : increases to
2. : remains equal to
3. : decreases to

As a consequence, no solution with can enter . Suppose and s.t. ; denote the value of when becomes zero

Existence of contradicts uniqueness of solutions, since at , there are then two solutions: that initiated in and that initiated with

Boundedness

positive quadrant (positively) invariant under flow of -

We could detail more precisely (positive IC ..) but this suffices here

From the invariance dans the boundedness of the total population , we deduce that solutions to - are bounded

Where things can become complicated...

• If , e.g., , what happens to the incidence?
• If , e.g., , solutions are unbounded

Computing the DFE

• Set all infected variables to zero, see what happens...
• Personnally: I prefer to set some infected variables to zero and see if I recuperate the DFE that way

Epidemic models

• Computation of
• Final size relation
• Examples

Computation of

A method for computing in epidemic models

• This method is not universal! It works in a relatively large class of models, but not everywhere. If it doesn't work, the next generation matrix method (see later) does work, but should be considered only for obtaining the reproduction number, not to deduce LAS (cf. my remark earler)
• Here, I change the notation in the paper, for convenience

Standard form of the system

Suppose system can be written in the form

where , and are susceptible, infected and removed compartments, respectively

IC are with at least one of the components of positive

• continuous function encoding recruitment and death of uninfected individuals
• diagonal with diagonal entries the relative susceptibilities of susceptible compartments, with convention that
• Scalar valued function represents infectivity, with, e.g., for mass action
• row vector of relative horizontal transmissions

• has entry the fraction of individuals in susceptible compartment that enter infected compartment upon infection
• diagonal with diagonal entries the relative susceptibilities of susceptible compartments, with convention that
• Scalar valued function represents infectivity, with, e.g., for mass action
• row vector of relative horizontal transmissions
• describes transitions between infected states and removals from these states due to recovery or death

• continuous function encoding flows into and out of removed compartments because of immunisation or similar processes
• has entry the rate at which individuals in the infected compartment move into the removed compartment

Suppose is a locally stable disease-free equilibrium (DFE) of the system without disease, i.e., an EP of

If no demopgraphy (epidemic model), then just , of course

Final size relations

Final size relations

Assume no demography, then system should be writeable as

For continuous, define

Define the row vector

then

Suppose incidence is mass action, i.e., and

Then for , express as a function of using

then substitute into

which is a final size relation for the general system when

If incidence is mass action and (only one susceptible compartment), reduces to the KMK form

In the case of more general incidence functions, the final size relations are inequalities of the form, for ,

where is the initial total population

Examples

The SLIAR model

• Paper we have already seen:
  • Arino, Brauer, PvdD, Watmough & Wu. Simple models for containment of a pandemic (2006)
• However, suppose additionally that are also infectious

Here, , and , so , and

Incidence is mass action so and thus

For final size, since , we can use :

Suppose , then

If , then

A model with vaccination

Fraction of are vaccinated before the epidemic; vaccination reduces probability and duration of infection, infectiousness and reduces mortality

with and

Here, , ,

So

and the final size relation is

Where things go awry, final size-wise!

• Summer 2021 work of Aaron Shalev (U of M)
• Consider the very simple 2-patch metapopulation (see Lecture 05)

• Unidirectional movement from patch 1 to patch 2
• cannot be scalar-valued here!

Endemic models

• Computation of using the next generation matrix method
• Examples

Computation of using the next generation matrix method

Next generation matrix/operator

Diekmann and Heesterbeek, characterised in ODE case by PvdD & Watmough (2002)

Consider only compartments with infected individuals and write

• flows into infected compartments because of new infections
• other flows (with sign)

Compute the (Frechet) derivatives and with respect to the infected variables and evaluate at the DFE

Then

where is the spectral radius

Preliminary setup

, , with the first compartments the infected ones

the set of all disease free states:

Distinguish new infections from all other changes in population

• rate of appearance of new infections in compartment
• rate of transfer of individuals into compartment by all other means
• rate of transfer of individuals out of compartment

Assume each function continuously differentiable at least twice in each variable

where

Some assumptions

• (A1) If , then for

Since each function represents a directed transfer of individuals, all are non-negative

• (A2) If then . In particular, if , then for

If a compartment is empty, there can be no transfer of individuals out of the compartment by death, infection, nor any other means

• (A3) if

The incidence of infection for uninfected compartments is zero

• (A4) If then and for

Assume that if the population is free of disease then the population will remain free of disease; i.e., there is no (density independent) immigration of infectives

One last assumption for the road

Let be a DFE of the system, i.e., a (locally asymptotically) stable equilibrium solution of the disease free model, i.e., the system restricted to . We need not assume that the model has a unique DFE

Let be the Jacobian matrix . Some derivatives are one sided, since is on the domain boundary

• (A5) If is set to zero, then all eigenvalues of have negative real parts

Note: if the method ever fails to work, it is usually with (A5) that lies the problem

Stability of the DFE as function of

Important to stress local nature of stability that is deduced from this result. We will see later that even when , there can be several positive equilibria

Direction of the bifurcation at

bifurcation parameter s.t. for and for and DFE for all values of and consider the system

Write

as block matrix

Write , , the entry of and let and be left and right eigenvectors of s.t.

Let

Examples

• The SLIRS model
• A tuberculosis model incorporating treatment
• An "issue" with the next generation method for
• Bistable states

The SLIRS model

Example of the SLIRS model

Variations of the infected variables described by

Thus

Then compute the Jacobian matrices of vectors and