(For a more detailed explanation, refer to this YouTube presentation, to this paper or also to this more recent one (here in a better format).

Imagine the virus outbreaks in a few different places. As a countermeasure, we can adopt different levels of lockdown. The video below

shows the virus spreading (*I* = number of infected
individuals) in a squared region (*x _{1}*
and

The role played and suffered by care homes (RSA in Italian) has come dramatically in the foreground. Below is a simulation of what might have happened in the area around 2 care homes.

*S* is the number of individuals that can be infected
(Susceptibles), *I* is the number of Infective
individuals, *H* is the number of quarantied individuals
and *R* is the number of the individuals that
Recovered. They are all functions of *t* (time) and space
(*x _{1},x_{2}*).

Let's see how people's movements affect the virus spreading. First, assume no one moves and a small group of infected individuals appears in the bottom left of the square we consider.

Now, we superimpose to the virus spreading the movement of the
commuters. Say that between 06:00 and 09:00, those who are not
in quarantine move towards the city centre, sited at
(*0,0*). Later, between 17:00 and 20:00, they all go back
home. Here is the resulting spreading. At about time 4 (meaning
the 4^{th} day), some commuters are infected and the
virus diffusion is enhanced by the commuters' travelling.

To quantify the differences in the virus diffusion, see the time
dependent *R _{0}* index, left in the case with
movement, right in the case no one commutes:

Careful! After day 9, it looks like the situation with movements
(on the left) were *"better"* than that with no commuting
(right), since the left one has a lower *R _{0}*
index, actually even lower than 1. Think about it and you'll see
why it is indeed a further sign of a much worse
situation. The

Indeed, a better, though dramatic, evaluation of
the *"costs"* corresponding to the two situations (with
and without commuting) is given by the graphs of the number of
deaths, here below. (Again, the case with commuting is on the
left, while on the right there is no movement).

Many models are able to describe features of the covid pandemic. The one used in this page is (relatively!) simple, it takes into consideration age differences as well as spatial distribution and the role of quarantine is highlighted. Here is it:

where the *"diffusion"* is described by the term

The dependent and independent variables are

while the meanings of the other quantities, all dependent on * (t,a,x)* are:

To fully understand this model, a little more than *Calculus
2* is necessary. Nevertheless,
in this
presentation you find a rather simple description. A similar
presentation, though shorter and in Italian,
is here.

The introduction of vaccines has been a majhor help in fighting the pandemic. It also raised several questions about the vaccines' efficacy and about the optial vccination strategies. A model comprising the time dependent efffects of a vaccine and finite time immunization after recovery is

Here, *ρ _{V}=ρ_{V}(τ)*
describes the efficacy of a vaccine after time

A possible qualitative behavior of the
function *ρ _{V}=ρ_{V}(τ)* is
below.

The value *ρ _{S}* corresponds to how
easy/difficult it is for a susceptible individual to get
infected, while

On the other hand, the effect of two consecutive doses of a vaccine might be described by the following diagram.

This model displays other features, such as pandemic waves, and allows tocompare different vaccination strategies. Refer to the paper.

The content of this page comes from

Rinaldo M. Colombo, Mauro Garavello, Francesca Marcellini,
Elena Rossi

*An Age and Space Structured SIR Model Describing the Covid-19
Pandemic*

*Journal of Mathematics in Industry*, 10, 22, 2020

Preprint, 2020.

Rinaldo M. Colombo, Mauro Garavello

*Infectious Diseases Spreading Fought by Multiple Vaccines
Having a Prescribed Time Effect*

*Acta Botheoretica*, 71:1, 2023

Preprint, 2021