# Mathematical Modelling of the Coronavirus

Senator John Kennedy of Louisiana grills Chad Wolf, acting Homeland Security Secretary, about the Coronavirus
Watch it here.

Here is a bit of the philosophy of mathematical modelling

## Definitions

Susceptible - A person who does not have the disease, can catch it, and is active in the population (not isolated)
Infective - Someone who is infected with the disease and is active in the population (again, not isolated)
Removed - Someone who has had the disease, recovered, active, but no longer capable of either catching or transmitting the disease
Rate - The number of events per unit time. The contact rate, lambda (?), is the number of contacts of a typical infective per unit time that are adequate to transmit the disease if the other person contacted is susceptible.

## Types of Diseases (course of the disease states)

SI - A person is susceptible, becomes infected and remains infective for the rest of their life (for example, herpes)
SIS - A person is susceptible, becomes infective, recovers and can catch the disease again (for example, the common cold if we group together all viruses that cause one)
SIR - A person is susceptible, catches the disease and becomes infective, recovers and stays immune (example coronavirus as far as we know - this will be our assumption. We will make more comments about this possibility later.)
SIRS - After catching the disease, the person is immune, but the immunity is lost after some period of time and the person becomes susceptible again

## Rates used

lambda (λ) - the number of contacts per unit time that a typical infective person will make (that are adequate to transmit the disease if the person contacted is susceptible (note that contacts with removeds are still contacts, but the disease is not transmitted)
beta (β) - birth rate per unit time (here, we will take the unit of time to be weeks)
delta (δ) - death rate not due to the disease. We will presume that the death rate for susceptibles is the same as for removeds. That is to say, having had the disease does not increase your probability of death.
deltaD (δD) - death rate of infected persons (which, we assume, is elevated with respect to the non-infected population)
rho (ρ) - recovery rate from the disease

## The Model

Definitions of Functions
S(t) - the fraction of the population at time t that is susceptible
I(t) - the fraction of the population at time t that is infective
R(t) - the fraction of the population at time t that is removed
N(t) - the number of people in the population at time t

Time, t, here measured in weeks

## Equations

(NS)′(t) = -λ(IN)S + βN - δSN
(NI)′(t) = +λ(IN)S - δDIN - ρIN
(NR)′(t) = + ρIN - δRN
N′(t) = βN - δSN - δDIN - δRN
S(0) + I(0) + R(0) = 1

## Remembering the product rule for derivatives:

(NS)' = N'S + NS', so replacing the first three equations' left-hand sides with the appropriate product rules, dividing by N and subtracting the terms with N'/N, we get the new system of equations:
S′(t) = -λIS + β - δS - (N'/N)S
I′(t) = +λIS - δDI - ρI - (N'/N)I
R′(t) = + ρI - δR - (N'/N)R
N′/N = β - δS - δDI - δR

Finally, we can eliminate the N from the system by replacing the N'/N in the first three equations by the right-hand side of the last equation, so the model becomes:

S′(t) = -λIS + β - δS - (β - δS - δDI - δR)S
I′(t) = +λIS - δDI - (β - δS - δDI - δR)I
R′(t) = + ρI - δR - (β - δS - δDI - δR)R
S(0) + I(0) + R(0) = 1

This is the final system for today. Of course, there are many more theoretical considerations that we have done but will not show here. This final form can be used by
1. Estimating the parameters (Greek letters), and
2. Plugging the equation into a computer program to see what happens

Review of and comments on the video:

From our model we would expect, after introducing the virus into totally susceptible population, a slight delay (depending on the contract rate and the initial number of infections, a rather explosive growth of the spread of the disease. A peak will occur after which, depending on the recovery rate, the disease will decline. This decline could be very gradual if the infectious period is long.

Get the lecture notes and r-programmes from the video here.

Contact me at: dtudor@germinalknowledge.com