SIR model equations - diffrences

The standard equation for SIR model is dS/dt=bSI, but I can find it in many places, devided by N (the total population size): dS/dt=bSI/N. Can somebody explain me why?

Thank you in advance!

Answer 1

[John D'Errico]

No. The standard equation is irrlevant, UNLESS you understand what the terms mean. There is no standard equation if you are lacking that. Othrwise, all you have is a list of characters.

Consider a population of people. That total population, call it N, could be broken down into 3 subsets

S = Number susceptible to infection

I = Number of people currently infected

R = number of people that were infected, and are recovered.

Then we would clearly have the relation

N = S + I + R

Assume that N is essentially fixed. This is an approximation, assuming the disease is generally not fatal, and the disease progression is fast enough that births are not a factor.

And in some cases, we might allow the set of people who have recovered to become re-infected. There may be some rate at which that happens, perhaps different from the normal infection rate.

How does infection occur? This is due to a person who is susceptible coming in contact with a person who is infected. How does the set of susceptbles change?

They move from susceptible to infected by contact with an infected person. So how does the set S change?

dS/dt = -beta*S*(I/N)

Here beta is a transmission rate. beta under 1 will be good. beta much over 1 is bad news. (This is why social distancing works, because it effectively reduces the parameter beta to become less than 1.) I/N is the FRACTION of people that are infected. And there is your problem. You need to have that as a FRACTION of currently infected people, because that is how you become infected. It reflects the probability that you (as a susecptible) will meet a person who is already infected.

Clearly some of the sources you have seen treat I as the FRACTION of people infected, but most sources treat I as a sub-population itself, which is totally sensible.

As well, there is a negative sign in there. Unless of course, you treat beta as a negative number.

Again, unless you understand what the terms in a model means, the model is just a bunch of characters on the page.

Next, we would have a differential equation to describe how those infected move to the recovered population, as well as a third equation that would describe the movement of recovered people back into the infected set.

So we might a parameter gamma, that describes the rate of infected people moving to the recovered subgroup. That allows us to write a differential equation for the infected population as

dI/dt = beta*S*(I/N) - gamma*I

And, of course, the is a complete mass balance system. So people move from one population to another, but do not exit the system. (We could consider a death rate too, so some people could move to a 4th set: D, as some fraction of the set I.)

dR/dt = gamma*I

If recovered people could move back into the infected set with some rate, then these last two equations would each gain a new term. Or, we could have a 4th sub-population D, allowing people to be removed from the total at some rate based on the number of infecteds. (Finally, if this was in the movies, while we normally think of the set D as an absorbing state in this system, we might have some population moving from the set D into a 5th subset Z. I assume you know what Z would then signify.) Regardless, you can add many new terms, each increasing complexity of the behavior you can see.

The point of all this is unless you understand what those variables mean, then the model has no meaning at all.

[Luiza]

Thank you very much, John! I really appreciate your answer, it is very well explained and it makes sense the fraction of I population, defined as (I/N).

I found the next paragraph in a book.

The first equation is presented like dS/dt = -beta*S*I instead of dS/dt = -beta*S*(I/N).

How should I interpret this? Should I consider I(infected) in this case as a sub-population? In the book, they refer as it as: N = S(t)+I(t)+R(t). In this case the value I(infected) should represent the total infected individuals at time t, and not a fraction of the population.

===================================================================================

The number of individuals in each of these classes changes with time, that is,

S(t), I(t), and R(t) are functions of time t. The total population size N is the sum of

the sizes of these three classes:

N = S(t)+I(t)+R(t).

S'(t) = −incidence.

How can we represent the incidence? Consider one infectious individual. Assume:

• cN is the number of contacts per unit of time this infectious individual makes.

Here we assume that the number of contacts made by one infectious individual

is proportional to the total population size with per capita contact rate c.

• (S/N) is the probability that a contact is with a susceptible individual. Thus,

• cN (S/N) is number of contacts with susceptible individuals that one infectious individual

makes per unit of time. Not every contact with a susceptible individual

necessarily leads to transmission of the disease. Suppose p is the probability that

a contact with a susceptible individual results in transmission. Then,

• pcS is number of susceptible individuals who become infected per unit of time

per infectious individual.

• β SI is the number of individuals who become infected per unit of time (incidence).

Here we have set β = pc.

If we define λ (t) = β I, then the number of individuals who become infected per

unit of time is equal to λ (t)S. The function λ (t) is called the force of infection. The

coefficient β is the constant of proportionality called the transmission rate constant.

The number of infected individuals in the population I(t) is called the prevalence of

the disease.

There are different types of incidence depending on the assumption made about

the form of the force of infection. One form is called mass action incidence. With

this form of incidence, we obtain the following differential equation for susceptible

individuals:

S'=(t) = −β IS.

The susceptible individuals who become infected move to the class I. Those individuals

who recover or die leave the infected class at constant per capita probability

per unit of time α, called the recovery rate. That is, αI is the number of infected

individuals per unit of time who recover. So,

I'=(t) =β IS−αI.

Individuals who recover leave the infectious class and move to the recovered class

R'=(t) =αI.

Thus, the whole model is given by the following system of ODEs:

S'=(t) = −β IS,

I'(t) = β IS−αI,

R'(t) = αI.

===================================================================================

Thank you very much again!

[Bjorn Gustavsson]

Play around with this model. Implement your coupled ODEs, integrate for a couple of different transition-rates look at the results, and have a think about what varies in which ways when you change the your beta and alpha. Nothing we tell you will give you the understanding you obtain by doing. If you've done that then we can discuss.

[John D'Errico]

Yes. I do agree with what Bjorn has said BTW. Try it. Play around. Get your hands dirty.

But as you asked, you do need to understand that equation:

N = S(t)+I(t)+R(t).

So we assume a fixed population of individuals. They move around between sub-populations. Susceptible individuals moving to infected, then to recovered. This is the simplest case, as I said.

But I(t) is indeed the number of individuals at time t, those that are currently infected. Think about what the product means, thus

-beta * S * (I/N)

Suppose you are a "susceptible" individual. Not infected currently. Never have been. No antibodies. You walk around outside. Talk to one of your normal set of friends. What is the probability you will catch the bug? First, what are the odds this random person you talk to is currently infected? Yes. I(t)/N. Once you do interact with them, will you catch it? beta reflects the how easily this is transmitted. If you do catch the bug, then you DECREASE S(t), because you move from sub-population S(t) into subpopulation I(t). Therefore the derivative of S(t) must be negative in this simple model. Eventually, at least in the simple model, you move into subpopulation R(t), which in the simplest model leaves you "safe" from re-infection.

Again, this is a VERY simple form of a model. Overly simplistic models may fail to predict well. It lies in the skill of the mathematical modelor to recognize which terms are safe to leave out from a more complex model, if you desire accurate predictive behavior. That skill is a very important one. For example:

1. Will some individuals die as a result of this disease? If so, then we need to consider an absorbing state D(t). I will not admit a sub-pop Z(t), since I am not making a movie.
2. Will there be new individuals born, or is the pandemic under study moving too fast that the fraction born will not impact the results sufficiently to be seen?
3. Will the transmission rate remain constant over time? In fact, we have seen this is often not true. As restaurants and sports events open and people become tired of staying home, the rate can suddenly increase. (Personally, I prefer not to wave a red flag at a charging bull, but that is my choice.)
4. If deaths are possible under the model assumptions, may that death rate change with time too? For example, it may be true that as ICU beds become overwhelmed, the death rate will increase. As new treatments are developed, the death rate may decrease.
5. Does a vaccine exist? If so, then we could view it as if some of the population can effectively move directly from the susceptible population S(t) into the recovered population R(t). Since not all people seem to be willing to take a vaccine, only a fraction of people will move into the R(t) population.
6. Can recovered individuals ever get re-infected? This is generally an important factor, that depends on how long antibodies persist. It also depends on multiple strains of the bug floating around. While you may be recovered and quasi-resistant to bug strain A, there are still strains B and C that may be out there too, slightly different enough that your body will not recognize them.
7. Is everyone equally susceptible to infection? That is generally not true. Some individuals will be immune compromised, etc.

A truly complete model may be very complex, with a huge number of terms in it and a vast number of interactions. An appreciation is necessary of which of the many ideas I discussed above are important to predict a pandemic. But you can still understand how a simpler model reacts, and it still may provide reasonable predictions.

So yes. I(t) is a sub-population itself. And therefore, I/N is a relative fraction.

[Luiza]

Thank you very much, again.I understand perfectly what you say.

And I understand very well the significance for S,I, R and how the equations are defined.

My initial question was quite diffrent.

I have 2 sistems.

The first one is:

dS/dt=-beta*S*I

dI/dt=beta*S*I-alpha*I

dR/dt=alpha*I

And the second is:

dS/dt=-beta*S*I/N

dI/dt=beta*S*I/N-alpha*I

dR/dt=alpha*I

And I try to understand a little bit the diffrences between them.

I think both of them are correct, but the calculations applied on each of them should be different, because for the first sistem we consider the subclasses, while for the second sistem we consider the fractions.

For example for sistem no1: R0=(beta/alpha)*N, while for the sistem no2, R should be: R0= beta/alpha.

And so on for the others calculations.

Please correct me if I am wrong.

Thank you very much again!

[Bjorn Gustavsson]

If you implement the two systems you can obviously check this assumption of yours rather easily.

Answer 2

[ZOUBAIR DAOUMA]

please is anyone a Matlab code for the numerical simulations on a square lattice of S.I.R Model