Medical researchers and mathematicians have developed a series of sophisticated mathematical models to describe the spread of infectious diseases. But even a simple model is useful to predict how long an outbreak of a disease, for example the flu, will last and how many people will be sickened by it.

The oldest and most common model is the SIR model which considers every person in a population to be in one of three conditions:

- S = Susceptible to becoming infected
- I = Infected through contact with someone already infected
- R = Recovered, no longer sick or infected.

Through time a person may move from being susceptible to infected to recovered, so the number of people in each condition changes, but the total of S + I + R is constant. S + I + R = N where N represents the entire population and is identified as closed system population.

This is a compartmental model, with S, I and R being compartments. Every person starts off in a compartment and many move to others over time. Graphically the compartment model looks like the figure below with the rates of movement between compartments given as Greek letters above the arrows indicating direction of movement.

**Assumptions & Parameters**

This is a steady-state model with no one dying or being born, to change the total number of people. In this model once someone recovers they are immune and can’t be infected again. More sophisticated models allow re-infections. The model also assumes that a disease is passed from person to person. The SIR model can’t be used for diseases that spread other ways, such as by insect bites.

To run this model, you need to know the following:

- initial population, S (initial number of people who are susceptible),
- initial number of infected people, I
- Infection rate, ß (Greek letter beta, the rate (#/day) that susceptible people become infected),
- recovery rate, γ (Greek letter gamma, the rate that infected people recover),
- time increment, T (the time interval or steps during which changes occur.) T is usually set at one day and because its value is 1, is often ignored. For rapidly spreading outbreaks T might be one hour or some other short interval.

How do you know the values of these parameters? The number of susceptible people (S) can be the population of a city or town where the outbreak occurs. The number of initially infected people (I) is a guess unless it is known, for example, that a single traveller brought the disease into a community. The infection rate (ß) and the recovery rate (γ) can be selected from rates determined from prior outbreaks, but they often vary for different outbreaks of the same disease.

**Equations**

With these parameters, the number of people at any time who are Susceptible, Infected, or Recovered can be calculated with these equations:

S_{n} = S_{n-1 }– ((S_{n-1}/S) ** _{*}** (ß

**I**

_{*}_{n-1}))

I_{n} = I_{n-1 }+ (S_{n-1}/S) ** _{*}** (ß

**I**

_{*}_{n-1}) – (I

_{n-1 }

**γ)**

_{*}R_{n} = R_{n-1 }+ (I_{n-1} ** _{*}** γ)

Because of subscripts and Greek letters these equations look complicated, but they aren’t really!

These equations calculate the number of people in each condition today (n), based on the number yesterday (n-1) and the rates of change, ß and γ. The subscript n means the number in one time interval, and n-1 means the number in the previous interval. So with a time interval of one day, then the first equation:

S_{n} = S_{n-1 }– ((S_{n-1}/S) ** _{*}** (ß

**I**

_{*}_{n-1}))

can be understood as:

*The number of susceptible people today (*S_{n}*) equals the number yesterday (*S_{n-1}*), MINUS the percentage of people who become infected today (which is yesterday’s number of susceptible people (*S_{n-1})* divided by the original number (/S)), times their rate of infection** (ß**)**, times how many people were infected (*I_{n-1}*) yesterday.*

The number of susceptible people today equals the number who were susceptible yesterday minus the number who become infected today. As long as the disease is spreading, the number not yet infected – the remaining susceptibles – decreases every day.

The number who become infected today equals yesterday’s number of susceptibles times the rate of infection, but it may seem odd that we also multiply that result by how many were infected yesterday. The reason is that *the rate of infection is for each infected person*. If 3 people are infected the chance of anyone else becoming infected is 3 times as high as when only 1 person is infected. So any estimate of the rate of spread of the disease requires knowledge of the infection rate and the numbers of initially infected and initially susceptible people.

The 2nd equation says:

I_{n} = I_{n-1 }+ (S_{n-1}/S) ** _{*}** (ß

**I**

_{*}_{n-1}) – (I

_{n-1 }

**γ)**

_{*}*The number of infected people today *(I_{n})* equals the number who were infected yesterday *(I_{n-1}),* PLUS the number of susceptible people who became infected today, MINUS the number of infected yesterday who recovered.*

At the beginning of an outbreak the number of people getting infected every day is probably larger than the number recovering, so the number of infected will keep rising until more people recover than get infected.

The 3rd equation says:

R_{n} = R_{n-1 }+ (I_{n-1} ** _{*}** γ)

*The number of recovered people today *(R_{n})* equals the previous number who had recovered, PLUS the number who of infected people yesterday who recovered today.*

The number of susceptible people always decreases, but the number of infected and recovered initially rise and then decline, as people get sick and then get better.

Let’s make a simple calculation with these realistic values for a flu outbreak:

initial susceptible population, S = 1000

initial infected people, I = 1

rate of infection, ß = 0.29/day

rate of recovery, γ = 0.15/day

time increment, T = 1 day

*Note that the transition rates tell you how many days it takes to double the number of people who are infected or recovered.*

*Since ß**= 0.29/day, the time to double the number of infected people is about 1/0.29 = 3.4 days. And γ** = 0.15 implies that it takes 1/0.15 = 6.7 days to recover. Infections quickly outnumber recoveries.*

To start your model you set the number of infected people for the previous day, I_{n-1} = 1, and the number of recovered R_{n-1} = 0.

Now you calculate numbers of people in each compartment at the end of day 1:

S_{n} = S_{n-1 }– ((S_{n-1}/S) ** _{*}** (ß

**I**

_{*}_{n-1}))

S_{1} = 999-((999/1000)** _{ *}** (0.29

**1)) = 998.7 are susceptible**

_{* }I_{1} = I_{n-1 }+ (S_{n-1}/S * ß * I_{n-1}) – (I_{n-1 }** _{*}** γ)

= 1 + (999/1000) *** _{ (}**0.29 * 1) – (1 * 0.15) = 1 + 0.29 – 0.15 = 1.14 people are infected

R_{1} = R_{n-1 }+ I_{n-1} ** _{*}** γ

= 0 + 1 ** _{*}** 0.15 = 0 + 0.15 = 0.15 people have recovered

For Day 2 (n = 2):

S_{n} = S_{n-1 }– ((S_{n-1}/S) ** _{*}** (ß

**I**

_{*}_{n-1}))

S_{2} = 998.7 – ((998.7/1000)** _{ *}** (0.29

**1.14)) = 998.4 are susceptible**

_{* }I_{2} = I_{n-1 }+ (S_{n-1}/S ** _{*}**ß) – (I

_{n-1 }

**γ)**

_{*} = 1.14 + (998.7/1000 ** _{* }**0.29) – (1

**0.15) = 1.14 + .29 – 0.15 = 1.30 people infected**

_{*}R_{1} = R_{n-1 }+ I_{n-1} ** _{*}** γ

= 0.15 + 1 ** _{*}** 0.15 = 0.15 + 0.15 = 0.3 people recovered.

The decrease in numbers of susceptibles, and the increase in numbers of infected and recovered, are all very small on day 1 and day 2. This shows that the disease is not spreading rapidly. But when you plot these data over the course of the outbreak you may be surprised!

**Let PHP Do it!**

These equations can be calculated by hand for each day of an outbreak, but it’s a lot easier to make a PHPl script to do it. So make one!

< ?php #http://www.pandemsim.com/data/index.php/make-your-own-sir-model/ #N number people #I0 First infected #Time increment = 1 day #Beta transmission #Gamma $N=1000; $I0=1; $Beta = 0.35; $Gamma = 0.035; $idays=200; $idays7=$idays-7; $S = array_fill(0, $N, $N-1); $I = array_fill(0, $N, 1); $R = array_fill(0, $N, 0); for ($i = 1; $i <= $idays; $i++) { $S[$i]=$S[$i-1]-(($S[$i-1]/$N)*$Beta*$I[$i-1]); $I[$i]=$I[$i-1]+($S[$i-1]/$N)*($Beta*$I[$i-1])-($I[$i-1]*$Gamma); $R[$i]=$R[$i-1]+($I[$i-1]*$Gamma); #Print #echo $i. », ».$S[$i]. », ».$I[$i]. », ».$R[$i]. », ». »

« ;

}

for ($i = 1; $i < = $idays7; $i++) { $Rt[$i]=$I[$i+7]/$I[$i]; #Print echo $i. », ».$Rt[$i]. », ». »

« ;

}