Tag: computational

The SIR Model for the Spread of Infectious Diseases

In the current Coronavirus crisis, everybody is talking about flattening “the curve”. In the news, you will often see graphs of the total number of cases or the total number of deaths over time. So you may be forgiven to think that these are the curves that everybody is trying to flatten. In fact, what epidemiologists mean by the curve is the graph of the number of actively infected people over time. This curve is important because it determines the load that is placed on the healthcare system of a country. The current number of cases determines how many hospital beds, how many ventilators, and how much healthcare personnel are needed.

Mathematics and computer simulations play an important role in estimating how the disease will spread, how many people will be affected, and how much resources are needed. They also allow predicting the effects of different measures to control the spread. For example, the current lockdown in many countries around the world is reducing the number of people that an infected individual can pass the virus on to. It is important to know how effective this measure is. One of the big questions is when it is safe to relax the isolation of people and how much it would affect the spread if individual businesses re-open.

Before continuing, I have to add a disclaimer. I am interested in mathematics but I am not an expert epidemiologist. The models I am showing you here are very simple starting points for simulating the spread of diseases. They can give you some idea on how parameters like the infection rate and recovery rate influence the overall number of infected individuals. But they should not be used to draw any quantitative conclusions.

The SIR Model

To get a basic feel for the way infections spread through a population, epidemiologists have developed simple mathematical models. Probably the first model you will hear about in this context is the SIR model. The SIR model is a version of a compartmental model. This means that the total population is divided up into separate compartments. The quantity $S$ denotes the number of susceptible individuals. These are the people that are not infected and also don’t have any immunity to the disease. $I$ is the number of infected individuals and $R$ is the number of individuals that are not infectious but also can’t get the disease. Most scientists denote the $R$ to mean removed as it includes both people who have recovered and are immune but also those that have died. Due to the current sensitivity of the subject, many people prefer to call $R$ the recovered population.

Compartmental models define rates by which individuals change from one population to another. The SIR model has two rates, the rate of infection and the rate of recovery. The absolute rate of infection is proportional to the number of infected people. On average, each infected individual will pass the infection to a number of people in a given time interval. This number is usually called $\beta$. However, if an infected individual passes the virus to a recovered person, the infection will not spread. The probability of passing the infection on is given by $S/N$ where $N$ is the total population $N=S+I+R$. Putting this together, the absolute rate of infection is

$$\frac{\beta I S}{N}.$$

The rate of recovery is slightly more simple. Each infected individual will recover with some probability $\gamma$ in a given time interval. The absolute rate of recovery is then expressed as

$$\gamma I.$$
The infection rate reduces the number of susceptible individuals $S$ and increases the number of infected individuals $I$. The recovery rate reduces the number of infected individuals $I$ and increases the number of recovered individuals $R$. The complete set of rate equations is then

$$\begin{eqnarray}
\frac{dS}{dt} &=& – \frac{\beta I S}{N}, \\
\frac{dI}{dt} &=& \frac{\beta I S}{N} – \gamma I, \\
\frac{dR}{dt} &=& \gamma I.
\end{eqnarray}$$

The ratio of the coefficients $\beta$ and $\gamma$ is known as the basic reproduction ratio.

$$R_0 = \frac{\beta}{\gamma}$$.

The $R_0$ is important because it determines whether the infection will spread exponentially or eventually die out.

I have implemented a little JavaScript app that integrates the SIR equations and shows the development of the populations over time. Feel free to play around with the sliders and explore how the parameters influence the spread.

Please enable JavaScript

I encourage you to play around with the parameters to see how the model behaves. For an infection rate of 1 and a recovery rate of 0.5, the populations stabilise when about 80% of the population has been infected and has recovered. The maximum of the infectious population, the $I$ curve, reaches about 16%. If you reduce the infection rate, the $I$ curve flattens, prolonging the time over which the disease is spreading but reducing the maximum number of infected individuals at any one time.

The SEIR Model

One of the major assumptions in the SIR model is that an infected individual can immediately spread the infection. A refinement of the model is the addition of a population, $E$, of exposed individuals. These are people that are infected but are not yet infectious. The SEIR model introduces another rate, $a$, at which exposed individuals turn infectious. The quantity $a$ can be understood as the inverse of the average incubation period. The absolute rate at which exposed individuals become infectious is

$$a E.$$

The complete set of equations of the SEIR model are then as follows.

$$\begin{eqnarray}
\frac{dS}{dt} &=& – \frac{\beta I S}{N}, \\
\frac{dE}{dt} &=& \frac{\beta I S}{N} – a E, \\
\frac{dI}{dt} &=& a E – \gamma I, \\
\frac{dR}{dt} &=& \gamma I.
\end{eqnarray}$$

The SEIR model is also implemented in the app above. Simply pick SEIR Model from the dropdown menu and start exploring.

The SEIR Model with Delay

The SEIR model above assumes that an individual, once exposed, can immediately turn infectious. The constant rate $a$ implies that the probability of changing from the exposed state to the infectious state is the same on day one of being exposed as it is on day ten. This might not be realistic because diseases typically have some incubation period. Only after some number of days after being exposed will an individual become infectious. One can model this kind of behaviour with a time delay. Let’s say that after a given incubation period $\tau$, every exposed individual will turn infectious. The absolute rate at which exposed individuals become infectious is then given by

$$\frac{\beta I(t-\tau) S(t-\tau)}{N}.$$

Here the $S(t-\tau)$ means taking the value of the susceptible individuals not at the current time, but at a time in the past with a delay of $\tau$. The complete set of equations of the SEIR model with delay are then as follows.

$$\begin{eqnarray}
\frac{dS}{dt} &=& – \frac{\beta I(t) S(t)}{N}, \\
\frac{dE}{dt} &=& \frac{\beta I(t) S(t)}{N} – \frac{\beta I(t-\tau) S(t-\tau)}{N}, \\
\frac{dI}{dt} &=& \frac{\beta I(t-\tau) S(t-\tau)}{N} – \gamma I(t), \\
\frac{dR}{dt} &=& \gamma I(t).
\end{eqnarray}$$

I have written the time dependence explicitly for all quantities on the right-hand side to make it clear how the time delay should be applied.

You can choose this model in the app above by selecting SEIR Model with Delay from the dropdown menu.

Some Conclusions

The SEIR model and the SEIR model with delay both introduce a population of exposed individuals that are not yet infectious. This draws out the spread of the disease over a longer time. It also slightly reduces the maximum of the infectious population curve $I$. Introducing a time delay doesn’t change the curves too much. But for long incubation periods, the curve of infectious individuals can have multiple maxima. So at some time, it may look like the disease has stopped spreading while in reality, a next wave is just about to start. The two versions of the SEIR model are two extremes and the truth lies somewhere in between these two.

I have to stress again that I am not an epidemiology expert and that the models presented here are very simple models. For any meaningful prediction of the spread of a real disease, much more complex models are needed. These models must include real data about the number of contacts that different parts of the population have between each other.

The code for the application above is available on
GitHub

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Computational Physics Basics: How Integers are Stored

Unsigned Integers

Computers use binary representations to store various types of data. In the context of computational physics, it is important to understand how numerical values are stored. To start, let’s take a look at non-negative integer numbers. These unsigned integers can simply be translated into their binary representation. The binary number-format is similar to the all-familiar decimal format with the main difference that there are only two values for the digits, not ten. The two values are 0 and 1. Numbers are written in the same way as decimal numbers only that the place values of each digit are now powers of 2. For example, the following 4-digit numbers show the values of the first four

Binary Counter

0 0 0 1   decimal value 20 = 1
0 0 1 0   decimal value 21 = 2
0 1 0 0   decimal value 22 = 4
1 0 0 0   decimal value 23 = 8

The binary digits are called bits and in modern computers, the bits are grouped in units of 8. Each unit of 8 bits is called a byte and can contain values between 0 and 28 − 1 = 255. Of course, 255 is not a very large number and for most applications, larger numbers are needed. Most modern computer architectures support integers with 32 bits and 64 bits. Unsigned 32-bit integers range from 0 to 232 − 1 = 4, 294, 967, 295 ≈ 4.3 × 109 and unsigned 64-bit integers range from 0 to 264 − 1 = 18, 446, 744, 073, 709, 551, 615 ≈ 1.8 × 1019. It is worthwhile noting that many GPU architectures currently don’t natively support 64-bit numbers.

The computer’s processor contains registers that can store binary numbers. Thus a 64-bit processor contains 64-bit registers and has machine instructions that perform numerical operations on those registers. As an example, consider the addition operation. In binary, two numbers are added in much the same way as using long addition in decimal. Consider the addition of two 64 bit integers 7013356221863432502 + 884350303838366524. In binary, this is written as follows.

  01100001,01010100,01110010,01010011,01001111,01110010,00010001,00110110
+ 00001100,01000101,11010111,11101010,01110101,01001011,01101011,00111100
---------------------------------------------------------------------------
  01101101,10011010,01001010,00111101,11000100,10111101,01111100,01110010

The process of adding two numbers is simple. From right to left, the digits of the two numbers are added. If the result is two or more, there will be a carry-over which is added to the next digit on the left.

You could add integers of any size using this prescription but, of course, in the computer numbers are limited by the number of bits they contain. Consider the following binary addition of (264 − 1) and 1 .

  11111111,11111111,11111111,11111111,11111111,11111111,11111111,11111111
+ 00000000,00000000,00000000,00000000,00000000,00000000,00000000,00000001
---------------------------------------------------------------------------
  00000000,00000000,00000000,00000000,00000000,00000000,00000000,00000000

If you were dealing with mathematical integers, you would expect to see an extra digit 1 on the left. The computer cannot store that bit in the register containing the result but stores the extra bit in a special carry flag. In many computer languages, this unintentional overflow will go undetected and the programmer has to take care that numerical operations do not lead to unintended results.

Signed Integers

The example above shows that adding two non-zero numbers can result in 0. This can be exploited to define negative numbers. In general, given a number a, the negative  − a is defined as the number that solves the equation
a + ( − a) = 0.
Mathematically, the N-bit integers can be seen as the group of integers modulo 2N. This means that for any number a ∈ {0, …, 2N − 1} the number  − a can be defined as
 − a = 2N − a ∈ {0, …, 2N − 1}.
By convention, all numbers whose highest value binary bit is zero are considered positive. Those numbers whose highest value bit is one are considered negative. This makes the addition and subtraction of signed integers straightforward as the processor does not need to implement different algorithms for positive or negative numbers. Signed 32-bit integers range from  − 2, 147, 483, 648 to 2, 147, 483, 647, and 64-bit integers range from  − 9, 223, 372, 036, 854, 775, 808 to 9, 223, 372, 036, 854, 775, 807.

This format of storing negative numbers is called the two’s complement format. The reason for this name becomes obvious when observing how to transform a positive number to its negative.

01100001,01010100,01110010,01010011,01001111,01110010,00010001,00110110 (7013356221863432502)
10011110,10101011,10001101,10101100,10110000,10001101,11101110,11001010 (-7013356221863432502)

To invert a number, first, invert all its bits and then add 1. This simple rule of taking the two’s complement can be easily implemented in the processor’s hardware. Because of the simplicity of this prescription, and the fact that adding a negative number follows the same algorithm as adding a positive number, two’s complement is de-facto the only format used to store negative integers on modern processors.

Exercises

  1. Show that taking the two’s complement of an N-bit number a does indeed result in the negative  − a if the addition of two numbers is defined as the addition modulo 2N.
  2. Find out how integers are represented in the programming language of your choice. Does this directly reflect the representation of the underlying architecture? I will be writing another post about this topic soon.
  3. Most processors have native commands for multiplying two integers. The result of multiplying the numbers in two N-bit registers are stored in two N-bit result registers representing the high and low bits of the result. Show that the resulting 2N bits will always be enough to store the result.
  4. Show how the multiplication of two numbers can be implemented using only the bit-shift operator and conditional addition based on the bit that has been shifted out of the register. The bit-shift operator simply shifts all bits of a register to the left or right.

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