## Simple Math (Arithmetic) for Modelling Infection Growth (and decline) in an Epidemic

### using only simple addition and multiplication

#### including some examples at different infection rates

(2020 Mar blog post)

SECTIONS BELOW:

DATA TABLES BELOW:

! Note ! Semi-log plot of daily new cases of
COVID-19 in top five countries
and in the world --- from the
Wikipedia Covid-19 pandemic page
(World Health Organization data) MATH FOR EXPONENTIAL GROWTH The math for exponential growth is based on a simple equation that says that the 'rate of change' of the 'population count' of an 'item' is changing in proportion to the current number of that item (examples: humans in an epidemic, atoms in a chain reaction). In compact math form: dx/dt = R * x where 'x' is a number representing the current count of the population and 'R' is a 'proportionality constant' that affects the rate of growth (or rate of decline if 'R' is negative). The symbol 'dt' represents a time step --- and 'dx' represents the change in the population count over one instance of that time step. The symbol 'dx/dt' represents a ratio --- the change of the population count per step in time. It can be thought of as the 'velocity or speed' of change of the population count. Let us consider the case of R = 2.0. It may help to think of dt as one unit of time. Then, for dt = 1, the equation above becomes dx = R * x. This equation says that 'dx' (the change in the population) is equal to 'R * x' --- over one time step. For R = 2.0, 'the population change' (dx) in one time step is double the current population count (x). We will be letting 'x' be the count of a number of 'infected' people in a population in a region, rather than a count of the entire population. (The 'total infected population count', x, will actually be tripling over each time step for R=2.0, as we will see below. For R=2.0, we will be modelling the fact that the 'new cases' --- that is, the 'CHANGE in the population of infecteds' --- over the next time step is given by doubling the current 'total infected population'. So, with R=2.0, we are modelling the case of, on average, each infected person causing the creation of two NEW infected people, over the next time step.) For the case of a virus epidemic, a suitable time step would be in 'weeks' rather than 'days' or 'hours' or 'minutes' --- because the data for determining the rate constant (R) is typically fluctuating quite a bit from day to day. Also, by using 'weeks' rather than 'days', we can 'minify' the number of computations as well as the size of the resulting data tables. In fact, the factor 'R' is typically not constant, but can vary over time as situations such as 'social distancing' and 'migration of infectives over boundaries' affect that rate. However, we can get quite useful 'ballpark' predictions based on an 'average' value of the factor 'R'. METHOD OF NUMERICAL COMPUTATION The 'rate equation' shown above is one of the simplest forms of what is known as a 'differential equation'. We think of the unknown 'x' as a 'function of time' --- typically denoted 'x(t)'. And we want to generate values for 'x' at various times 't'. To use that 'rate equation' to make numerical predictions, we actually use it in a different form: dx = R * x(t) * dt This equation says that the 'change in x' (near a given time 't') is the product of R and x(t) and dt. For predictions of growth/change in epidemics, it is typical to use dt = 1 --- such as one week (or one day or one month). So, for computational purposes, we use the simple equation: dx = R * x(t) where 'R' must be based on the same time-units as 'dt'. The equation above gives a number 'dx' representing a change in 'x' near a time 't'. However, that does not give us 'x' at a next time step. For that, we need an additional very simple equation: x(t+dt) = x(t) + dx This equation simply says that the value of 'x' at a 'next time step' is given by the value of 'x' at the 'previous time step' PLUS the change in 'x' that we got from the rate equation: dx = R * x(t) STARTING THE COMPUTATION OK. So now we have the two simple equations that we will use to generate x(t) at various times --- t, t + dt, t + 2*dt, t + 3*dt, ... The two simple equations are dx = R * x(t) x(t+dt) = x(t) + dx But now we need a bit of data to start the computation. This bit of data is called an 'initial value' or 'initial condition'. In general, we can think of wanting to generate a table of values of 'x' at times t0, t1, t2, t3, ... And, in our 'constant-time-step' case: t1 = t0 + dt, t2 = t1 + dt, t3 = t2 + dt, ... where dt = 1 (week, say). Then, to start off our computation, we need a value of 'x' at initial time 't0' --- denoted x(t0). Then we simply start computing, using the pair of equations above, over and over: dx = R * x(t0) x(t1) = x(t0) + dx   dx = R * x(t1) x(t2) = x(t1) + dx   dx = R * x(t2) x(t3) = x(t2) + dx   and so on. For simplicity, we will let t0 = 0. Then as we successively add dt = 1 to t0, we get t1 = 1, t2 = 2, t3 = 3, ... Then, with this 'one unit time step', the pairs of equations above become: dx = R * x(0) x(1) = x(0) + dx   dx = R * x(1) x(2) = x(1) + dx   dx = R * x(2) x(3) = x(2) + dx   and so on.

 A PREDICTION BASED ON R=2.0 The following table is one that I generated based on the fact that, in mid-March 2020, there were said to be about 5,000 reported cases of infections, in the United States, from the 2019-coronavirus (COVID-19). From some of the limited data at that time, it looked like the number of infections were (at the very least) doubling every week. So I decided to see what the initial growth rate of infections would look like for R = 2.0 in our 'computational equations' above. In this case, our 'x(t)' will denote the number of total reported COVID-19 infections at time 't' --- in the United States. Note that at week-zero, we start with the value x(0) = 5K. We double that to get dx = R * x(0) = 2.0 * 5K = 10K. Then, to get x(1), we use x(1) = x(0) + dx = 5K + 10K. And we continue that pattern. To make these computations look simpler, we could combine our 2 'computational equations' into one: x(i+1) = x(i) + dx = x(i) + R * x(i) So x(i+1) = x(i) + 2.0 * x(i) Note that this computation can be simplified from one multiplication and one addtion to a single multiplication: x(i+1) = (1 + R) * x(i) So x(i+1) = 3.0 * x(i) So each entry in the last column of the table is simply 3.0 times the previous entry in that column.

``````
---------------------------------------------------------------
Virus Infection Simulation
(Rate R = 2.0 per week ; i.e. doubling 'x' gives 'dx')

(dx=R*x)         ( x(i+1)=x(i)+R*x(i) )
Week   Infections       Total Infections
Number  (2xPrev.week)  (Prev.week + This week's increase)
------  ----------   -----------------------------------
0        0            5K                      mid-March
1       10K           5K +     10K = 15K
2       30K          15K +     30K = 45K
3       90K          45K +     90K = 135K
4      270K         135K +    270K = 405K     mid-April
5      810K         405K +    810K = 1215K
6     2430K        1215K +   2430K = 3645K
7     7290K        3645K +   7290K = 10935K
8     21870K      10935K +  21870K = 32805K   mid-May
9     65610K      32805K +  65610K = 98415K
10    196830K      98415K + 196830K = 295245K
nearly 330000K = U.S. population
```
```

The lower graph is the NEW cases,
i.e. new infections for each week.
The upper graph is the CUMULATIVE cases,
i.e. the cumulative infections.

 Note that at 8 weeks (two months after mid-March = mid-May 2020), this table predicts that there may be on the order of 33 million reported infections in the United States --- out of a population of about 330 million. So, at 8 weeks (mid-May 2020), about 33/330 or about 10 percent of the population of the U.S. may have experienced infection. Note that this means that about 90% of the population may still be susceptible to infection. And, stepping back a month, this table predicts that there may be on the order of 405 thousand reported infections in the United States in mid-April. So, at 4 weeks (mid-April 2020), only 405,000/330,000,000 = 0.0012 --- or ONLY about one-tenth of one percent of the population of the U.S. may have experienced infection --- EVEN THOUGH 405,000 infections sounds like a LOT OF INFECTIONS. This means that more than 99% of the population may still be susceptible to infection. In sections below, a modification of our 'computational equations' will be presented to take into account that this exponential growth cannot go on forever. There are a limited amount of 'susceptibles' in the population as more and more of the population becomes infected. But, before we take on that 'enhancement' of our predictive equations, let us consider the 'initial' form of the 'curve of deaths'. DEATH PREDICTIONS Some of the initial data from the United States indicated that the number of deaths per number of infected was about one percent. This 'death-percentage' may be rather optimistic. In some areas of the U.S. and in some countries, it looks like the death-percentage may be more like 3 or 6 percent. (In some nursing homes, the 'death-percentage' is on the order of 50 percent.) Using that fact (rough estimate), we can generate the following table to provide a 'curve of deaths'.

``````
-----------------------------------------------------------
Deaths resulting from the virus with Infection Rate
R = 2.0 per week (i.e. doubling of 'x' gives 'dx')

End of  ( x(i+1)=(1+R)*x(i) )
Week     Cumulative Total    Cumulative Total Deaths
Number      Infections        (1% of Cum. Infections)
------   -----------------    -----------------------
0         5K                 50         mid-March
1         15K                150
2         45K                450
3         135K               1350
4         405K               4050       mid-April
5         1215K              12,150
6         3645K              36,450
7         10935K             109,350
8         32805K             328,050    mid-May
9         98415K             984,150
10         295245K            2,952,450

```
```

The lower graph (near the x-axis) is the
1 percent (1/100th) of infections result in death.
The upper graph is the CUMULATIVE CASES,
i.e. the cumulative infections.

The cumulative deaths curve looks very low
relative to the cumulative-infections curve
--- but deaths will trend toward 2 million total,
not an insignificant figure for those 2 million
people --- and their relatives and friends.

The cumulative infections curve gives an idea
of how badly hospitals could be overwhelmed.

NOTE:
The wearing of masks and social distancing would
result in a much lower infection rate --- thus
'flattening' the cumulative infections curve ---
as it has been flattened in countries like South Korea
and Japan, countries that learned some lessons
from past SARS outbreaks.

 This table suggests that the number of deaths in the United States from COVID-19 would be on the order of 4,000 by mid-April 2020. And, by mid-May 2020, the number of deaths could be on the order of 330,000. A PREDICTION BASED ON R=4.0 Soon after I generated the table above (for doubling of infections every week), I noticed --- in the data (bar graphs) for cumulative infections in the United States (at the U.S. coronavirus page at Wikipedia) --- that the infections were doubling about every 3 to 4 days --- not every 7 days. So I decided to generate a table for that situation --- of a doubling in infections every 3.5 days. That situation implies that we should, perhaps, use R = 4.0 in the 'computational equations' above. In case you ask 'why 4.0?', here is why: Say you have 100 infected people at the start of the week. Then 3.5 days later, you will have 2 * 100 = 200 infected people. And then 3.5 days later, you will have 2 * 200 = 400 infected people. So you started at the beginning of the week with 100 infected people --- and you end up at the end of the week with 400 infected people. Hence, every week, the number of infected goes up a factor of 4.0. (Actually, we should determine R by noting that dx/dt = R * x can be rearranged to R = dx / x     when dt = 1. So we should evaluate R as the 'change in x over x'. In this case, it would be R = dx / x = (x(i+1) - x(i)) / x(i) = (400 - 100) / 100 = 300 / 100 So R = 3.0 at that one week. But let us 'go big' and use R = 4.0.) In generating this table, we will reduce the number of operations necessary by noting what we observed above: Namely, the two 'computational equations' dx = R * x(t) x(t+dt) = x(t) + dx which involve a multiplication and an addition, can be simplified to a single equation x(i+1) = x(i) + R * x(i) = (1 + R) * x(i) For R = 4.0, we get the equation x(i+1) = 5.0 * x(i) which involves a single multiplication. And that gives us the following table.

``````
-------------------------------------------------------------------
Virus Infection Simulation
(Rate R = 4.0 per week ; i.e. quadrupling 'x' gives 'dx')

( x(i+1)=(1+R)*x(i) )
End of          Cumulative                  Cumulative
Week        Total Infections              Total Deaths
Number    (5.0 times the Prev.week)     (1% of Cum. Infections)
------   ----------------------------   -----------------------
0                 5K                   50         mid-March
1                25K                   250
2               125K                   1,250
3               625K                   6,250
4             3,125K                   31,250     mid-April
5            15,625K                   156,250
6            78,125K                   781,250
7           390,625K  <--- past the
8         1,953,125K      population of U.S.
9         9,765,625K
10        48,828,125K

```
```

The lower graph (near the x-axis) is the
1 percent (1/100th) of infections result in death.
The upper graph is the CUMULATIVE CASES,
i.e. the cumulative infections.

NOTE:
This R=4.0 infection rate is quite high, perhaps
even higher than the initial rate in New York City.
With wearing of masks and social distancing,
a cumulative curve this steep can be avoided.

``````
------------------------------------------------------------------
Virus Infection Simulation
(Rate R = 5.0 per week ; i.e. quintupling 'x' gives 'dx')

( x(i+1)=(1+R)*x(i) )
End of           Cumulative                  Cumulative
Week         Total Infections              Total Deaths
Number      (6.0 times the Prev.week)     (1% of Cum. Infections)
------   ----------------------------   -----------------------
0                 5K                   50         mid-March
1                30K                   300
2               180K                   1,800
3             1,080K                   10,800
4             6,480K                   64,800     mid-April
5            38,880K                   388,800
6           233,280K                   2,332,800
7         1,399,680K  <--- far
past population
of U.S. which is
```
```

The lower graph (near the x-axis) is the
1 percent (1/100th) of infections result in death.
The upper graph is the CUMULATIVE CASES,
i.e. the cumulative infections.

NOTE:
This R=5.0 infection rate is quite high,
perhaps higher than will be experienced
anywhere in the world with Covid-19.
But a future mutation of this virus --- or
an even more virulent virus like MERS ---
could have an infection rate this high.

 For further information : In case I do not return to update this page, here are a few keyword WEB SEARCHES that you can use to provide updates. 'Simple Math (Arithmetic) for
Modelling Epidemic Infection Growth'

--- including several tables of example data
for various infection rates.

OR, ...

##### < Go to Top of Page, above. >

Page history:

Page was created 2020 Mar 31.

Page was changed 2020 Apr 01.
(Added tables for R=4.0 and R=5.0.)

Page was changed 2020 Apr 02.

Page was changed 2020 Apr 03.
(Added death estimates to R=4.0 and 5.0 tables. Added a link to a separate GROWTH-and-DECLINE web page.)

Page was changed 2020 Apr 04.
(Added a few miscellaneous text and math items, for clarification.)

Page was changed 2020 Apr 05.
(Made a few text changes in table headings and the 'MATH FOR EXPONENTIAL GROWTH' section, for clarification.)

Page was changed 2020 Apr 06.
(Made a few more text changes, for clarification.)

Page was changed 2020 Apr 07.

Page was changed 2020 Apr 13.
(Added 2 cartoon images, to introduce some pictorial levity --- along with messaging.)

Page was changed 2020 Apr 21.