# Pst!, SIR

The *SIR* model for epidemics is one of the simplest models for
contagion processes. It is based on the following three assumptions,

- There are a fixed number \(N\) of individuals.

At each time \(t\), the \(N\) individuals are clasified into three mutually exclusive groups: \((S,I,R)\), thus, \(N=S+I+R\) is constant for all \(t\) but,

\(S=S(t)\) are the number of individuals that at time \(t\) fall into
the category of *Susceptible* individuals. These
are the ones that are able to become infected.

\(I=I(t)\) are the *Infective* individuals at time \(t\), that are
able to infect others.

\(R=R(t)\) are the *Removed* ones. These account for the individuals
that die due to the virus or that become inmune after catching the virus.

The basic *SIR* model is a system of three difference equations for
the changes \(\delta S, \delta I, \delta R\). Where for \(U=U(t)\) we
define \(\delta U = U(t+1)-U(t)\) so that \(U(t+1) = U(t) + \delta U\)
provides the rule to evolve the process from time \(t\) to time \(t+1\).
The first equation just says that \(N\) is constant, i.e. \(\delta N =0\),
and since \(N=S+I+R\), and the \(\delta\) operation is linear we get,

\[ \delta S + \delta I + \delta R = 0\]

- The second equation specifies the basic nonlinearity of the
*SIR*model.*Susceptible*individuals can only move into the*Infective*group by catching the virus from an*Infective*individual. There are \(S\cdot I\) possible matches and a proportion \(a > 0\) of them will become infected. Thus,

\[ \delta S = -a\cdot (S\cdot I) \]

- The third, and final equation of the
*SIR*model just says that all the changes in*Removed*come from*Infectives*either by bocoming inmune after catching the virus, or by dying due to the infection. Therefore, the number of*Removed*indivuals can only increase by another proportion \(b > 0\) of the*Infectives*at that time,

\[\delta R = b\cdot I\]

Using the first equation, \(\delta I = -\delta S - \delta R\).
Substituting equations (2) and (3) of the *SIR* model into the
right hand side we get,

\[ \delta I = \left(\frac{a}{b}S - 1\right) b \cdot I \]

Notice that since \(b>0\), the *Infectives* increase (\(\delta I > 0\)),
decrease (\(\delta I < 0\)) or stay put (\(\delta I = 0\)),
depending on \(R_{0} = a S/b\) being grater than \(1\), or less than \(1\),
or equal to \(1\).

**Definition 1**\(R_{0}\) is called the

*reproductive function*associated to a given

*SIR*epidemic. \(R_{0} = a S/b = R_{0}(t)\) is a function of \(t\) since \(S\) and possibly \(a\) and \(b\) also are themselves functions of \(t\).

Now if we assume that these functions of \(t\) are smooth, we can integrate the last equation. Make \(\delta t\) (which is often \(1\) day) infinitesimally small (one hour, minute,second \(\ldots\)) so that when \(\delta t \rightarrow 0\) the last equation becomes,

\[ \int \, \frac{dI}{I} = \int \left(R_{0}-1\right) b\, d\tau \] where the integration goes from \(\tau=0\) to \(\tau=t\). Hence,

\[\log I(t) - \log I(0) = \int \left(R_{0}-1\right) b\, d\tau\] and therefore,

\[ I(t) = \exp\left\{ \int_{\tau=0}^{t}
\left(R_{0}-1\right) b\, d\tau \right\}\, I(0)\]
and assume the epidemic starts with only one infective so \(I(0)=1\).
We deduce that the number of *Infectives* increases exponentially,
decreases exponentially or stays constant, depending on the sign
of \(R_{0}(t)-1\). In the event that \(R_{0}\) and \(b\) are independent
of \(t\) or for \(t\approx 0\) we get,

\[ I(t) = \exp\left\{ (R_{0}-1) b t\right\}. \] In general, we have,

\[ R_{0}(t) = 1 + \frac{d I}{d R}\] which provides an easy way to estimate \(R_{0}(t)\) by replacing the infinitessimals with the daily observed increments,

\[ R_{0}(t) = 1 + \frac{\delta I}{\delta R}(t) \]
This last equation also shows that \(R_{0}\) is *independent*
of the size \(N\) of the population. It is obtained by
adding \(1\) to the ratio of increments of *Infective*
to *Removed*. No \(N\). Just increments of \(I\) and \(R\).

More precisely we would like to be able to prove the following,

**Proposition 1**\(R_{0}(t)\) is the average number of secondary infections at time \(t\).

Secondary infections are the infections produced by an infected individual. We can check the plausibility of the proposition by noticing that \(R_{0}\)s combine like averages combine. But, we do not have a probabilistic model yet, so what exactly does it mean to get the average at time \(t\)?

If we split \(N=N_{1}+N_{2}\) into two subpopulations with corresponding \(S_{1},I_{1},R_{1}\) and \(S_{2},I_{2},R_{2}\) then,

\[ R_{0} = \frac{N_{1}}{N_{1}+N_{2}}\, R_{0,1} +
\frac{N_{2}}{N_{1}+N_{2}}\, R_{0,2} \]
provided \(R_{1}/R_{2} = N_{1}/N_{2}\), i.e. the splitting is not
done maliciously, for example collecting all the removed individuals into the
second group and none in the first group. In other words, if the
splitting produces two groups operating the same *SIR* epidemic we will
expect \(R_{i}(t) = c(t) N_{i}\) with *the same* \(c(t)\) for \(i=1,2\). In this case,
the above equation becomes,

\[ R_{0} = 1 + \frac{d I}{d R} = 1 + \frac{d R_{1}}{d R}\, \frac{d I_{1}}{d R_{1}} + \frac{d R_{2}}{d R}\, \frac{d I_{2}}{d R_{2}} = \frac{d R_{1}}{d R}\, R_{0,1} + \frac{d R_{2}}{d R}\, R_{0,2}\] where we have replaced \(1=dR_{1}/dR + dR_{2}/dR\). More over, for \(i=1,2\)

\[ \frac{d R_{i}}{dR} = \frac{N_{i} dc}{N_{1} dc + N_{2} dc} = \frac{N_{i}}{N_{1}+N_{2}}. \] Now \(R_{0}(t)\) is an average over the population that makes the infectives explode or collapse exponentially depending on the sign of \(R_{0}(t)-1\). This cannot be anything other than what is claimed. The average number of secondary infections. I am open to suggestions on how to tide this up. This will only make complete sense after assuming a probabilistic mechanism for the evolution of the pandemic process. For suppose we allow the possibility of \(N=\infty\) then the mean could become infinity itself and thus, the mean of the increments will become undefined.

The reproductive function \(R_{0}(t)\) is an invaluable overall summary of the evolution of the epidemic, that because is independent of the size of the population, can be used for making comparisons across scales, from small villages to whole continents. For approximating \(R_{0}(t)\) we need only daily estimates of the ratio: \(\delta I/\delta R\).

## Exploratory estimation of \(R_{0}(t)\) for the Covid19 epidemic

`source("https://omega0.xyz/continentalR0.R")`

```
-################### plots:
-#
-# plot 2 countries. Ver 1:
-# plot2R01('Chile','United States')
-# plot_country <- function(df,country,lead=5,dof=15)
-# R0 country:
-# plotR0 <- function(pais) plot_country(df,pais)
-#
-# R0 continent:
-# plotr0c <- function(df_continent,lead=5,dof=15)
-# the 6 continents:
-# plotR06c <- function(df=dfc)
-# Continents with variable lead and dof:
-# plot_R0sC <- function(lead=5,dof=15)
-# plot_mAUCc <- function(lead=5,dof=15) {
-# plot_median_AUCc(lead=5,dof=10)
-# mean AUC of continents:
-# plot_dfca <- function()
-# World:
-# plot_world <- function(lead=5,dof=15)
-# Population percents:
-# plot_pops <- function(continent='Africa')
```

### Suggestion to the reader:

Copy and paste the line of code above in an R session and follow along this document in a window next to your browser.

The list of some of the plots available includes the \(R_{0}(t)\) for the entire world,

`plot_world()`

using current data. The estimation is done as follows:

The

*Infectives*at day \(t\), \(I(t)\) is estimated as the*new cases*discovered day \(t\) plus all the new cases discovered a*lead*number of days into the future.The

*Removed*, \(R(t)\), are simply all the observed*new cases*upto day \(t\) plus all the observed*new deaths*upto day \(t\).\[ R_{0}(t) = 1 + \frac{\delta I}{\delta R} (t)\]

\[ 1 + \cdot /0 = 1\]

The smoothing spline with

*dof*degrees of freedom of the raw estimator is the curve shown above.

For example a drastic smoother is just the overall trend (line) obtained with,

`plot_world(dof=2)`

and with very little smoothing,

`plot_world(dof=80)`

The current picture of the six continents is,

`plot_R0sC()`

or just the current trends,

`plot_R0sC(dof=2)`

#### Compare two countries

`plot2R01('Spain','Italy')`

OMG! look at Italy.

Overall sizes for the \(R_{0}\) of the six continents,

`plot_median_AUCc()`

##### Names of countries

`countries`

```
## [1] "Afghanistan" "Albania"
## [3] "Algeria" "Andorra"
## [5] "Angola" "Anguilla"
## [7] "Antigua and Barbuda" "Argentina"
## [9] "Armenia" "Aruba"
## [11] "Australia" "Austria"
## [13] "Azerbaijan" "Bahamas"
## [15] "Bahrain" "Bangladesh"
## [17] "Barbados" "Belarus"
## [19] "Belgium" "Belize"
## [21] "Benin" "Bermuda"
## [23] "Bhutan" "Bolivia"
## [25] "Bonaire Sint Eustatius and Saba" "Bosnia and Herzegovina"
## [27] "Botswana" "Brazil"
## [29] "British Virgin Islands" "Brunei"
## [31] "Bulgaria" "Burkina Faso"
## [33] "Burundi" "Cambodia"
## [35] "Cameroon" "Canada"
## [37] "Cape Verde" "Cayman Islands"
## [39] "Central African Republic" "Chad"
## [41] "Chile" "China"
## [43] "Colombia" "Comoros"
## [45] "Congo" "Costa Rica"
## [47] "Cote d'Ivoire" "Croatia"
## [49] "Cuba" "Curacao"
## [51] "Cyprus" "Czech Republic"
## [53] "Democratic Republic of Congo" "Denmark"
## [55] "Djibouti" "Dominica"
## [57] "Dominican Republic" "Ecuador"
## [59] "Egypt" "El Salvador"
## [61] "Equatorial Guinea" "Eritrea"
## [63] "Estonia" "Ethiopia"
## [65] "Faeroe Islands" "Falkland Islands"
## [67] "Fiji" "Finland"
## [69] "France" "French Polynesia"
## [71] "Gabon" "Gambia"
## [73] "Georgia" "Germany"
## [75] "Ghana" "Gibraltar"
## [77] "Greece" "Greenland"
## [79] "Grenada" "Guam"
## [81] "Guatemala" "Guernsey"
## [83] "Guinea" "Guinea-Bissau"
## [85] "Guyana" "Haiti"
## [87] "Honduras" "Hong Kong"
## [89] "Hungary" "Iceland"
## [91] "India" "Indonesia"
## [93] "Iran" "Iraq"
## [95] "Ireland" "Isle of Man"
## [97] "Israel" "Italy"
## [99] "Jamaica" "Japan"
## [101] "Jersey" "Jordan"
## [103] "Kazakhstan" "Kenya"
## [105] "Kosovo" "Kuwait"
## [107] "Kyrgyzstan" "Laos"
## [109] "Latvia" "Lebanon"
## [111] "Lesotho" "Liberia"
## [113] "Libya" "Liechtenstein"
## [115] "Lithuania" "Luxembourg"
## [117] "Macedonia" "Madagascar"
## [119] "Malawi" "Malaysia"
## [121] "Maldives" "Mali"
## [123] "Malta" "Mauritania"
## [125] "Mauritius" "Mexico"
## [127] "Moldova" "Monaco"
## [129] "Mongolia" "Montenegro"
## [131] "Montserrat" "Morocco"
## [133] "Mozambique" "Myanmar"
## [135] "Namibia" "Nepal"
## [137] "Netherlands" "New Caledonia"
## [139] "New Zealand" "Nicaragua"
## [141] "Niger" "Nigeria"
## [143] "Northern Mariana Islands" "Norway"
## [145] "Oman" "Pakistan"
## [147] "Palestine" "Panama"
## [149] "Papua New Guinea" "Paraguay"
## [151] "Peru" "Philippines"
## [153] "Poland" "Portugal"
## [155] "Puerto Rico" "Qatar"
## [157] "Romania" "Russia"
## [159] "Rwanda" "Saint Kitts and Nevis"
## [161] "Saint Lucia" "Saint Vincent and the Grenadines"
## [163] "San Marino" "Sao Tome and Principe"
## [165] "Saudi Arabia" "Senegal"
## [167] "Serbia" "Seychelles"
## [169] "Sierra Leone" "Singapore"
## [171] "Sint Maarten (Dutch part)" "Slovakia"
## [173] "Slovenia" "Somalia"
## [175] "South Africa" "South Korea"
## [177] "South Sudan" "Spain"
## [179] "Sri Lanka" "Sudan"
## [181] "Suriname" "Swaziland"
## [183] "Sweden" "Switzerland"
## [185] "Syria" "Taiwan"
## [187] "Tajikistan" "Tanzania"
## [189] "Thailand" "Timor"
## [191] "Togo" "Trinidad and Tobago"
## [193] "Tunisia" "Turkey"
## [195] "Turks and Caicos Islands" "Uganda"
## [197] "Ukraine" "United Arab Emirates"
## [199] "United Kingdom" "United States"
## [201] "United States Virgin Islands" "Uruguay"
## [203] "Uzbekistan" "Vatican"
## [205] "Venezuela" "Vietnam"
## [207] "Western Sahara" "Yemen"
## [209] "Zambia" "Zimbabwe"
## [211] "World" "International"
```