Editing Compartmental models (epidemiology) (section)

====Exact analytical solutions to the SIR model====
In 2014, Harko and coauthors derived an exact so-called analytical solution (involving an integral that can only be calculated numerically) to the SIR model.<ref name="Harko" /> In the case  without vital dynamics setup, for <math>\mathcal{S}(u)=S(t)</math>, etc., it corresponds to the following time parametrization

:<math>\mathcal{S}(u)= S(0)u </math>
:<math>\mathcal{I}(u)= N -\mathcal{R}(u)-\mathcal{S}(u) </math>
:<math>\mathcal{R}(u)=R(0) -\rho \ln(u)</math>

for

:<math>t= \frac{N}{\beta}\int_u^1 \frac{du^*}{u^*\mathcal{I}(u^*)} , \quad \rho=\frac{\gamma N}{\beta},</math>

with initial conditions

:<math>(\mathcal{S}(1),\mathcal{I}(1),\mathcal{R}(1))=(S(0),N -R(0)-S(0),R(0)), \quad  u_T<u<1,</math>

where <math>u_T</math> satisfies <math>\mathcal{I}(u_T)=0</math>. By the transcendental equation for <math>R_{\infty}</math> above, it follows that <math>u_T=e^{-(R_{\infty}-R(0))/\rho}(=S_{\infty}/S(0)</math>, if <math>S(0) \neq 0)</math> and <math>I_{\infty}=0</math>.

An equivalent so-called analytical solution (involving an integral that can only be calculated numerically) found by Miller<ref>{{cite journal | vauthors = Miller JC | title = A note on the derivation of epidemic final sizes | journal = Bulletin of Mathematical Biology | volume = 74 | issue = 9 | pages = 2125–2141 | date = September 2012 | pmid = 22829179 | pmc = 3506030 | doi = 10.1007/s11538-012-9749-6 | quote = Section 4.1 }}</ref><ref>{{cite journal | vauthors = Miller JC | title = Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes | journal = Infectious Disease Modelling | volume = 2 | issue = 1 | pages = 35–55 | date = February 2017 | pmid = 29928728 | pmc = 5963332 | doi = 10.1016/j.idm.2016.12.003 | quote = Section 2.1.3 }}</ref> yields

:<math>
\begin{align}
S(t) & = S(0) e^{-\xi(t)} \\[8pt]
I(t) & = N-S(t)-R(t) \\[8pt]
R(t) & = R(0) + \rho \xi(t) \\[8pt]
\xi(t) & = \frac{\beta}{N}\int_0^t I(t^*) \, dt^*
\end{align}
</math>

Here <math>\xi(t)</math> can be interpreted as the expected number of transmissions an individual has received by time <math>t</math>.  The two solutions are related by <math>e^{-\xi(t)} = u</math>.

Effectively the same result can be found in the original work by Kermack and McKendrick.<ref name="Kermack–McKendrick">{{cite journal |vauthors=Kermack WO, McKendrick AG |date=1927 |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |volume=115 |issue=772 |pages=700–721 |bibcode=1927RSPSA.115..700K |doi=10.1098/rspa.1927.0118 |doi-access=free}}</ref>

These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to <math>I</math>. The equations may thus be divided through by <math>I</math>, and the time rescaled so that the differential operator on the left-hand side becomes simply <math>d/d\tau</math>, where <math>d\tau=I dt</math>, i.e. <math>\tau=\int I dt</math>. The differential equations are now all linear, and the third equation, of the form <math>dR/d\tau =</math> const., shows that <math>\tau</math> and <math>R</math> (and <math>\xi</math> above) are simply linearly related.

A highly accurate analytic approximant of the SIR model as well as exact analytic expressions for the final values <math>S_{\infty}</math>, <math>I_{\infty}</math>, and <math>R_{\infty}</math> were provided by [[Martin Kröger|Kröger]] and Schlickeiser,<ref name="KrogerSchlickeiser"/> so that there is no need to perform a numerical integration to solve the SIR model (a simplified example practice on [[COVID-19]] numerical simulation using [[Microsoft Excel]] can be found here <ref name="SIR_COVID19">{{cite journal | vauthors = Hart KD, Thompson C, Burger C, Hardwick D, Michaud AH, Bulushi A, Pridemore C, Ward C, Chen J |title = Remote Learning of COVID-19 Kinetic Analysis in a Physical Chemistry Laboratory Class | journal = ACS Omega | volume = 6 | issue= 43 | pages = 29223–29232 |year=2021 |doi = 10.1021/acsomega.1c04842|pmid = 34723043 |pmc = 8547164 }}</ref>), to obtain its parameters from existing data, or to predict the future dynamics of an epidemics modeled by the SIR model. The approximant involves the [[Lambert W function|Lambert {{mvar|W}} function]] which is part of all basic data visualization software such as Microsoft Excel, [[MATLAB]], and [[Wolfram Mathematica|Mathematica]].

While [[David George Kendall|Kendall]]<ref name="Kendall">{{cite book |title=Contributions to Biology and Problems of Health |vauthors=Kendall DG |date=1956 |publisher=University of California Press |volume=4 |pages=149–165 |chapter=Deterministic and Stochastic Epidemics in Closed Populations |doi=10.1525/9780520350717-011 |mr=0084936 |zbl=0070.15101 |chapter-url=http://projecteuclid.org/euclid.bsmsp/1200502553}}</ref> considered the so-called all-time SIR model where the initial conditions <math>S(0)</math>, <math>I(0)</math>, and <math>R(0)</math> are coupled through the above relations, Kermack and McKendrick<ref name="Kermack–McKendrick"/> proposed to study the more general semi-time case, for which <math>S(0)</math> and <math>I(0)</math> are both arbitrary. This latter version, denoted as semi-time SIR model,<ref name="KrogerSchlickeiser"/> makes predictions only for future times <math>t>0</math>. An analytic approximant and exact expressions for the final values are available for the semi-time SIR model as well.<ref name="KrogerSchlickeiser_partB"/>