## Journal of Applied Probability

- J. Appl. Probab.
- Volume 53, Number 4 (2016), 1031-1040.

### The deterministic Kermack‒McKendrick model bounds the general stochastic epidemic

Robert R. Wilkinson, Frank G. Ball, and Kieran J. Sharkey

#### Abstract

We prove that, for Poisson transmission and recovery processes, the classic susceptible→infected→recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time *t*>0, a strict lower bound on the expected number of susceptibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.

#### Article information

**Source**

J. Appl. Probab., Volume 53, Number 4 (2016), 1031-1040.

**Dates**

First available in Project Euclid: 7 December 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1481132834

**Mathematical Reviews number (MathSciNet)**

MR3581239

**Zentralblatt MATH identifier**

1353.92105

**Subjects**

Primary: 92D30: Epidemiology

Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J22: Computational methods in Markov chains [See also 65C40] 05C80: Random graphs [See also 60B20]

**Keywords**

General stochastic epidemic deterministic general epidemic SIR Kermack‒McKendrick message passing bound

#### Citation

Wilkinson, Robert R.; Ball, Frank G.; Sharkey, Kieran J. The deterministic Kermack‒McKendrick model bounds the general stochastic epidemic. J. Appl. Probab. 53 (2016), no. 4, 1031--1040. https://projecteuclid.org/euclid.jap/1481132834