Matrix-Geometric Methods for the General Stochastic Epidemic

On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections

“…In the stochastic setting, three important quantities of the SIR-model include the final size distribution, the expected duration of an epidemic, and the basic reproduction number R 0 , which are analyzed by using a variety of tools in applied probability; see, for example, the articles by Allen [3], Allen and Burgin [4], Artalejo et al [9], and Artalejo and López-Herrero [12], among others. Gani and Purdue [27] outline a matrix-geometric method for determining the total size distribution in a recursive manner, and El Maroufy et al [25] explore how this method can be used to study SIR-models with a generalized mechanism of infection. A recursive algorithm used to compute the distribution of the epidemic size, under the assumption of a large finite initial population of susceptibles, is described by Gordillo et al [30], who also derive a diffusion limit in the near-critical case.…”

“…The rates λ i,s and µ i can be specified in infinitely many ways. For example, the choice λ i,s = βis and µ i = ηi yields the general stochastic epidemic (Bailey [16,Chapter 6], Gani and Purdue [27]), and Models 1 and 2 analyzed by Neuts and Li [40] are related to the respective infection rates λ i,s = βi α s and λ i,s = βi min{s, n}, where the parameter α ∈ (0, 1) quantifies the degree of interaction between infectives and susceptibles, and the value specifies the fraction of susceptible population that is exposed to each infective; the transmission of myxomatosis among rabbits is analyzed by Saunders [43] by selecting the infection rates λ i,s = (i + s) −1/2 is.…”

“…Numerical experiments and discussion. In our numerical examples, we focus on the general stochastic epidemic (Gani and Purdue [27]) at an invasion time, and we assume rates λ i,s = βis and µ i = ηi with β = (1 + n) −1 β , β > 0, η = 1.0 and initial size 1+n ∈ {10, 50, 100, 150} of infectives and susceptibles. The aim is to study the dynamics of Markov-modulated versions of the basic SIR-model when the exponential regime on the underlying processes -governing a new infection and the removal of an infective -is replaced by other more general distributional assumptions.…”

Modelling Epidemic Diseases

“…I was lucky enough to solve this equation using Laplace transforms and some matrix algebra in Gani (1967) at the 5th Berkeley Symposium for Mathematical Statistics and Probability. Previously, Whittle (1955) had obtained the stochastic equivalent of the Kermack–McKendrick Threshhold Theorem for the general stochastic epidemic in continuous time, thus completing the theory of this process.…”

Comment on the Paper by Gani

“…In his very important 1965 Berkeley Symposium paper (Gani 1967), Joe solved a partial differential equation and expressed the solution in terms of matrices. To illustrate the solution, he considered a 2‐person epidemic.…”