The simple stochastic epidemic for small populations with one or more initial infectives

Markovian dynamics on complex reaction networks

“…Recently, several articles have appeared in the literature introducing adaptive networks that take into account the interplay between network topology and dynamics (Ehrhardt et al, 2006;Gross and Blasius, 2008;Gross et al, 2006;Holme and Newman, 2006;Ren et al, 2010;Yuan and Zhou, 2011;Zhang et al, 2010b;Zhou and Kurths, 2006). These preliminary works clearly demonstrate that a number of intriguing properties emerge, not previously observed in nonadaptive networks: formation of complex topologies, spontaneous emergence of modular organization, more complex dynamics than the ones observed in nonadaptive models, and self-organization towards a highly robust critical behavior characterized by power-law distributions.…”

“…In parallel to the previous developments, substantial effort has been independently focused on modeling and analyzing stochastic behavior in problems of epidemiology (Bailey, 1950(Bailey, , 1957(Bailey, , 1963Bartlett, 1949Bartlett, , 1957Bartlett, , 1960Black and McKane, 2010;Chen and Bokka, 2005;Haskey, 1954;Hill and Severo, 1969;Jenkinson and Goutsias, 2012;van Kampen, 1973van Kampen, , 1976Keeling andRoss, 2008, 2009;Youssef and Scoglio, 2011), ecology (Bartlett, 1960;Black and McKane, 2012;Datta et al, 2010;Dilão and Domingos, 2000;Li et al, 2011), sociology (Haken, 1975;Weidlich, 1972Weidlich, , 1991Weidlich, , 2006Weidlich and Haag, 1983), and theoretical neuroscience (Benayoun et al, 2010;Bressloff, 2009Bressloff, , 2010Buice and Cowan, 2007;Buice et al, 2010;Cowan, 1991;El Boustani and Destexhe, 2009;Haken, 1975;Ohira and Cowan, 1993;Soula and Chow, 2007). The main premise underlying this effort is the realization that environmental, demographic, behavioral, and biological factors fluctuate randomly and that the resulting stochasticity can cause dramatic deviation from what is predicted by deterministic approaches.…”

On Estimating the Infection Rate of the Simple Stochastic Epidemic

“…On the other hand, (10) can be derived from (5) in an obvious way. Hill & Severo (1969) suggest an approximation to the maximum likelihood estimate of the infection rate of the simple stochastic epidemic model. In this note, a refinement to this approximation is presented.…”

Approximation of Estimates in the Susceptible-Infectious-Removed Epidemic Model

“…For the discrete sampling scheme (not necessarily using the equidistant constraint), several studies have investigated the approximation of maximum likelihood estimates for the infection rate of the simple stochastic epidemic model (Hill and Severo, 1969;Kryscio, 1972;Choi and Severo, 1988). Oh et al (1991) presented approximations of the maximum likelihood estimate for the birth rate in a class of birth processes.…”