## 马尔科夫疾病传播

The modeling framework presented in the previous sections is mostly based on the Poisson approximation [1] for both the transmission and recovery processes. The Poisson approximation assumes that the probabilities per unit time of transmitting the disease through a given edge, or recovering for a given infected node, are constant, and equal to $\beta$ and $\mu$, respectively. Equivalently, the total time $\tau_i$ that a given node $i$ remains infected is a random variable with an exponential distribution $P_i(\tau_i)=\mu e^{-\tau_i \mu}$, and that the time $\tau_a$ for an infection to propagate from an infected to a susceptible node along a given edge (the interevent time) is also exponentially distributed $P_a(\tau_a)=\mu e^{-\tau_a \mu}$.

From a practical point of view, the Poisson assumption leads to an increased mathematical tractability. Indeed, since the rates of transmission and recovery are constant, they do not depend on the previous history of the individual, and thus lead to memoryless, Markovian processes [2][3][4][5].

## 非马尔科夫疾病传播

While the Poisson approximation may be justified when only the average rates are known [6], it is at odds with empirical evidence for the time duration of the infective period in most diseases [7], whose distribution usually features a peak centered on the average value but exhibits strongly nonexponential tails. Furthermore, the interest in nonexponential transmission processes has also been fueled by the recent evidence on the patterns of social and communication contacts between individuals, which have been observed to be ruled by heavy-tailed distributions of interevent times.

The framework of non-Poissonian infection and recovery processes can be set up as follows, for either the SIS or SIR model [8]: Infected individuals remain infective for a period of time $\tau_i$, after which they recover, that follows the (nonexponential)$P_i(\tau_i)$ distribution. For simplicity, it is assumed that this distribution is the same for all nodes. Infection events take place along active links, connecting an infected to a susceptible node. Active links transmit the disease at times following the interevent distribution $P_a(\tau_a)$, i.e., a susceptible individual connected to an infected node becomes infected at a time $\tau_a$, measured from the instant the link became active. If a susceptible node is connected to more than one infected node, it becomes infected at the time of the first active link transmitting the disease. The complexity of this non-Markovian process is now evident: the infection of a node depends not only on the number of neighbors, but also on the time at which each connection became active.

Numerical results on non-Poissonian epidemics in networks are relatively scarce. Simple event-driven approaches rely on a time ordered sequence of events (tickets) that represent actions to be taken (recovery or infection) at given fixed times, which are computed from the interevent distributions $P_i(\tau_i)$ and $P_a(\tau_a)$. These approaches are quite demanding, so only small system sizes can be considered. For example, Van Mieghem and van de Bovenkamp (2013)[9] reported results for the SIS model with Poissonian recovery, with rate $\mu$, while infection happens with a nonexponential distribution following the Weibull form $P_a(\tau_a)~(x/b)^{\alpha-1}e^{-(x/b)^\alpha}$. In this case, strong variations in the value of the prevalence and of the epidemic threshold are found when varying the parameter $\alpha$. A promising approach is provided by the general simulation framework proposed by $\mathrm{Bogu\tilde{n}\acute{a}}$ et al. (2014)[10], based on the extension of the Gillespie algorithm for Poissonian processes [11]. This algorithm allows the simulation of much larger network sizes.

The consideration of non-Poissonian infection or recovery processes does not lend itself easily to analytical approaches [12]. Some simple forms for the distribution of infectious periods, such as the Erlang distribution, which can be described as the convolution of identical Poisson processes [13], can be tackled analytically by postulating an extended epidemic model with different infective phases and Poissonian transitions among them [14]. However, general non Poissonian forms lead to convoluted sets of integrodifferential equations [15]. As a consequence there are not many analytical results for non-Poissonian transitions in complex networks. We mention the results of Min, Goh, and Kim (2013)[16] which consider the SIR process on a network in which infection events follow an interevent distribution P aðτaÞ. Assuming that infected nodes remain in that state for a fixed amount of time τi, it is possible to compute [16] the disease transmissibility as

For a power-law distribution $P_a(\tau_a)~\tau_a^{-\alpha}$, it is found that $\tau_{ic}$ diverges as $\alpha\to2$, implying that only diseases without recovery are able to spread through the network [16]. An important step forward in the treatment of generic nonexponentially distributed recovery and transmission times in the SIR model is the application of a message-passing method, as reported by Karrer and Newman (2010)[17]. This approach leads to an exact description in terms of integrodifferential equations for trees and locally treelike networks, and to exact bounds for non-tree-like networks, in good agreement with simulations.

Finally, Cator, van de Bovenkamp, and Van Mieghem (2013)[18] proposed an extension of the SIS IBMF theory for nonexponential distributions of infection or healing times.Using renewal theory, their main result is the observation that the functional form of the prevalence in the metastable state is the same as in the Poissonian SIS model, when the spreading rate $\lambda=\beta/\mu$ is replaced by the average number of infection attempts during a recovery time. The theory by Cator, van de Bovenkamp, and Van Mieghem (2013)[18] also allows one to estimate the epidemic threshold in non-Markovian SIS epidemics.

## 参考文献 References

1. Tijms H C. A first course in stochastic models[M]. John Wiley and sons, 2003.
2. Van Kampen N G. Stochastic processes in chemistry and physics[J]. Chaos, 1981.
3. Ross S M. Stochastic Processes. John Wiley & Sons[J]. New York, 1996.
4. Tijms H C. A first course in stochastic models[M]. John Wiley and sons, 2003.
5. Van Mieghem P. Performance analysis of complex networks and systems[M]. Cambridge University Press, 2014.
6. Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.
7. Blythe S P, Anderson R M. Variable infectiousness in HFV transmission models[J]. Mathematical Medicine and Biology: A Journal of the IMA, 1988, 5(3): 181-200.
8. Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.
9. Van Mieghem P, Van de Bovenkamp R. Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks[J]. Physical review letters, 2013, 110(10): 108701.
10. Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.
11. Gillespie D T. Exact stochastic simulation of coupled chemical reactions[J]. The journal of physical chemistry, 1977, 81(25): 2340-2361.
12. Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.
13. Cox D R. Renewal Theory, 2-nd Edn[J]. 1967.
14. Lloyd A L. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods[J]. Proceedings of the Royal Society of London. Series B: Biological Sciences, 2001, 268(1470): 985-993.
15. Keeling M J, Grenfell B T. Disease extinction and community size: modeling the persistence of measles[J]. Science, 1997, 275(5296): 65-67.
16. Min B, Goh K I, Kim I M. Suppression of epidemic outbreaks with heavy-tailed contact dynamics[J]. EPL (Europhysics Letters), 2013, 103(5): 50002.
17. Karrer B, Newman M E J. Message passing approach for general epidemic models[J]. Physical Review E, 2010, 82(1): 016101.
18. Cator E, Van de Bovenkamp R, Van Mieghem P. Susceptible-infected-susceptible epidemics on networks with general infection and cure times[J]. Physical Review E, 2013, 87(6): 062816.