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Diffusion limit for the partner model at the critical value. (English) Zbl 1402.60123
Summary: The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. In [Ann. Appl. Probab. 26, No. 3, 1297–1328 (2016; Zbl 1345.60115)], the last author et al. found the critical value and studied the subcritical and supercritical regimes. In [ibid. 26, No. 5, 2824–2859 (2016; Zbl 1353.60086)], the last author has shown that (if there are enough initial infecteds \(I_0\)) the extinction time in the critical model is of order \(\sqrt{N} \). Here we improve that result by proving the convergence of \(i_N(t)=I(\sqrt{N} t)/\sqrt{N} \) to a limiting diffusion. We do this by showing that within a short time, this four dimensional process collapses to two dimensions: the number of \(SI\) and \(II\) partnerships are constant multiples of the the number of infected singles. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude \(\sqrt{N} \) on the original time scale and averages out of the limit theorem for \(i_N(t)\). As a by-product of our proof we show that if \(\tau_N\) is the extinction time of \(i_N(t)\) (on the \(\sqrt{N} \) time scale) then \(\tau_N\) has a limit.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F17 Functional limit theorems; invariance principles
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References:
[1] Bramson, M. (1998) State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems. 30, 89–140. · Zbl 0911.90162
[2] Durrett, R. (2010) Probability: Theory and Examples. Fourth Edition, Cambridge U. Press. · Zbl 1202.60001
[3] Ethier, S.N., and Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. John Wiley and Sons, New York. · Zbl 0592.60049
[4] Foxall, E. (2016) Critical behaviour of the partner model. Ann. Appl. Probab., 26(5), 2824–2859. · Zbl 1353.60086
[5] Foxall, E. (2016) Stochastic calculus and sample path estimation for jump processes. arXiv preprintarXiv:1607.07063. · Zbl 1341.60125
[6] Foxall, E. (2017) The naming game on the complete graph. arXiv preprintarXiv:1703.02088.
[7] Foxall, E., Edwards, R., and van den Driessche, P. (2016) Social contact processes and the partner model. Ann. Appl. Probab. 26(3), 1297–1328. · Zbl 1345.60115
[8] Harrison, J.M. and Van Mieghem, J.A. (1997) Dynamic control of Brownian networks: state space collapse and equivalent workload formulations. Ann. Appl. Probab. 7, 747–771. · Zbl 0885.60080
[9] Jacod, J. and Shiryaev, A.N. (2002). Limit theorems for stochastic processes. Springer Science and Business Media. · Zbl 0635.60021
[10] Jagers, P. and Nerman, O. (1984) The growth and composition of branching populations. Advances in Applied Prob. 16(2) 221–259. · Zbl 0535.60075
[11] Kallenberg, O. (1997) Foundations of modern probability. Springer Science and Business Media. · Zbl 0892.60001
[12] Kang, H.W., and Kurtz, T.G. (2013) Separation of time-scales and model reduction for stochastic reaction networks. Ann. Appl. Probab. 23(2), 529–583. · Zbl 1377.60076
[13] Kang, H.W., Kurtz, T.G., and Popovic (2014) Central limit theorems and diffusion approximations for multiscale Markov chain models. Ann. Appl. Probab. 24(2), 721–759. · Zbl 1319.60045
[14] Karlin, S., and Taylor, H.M. (1975) A First Course in Stochastic Processes, Second Edition. Academic Press, New York. · Zbl 0315.60016
[15] Kurtz, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7(1), 49–58. · Zbl 0191.47301
[16] Kurtz, T.G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability. 8(2), 344–356. · Zbl 0219.60060
[17] Kurtz, T.G. (1978) Strong approximation theorems for density-dependent Markov chains. Stochastic Processes and their Applications. 6(3), 223–240. · Zbl 0373.60085
[18] Protter, P. (2005) Stochastic Integration and Differential Equations, Second Edition. Springer-Verlag.
[19] Shah, D. and Wischik, D. (2012). Switched networks with maximum weight policies: fluid approximation and multiplicative state space collapse. Ann. Appl. Probab. 22(1), 70–127. · Zbl 1242.90066
[20] Stolyar, A. L. (2004). MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic. Ann. Appl. Probab. 14(1), 1–53. · Zbl 1057.60092
[21] Williams, R. J. (1998) Diffusion approximations for open multiclass queueing metworks: Sufficient conditions involving state space collapse. Queueing Systems: Theory and Applications. 30(1), 27–88. · Zbl 0911.90171
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