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. Foxall, Edwards, and van den Driessche [7] found the critical value and studied the subcritical and supercritical regimes. Recently Foxall [4] 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.
Electron. J. Probab.
23:
1-42
(2018).
DOI: 10.1214/18-EJP229
[1] Bramson, M. (1998) State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems. 30, 89–140. 0911.90162 10.1023/A:1019160803783[1] Bramson, M. (1998) State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems. 30, 89–140. 0911.90162 10.1023/A:1019160803783
[2] Durrett, R. (2010) Probability: Theory and Examples. Fourth Edition, Cambridge U. Press. 1202.60001[2] Durrett, R. (2010) Probability: Theory and Examples. Fourth Edition, Cambridge U. Press. 1202.60001
[3] Ethier, S.N., and Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. John Wiley and Sons, New York. 0592.60049[3] Ethier, S.N., and Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. John Wiley and Sons, New York. 0592.60049
[4] Foxall, E. (2016) Critical behaviour of the partner model. Ann. Appl. Probab., 26(5), 2824–2859. 1353.60086 10.1214/15-AAP1163 euclid.aoap/1476884305[4] Foxall, E. (2016) Critical behaviour of the partner model. Ann. Appl. Probab., 26(5), 2824–2859. 1353.60086 10.1214/15-AAP1163 euclid.aoap/1476884305
[5] Foxall, E. (2016) Stochastic calculus and sample path estimation for jump processes. arXiv preprint arXiv:1607.07063. 1607.07063[5] Foxall, E. (2016) Stochastic calculus and sample path estimation for jump processes. arXiv preprint arXiv:1607.07063. 1607.07063
[6] Foxall, E. (2017) The naming game on the complete graph. arXiv preprint arXiv:1703.02088. 1703.02088[6] Foxall, E. (2017) The naming game on the complete graph. arXiv preprint arXiv:1703.02088. 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. 1345.60115 10.1214/15-AAP1117 euclid.aoap/1465905004[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. 1345.60115 10.1214/15-AAP1117 euclid.aoap/1465905004
[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. 0885.60080 10.1214/aoap/1034801252 euclid.aoap/1034801252[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. 0885.60080 10.1214/aoap/1034801252 euclid.aoap/1034801252
[9] Jacod, J. and Shiryaev, A.N. (2002). Limit theorems for stochastic processes. Springer Science and Business Media. MR959133 0635.60021[9] Jacod, J. and Shiryaev, A.N. (2002). Limit theorems for stochastic processes. Springer Science and Business Media. MR959133 0635.60021
[10] Jagers, P. and Nerman, O. (1984) The growth and composition of branching populations. Advances in Applied Prob. 16(2) 221–259. 0535.60075 10.2307/1427068[10] Jagers, P. and Nerman, O. (1984) The growth and composition of branching populations. Advances in Applied Prob. 16(2) 221–259. 0535.60075 10.2307/1427068
[11] Kallenberg, O. (1997) Foundations of modern probability. Springer Science and Business Media. 0892.60001[11] Kallenberg, O. (1997) Foundations of modern probability. Springer Science and Business Media. 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. 1377.60076 10.1214/12-AAP841 euclid.aoap/1360682022[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. 1377.60076 10.1214/12-AAP841 euclid.aoap/1360682022
[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. 1319.60045 10.1214/13-AAP934 euclid.aoap/1394465370[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. 1319.60045 10.1214/13-AAP934 euclid.aoap/1394465370
[14] Karlin, S., and Taylor, H.M. (1975) A First Course in Stochastic Processes, Second Edition. Academic Press, New York. 0315.60016[14] Karlin, S., and Taylor, H.M. (1975) A First Course in Stochastic Processes, Second Edition. Academic Press, New York. 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. MR0254917 0191.47301 10.2307/3212147[15] Kurtz, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7(1), 49–58. MR0254917 0191.47301 10.2307/3212147
[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. 0219.60060 10.2307/3211904[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. 0219.60060 10.2307/3211904
[17] Kurtz, T.G. (1978) Strong approximation theorems for density-dependent Markov chains. Stochastic Processes and their Applications. 6(3), 223–240. 0373.60085 10.1016/0304-4149(78)90020-0[17] Kurtz, T.G. (1978) Strong approximation theorems for density-dependent Markov chains. Stochastic Processes and their Applications. 6(3), 223–240. 0373.60085 10.1016/0304-4149(78)90020-0
[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. 1242.90066 10.1214/11-AAP759 euclid.aoap/1328623696[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. 1242.90066 10.1214/11-AAP759 euclid.aoap/1328623696
[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. 1057.60092 10.1214/aoap/1075828046 euclid.aoap/1075828046[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. 1057.60092 10.1214/aoap/1075828046 euclid.aoap/1075828046
[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. 0911.90171 10.1023/A:1019108819713[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. 0911.90171 10.1023/A:1019108819713