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We study the following microscopic model of infection or epidemic reaction:red and blue particles perform independent nearest-neighbor continuous-timesymmetric random walks on the integer lattice $\mathbb{Z}$ with jump rates$D_R$ for red particles and $D_B$ for blue particles, the interaction rulebeing that blue particles turn red upon contact with a red particle. Theinitial condition consists of i.i.d. Poisson particle numbers at each site,with particles at the left of the origin being red, while particles at theright of the origin are blue. We are interested in the dynamics of the front,defined as the rightmost position of a red particle. For the case $D_R=D_B$,Kesten and Sidoravicius established that the front moves ballistically, andmore precisely that it satisfies a law of large numbers. Their proof is basedon a multi-scale renormalization technique, combined with approximatesub-additivity arguments. In this paper, we build a renewal structure for thefront propagation process, and as a corollary we obtain a central limit theoremfor the front when $D_R=D_B$. Moreover, this result can be extended to the casewhere $D_R>D_B$, up to modifying the dynamics so that blue particles turn redupon contact with a site that has previously been occupied by a red particle.Our approach extends the renewal structure approach developed by Comets,Quastel and Ram\'{{\i}}rez for the so-called frog model, which corresponds tothe $D_B=0$ case.

Fluctuations of the front in a one-dimensional model for the spread of an infection