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For convergent initial proportions, the scaled SIRS Markov chain converges uniformly in probability on each fixed finite interval to the solution of these equations. The Poisson time-change representation of a Markov chain writes it as its integrated drift plus centered Poisson errors. Bounded rates, Poisson concentration and the Gronwall inequality control the error. This finite-time limit does not by itself justify an approximation over time intervals growing with the population size.

Use the compartment counts $Z_n=(S_n,I_n,R_n)$ as the state of the SIRS Markov chain. Its state space is $\{(S,I,R)\in {\mathbb{Z}}_{\ge 0}^3:S+I+R=n\}$. The transition rates areEach susceptible has infection hazard $\lambda I/n$, so summing over the $S$ susceptibles gives the first rate. The memorylessness of the exponential distribution makes each of the $I$ current infectives have recovery hazard $\gamma$, irrespective of its infection time. Each removed individual has immunity-loss hazard $\nu$. These are all the possible off-diagonal transitions. The diagonal entry of the transition intensity matrix is minus their sum, and a transition with no eligible individual has rate zero. Thus the three total event rates are $\lambda SI/n$, $\gamma I$ and $\nu R$, and every event conserves population size.

Scale the SIRS Markov chain by population size: $U_n=Z_n/n=(s_n,i_n,r_n)$. Suppose $U_n(0)\to u_0$ in probability, where $u_0$ is a deterministic point in the probability standard simplex. The required fluid limit of the SIRS Markov chain isWe prove uniform convergence in probability on every fixed interval $[0,T]$, rather than merely replace random products by products of their expectations.

Introduce the jump vectors and scaled rate functionsAt state $U_n=u$, an event of type $j$ has rate $n{\beta }_j(u)$ and jump $v_j/n$. Let $N_1,N_2,N_3$ be independent unit-rate Poisson processes. The Poisson time-change representation of a Markov chain givesThis representation follows by giving each event type an independent exponential clock in its accumulated hazard. Conditional on the current state its firing rate is $n{\beta }_j$, hence it has exactly the transition rates from part (a). The finite state space and bounded rates rule out explosion. Values at the jump times do not affect the time integrals.

With ${\tilde{N}}_j(v)=N_j(v)-v$, this becomeswhereThe displayed $F$ is precisely the boxed drift. On the simplex, ${\beta }_j\le b_j$ with $b_1=\lambda$, $b_2=\gamma$, $b_3=\nu$. Thus every Poisson clock used before $T$ is at most $n{b}_{j}T$. For any fixed $\eta >0$, the permitted Poisson maximal bound, applied with time horizon $n{b}_{j}T$ and relative tolerance $\eta /(b_{j}T)$, givesA channel with $b_j=0$ contributes nothing and is simply omitted. Since $h(a)>0$ for $a>0$, these probabilities even decrease exponentially in $n$. Each $v_j$ has ${\ell }^1$ norm two. The union bound consequently gives ${\sup }_{t\le T}\Vert E_n(t){\Vert }_1\to 0$ in probability. Notice that the clock is random, but its deterministic upper bound makes the Poisson maximal estimate directly applicable.

The drift $F$ is Lipschitz on the simplex. Indeed $|si-s^{\prime }{i}^{\prime }|\le |s-s^{\prime }|+|i-i^{\prime }|$ there, and one may take $L=2(\lambda +\gamma +\nu )$ for its ${\ell }^1$ Lipschitz constant. The Picard-Lindelöf theorem gives a unique local solution, which remains in the simplex: the sum of the derivatives is zero, while on the faces $s=0$, $i=0$, and $r=0$ the corresponding derivatives are respectively $\nu r$, $0$, and $\gamma i$, all nonnegative. More explicitly, $i(t)=i(0)\exp ({\int }_0^t(\lambda s(a)-\gamma )da)$ is nonnegative, and the linear equations for $r$ and $s$ have nonnegative forcing terms; their integrating-factor formulas keep both nonnegative. Conservation then bounds every coordinate by one, so the local solution extends globally.

Subtract its integral equation from that of $U_n$. The Gronwall inequality yieldsThis proves the Poisson concentration proof of a density-dependent fluid limit for this model, including the initial-condition requirement and uniform finite-time approximation. In terms of population counts, the associated deterministic approximation is $\dot{S}=-\lambda SI/n+\nu R$, $\dot{I}=\lambda SI/n-\gamma I$, $\dot{R}=\gamma I-\nu R$. The limit concerns macroscopic initial proportions and fixed times; a single initial infective has limiting fraction zero, so early stochastic invasion on a growing time scale is not described by this finite-time limit alone.