ii) For any $v < v_*$ and $\varepsilon > 0$,

$$ \lim_{T \to \infty} \limsup_{K \to \infty} \mathbb{P} \left( \sup_{t \in [T, \tau_I^K(v_K)]} \left| \frac{I^K(t)}{K x_I^{K,T}(t)} - 1 \right| \ge \varepsilon; I^K(T) > 0 \right) = 0, $$

where $(x_S^{K,T}(t), x_I^{K,T}(t))$ is the solution of

$$ \begin{cases} \frac{d}{dt} x_S^{K,T}(t) = -\beta x_S^{K,T}(t) x_I^{K,T}(t), \\ \frac{d}{dt} x_I^{K,T}(t) = (\beta x_S^{K,T}(t) - \gamma) x_I^{K,T}(t). \end{cases} $$

such that $(x_S^{K,T}(0), x_I^{K,T}(0)) = (S^K(T)/K, I^K(T)/K)$.

Point i) is a consequence of Theorem 2.2 and point ii) comes from Theorem 3.6. These two parts have an intersection: in the time window when the number of infected individuals is large but negligible compared to $K/(\log K)^2$, we have informally

$$ I^K(t) \sim I(t) \sim We^{(\beta-\gamma)t} \sim K x_I^{K,T}(t-T). $$

We also obtain an approximation of the time when the epidemics reaches a given level (by Theorem 4.1):

**Corollary 5.4.** Let $(\zeta_K)_K$ be sequence which tends to infinity and satisfies $\zeta_K/K \to v$ as $K \to \infty$, for some $v \in [0, v_*)$. Then, for any $\varepsilon > 0$,

$$ \lim_{K \to \infty} \mathbb{P} \left( \left| \tau_I^K(\zeta_K) - \frac{\log(\zeta_K/W)}{r_*} - \tau(v) \right| \ge \varepsilon, W > 0 \right) = 0, $$

where $\tau$ is the increasing continuous function such that $\tau(0) = 0$ defined in Proposition 3.2.

This question of how long it takes for an epidemic to reach a high level has several motivations. Indeed, sanitary capacities for managing the infected individuals (hospital beds, isolation...) are limited. The time at which sanitary capacities reach saturation is an important factor in managing epidemics and taking control measures (social distancing, confinement, etc.). In addition, we control here the period during which the stochasticity of the process still plays a role, before deterministic predictions are valid with sufficient accuracy.

# 6. Appendix

## 6.1. Minimum of the martingale associated with the branching process Z

Recalling the notation of Section 2, we are interested in the minimal value of the martingale $W(t) := Z(t) \exp(-r_*t)$. Such object for simple branching processes has already attracted a lot of attention. We are not aware in the literature of the estimates given here and provide the proof for completeness. This relies on classical $L^2$ estimates to control the speed of the convergence of the martingale.

**Lemma 6.1.** For any $\varepsilon > 0$, and any $Z$-adapted stopping time $\tau$,

$$ \mathbb{P}(W(\tau) \le \varepsilon; W > 0) \le C \left( \varepsilon + \varepsilon^{-1} \mathbb{E} \left[ \mathbf{1}_{\{\tau < \infty\}} e^{-r_* \tau / 2} \right] \right), $$

where $W$ is the almost sure limit of $W(t)$ as $t$ goes to infinity, and $C$ is some positive constant independent of $\varepsilon$ and $\tau$. We use the convention $W(+\infty) := W$ for the inequality above to make sense.

*Proof.* To simplify the proof, we firstly treat the case where $\tau$ is deterministic and finite. To begin with,

$$ W(\tau + 1) - W(\tau) = e^{-r_* \tau} \left( \sum_{i=1}^{Z(\tau)} N_i e^{-r_*} - Z(\tau) \right), $$
ii) For any $v < v_*$ and $\varepsilon > 0$,

$$ \lim_{T \to \infty} \limsup_{K \to \infty} \mathbb{P} \left( \sup_{t \in [T, \tau_I^K(v_K)]} \left| \frac{I^K(t)}{K x_I^{K,T}(t)} - 1 \right| \ge \varepsilon; I^K(T) > 0 \right) = 0, $$

where $(x_S^{K,T}(t), x_I^{K,T}(t))$ is the solution of

$$ \begin{cases} \frac{d}{dt} x_S^{K,T}(t) = -\beta x_S^{K,T}(t) x_I^{K,T}(t), \\ \frac{d}{dt} x_I^{K,T}(t) = (\beta x_S^{K,T}(t) - \gamma) x_I^{K,T}(t). \end{cases} $$

such that $(x_S^{K,T}(0), x_I^{K,T}(0)) = (S^K(T)/K, I^K(T)/K)$.

Point i) is a consequence of Theorem 2.2 and point ii) comes from Theorem 3.6. These two parts have an intersection: in the time window when the number of infected individuals is large but negligible compared to $K/(\log K)^2$, we have informally

$$ I^K(t) \sim I(t) \sim W e^{(\beta - \gamma)t} \sim K x_I^{K,T}(t - T). $$

We also obtain an approximation of the time when the epidemics reaches a given level (by Theorem 4.1):

**Corollary 5.4.** Let $(\zeta_K)_K$ be sequence which tends to infinity and satisfies $\zeta_K/K \to v$ as $K \to \infty$, for some $v \in [0, v_*)$. Then, for any $\varepsilon > 0$,

$$ \lim_{K \to \infty} \mathbb{P} \left( \left| \tau_I^K(\zeta_K) - \frac{\log(\zeta_K/W)}{r_*} - \tau(v) \right| \ge \varepsilon, W > 0 \right) = 0, $$

where $\tau$ is the increasing continuous function such that $\tau(0) = 0$ defined in Proposition 3.2.

This question of how long it takes for an epidemic to reach a high level has several motivations. Indeed, sanitary capacities for managing the infected individuals (hospital beds, isolation...) are limited. The time at which sanitary capacities reach saturation is an important factor in managing epidemics and taking control measures (social distancing, confinement, etc.). In addition, we control here the period during which the stochasticity of the process still plays a role, before deterministic predictions are valid with sufficient accuracy.

# 6. Appendix

## 6.1. Minimum of the martingale associated with the branching process Z

Recalling the notation of Section 2, we are interested in the minimal value of the martingale $W(t) := Z(t) \exp(-r_*t)$. Such object for simple branching processes has already attracted a lot of attention. We are not aware in the literature of the estimates given here and provide the proof for completeness. This relies on classical $L^2$ estimates to control the speed of the convergence of the martingale.

**Lemma 6.1.** For any $\varepsilon > 0$, and any $Z$-adapted stopping time $\tau$,

$$ \mathbb{P}(W(\tau) \le \varepsilon; W > 0) \le C \left( \varepsilon + \varepsilon^{-1} \mathbb{E} \left[ \mathbf{1}_{\{\tau < \infty\}} e^{-r_* \tau / 2} \right] \right), $$

where $W$ is the almost sure limit of $W(t)$ as $t$ goes to infinity, and $C$ is some positive constant independent of $\varepsilon$ and $\tau$. We use the convention $W(+\infty) := W$ for the inequality above to make sense.

*Proof.* To simplify the proof, we firstly treat the case where $\tau$ is deterministic and finite. To begin with,

$$ W(\tau + 1) - W(\tau) = e^{-r_* \tau} \left( \sum_{i=1}^{Z(\tau)} N_i e^{-r_*} - Z(\tau) \right), $$
where, conditionally on $Z(\tau)$, the $N_i$ are i.i.d. with the same distribution as $Z(1)$. This implies that

$$ 
\begin{aligned}
\mathbb{E}[(W(\tau + 1) - W(\tau))^2] &= e^{-2r_{\star}\tau} \mathbb{E} \left[ \mathrm{Var} \left[ \sum_{i=1}^{Z(\tau)} (N_i e^{-r_{\star}} - 1) |Z(\tau)| \right] \right] \\
&= e^{-2r_{\star}\tau} \mathbb{E}[Z(\tau)] \mathrm{Var} [N_i e^{-r_{\star}} - 1] \leq C e^{-r_{\star}\tau}. \quad (6.1)
\end{aligned}
$$

On the other hand, we write

$$ 
\begin{aligned}
\mathbb{P}(W(\tau) \le \varepsilon; W > 0) \le \mathbb{P}(W \le 2\varepsilon; W > 0) + \mathbb{P}(W(\tau) \le \varepsilon; W > 2\varepsilon) \\
\le \mathbb{P}(W \le 2\varepsilon; W > 0) + \mathbb{P}(W - W(\tau) > \varepsilon).
\end{aligned}
$$

Recalling that, on the event $\{W > 0\}$, $W$ follows an exponential distribution, we can bound the first term of the sum above by $1 - e^{-C\varepsilon} \le C\varepsilon$.

The second term of the sum can be handled as follows:

$$ 
\begin{aligned}
\mathbb{P}(W - W(\tau) > \varepsilon) &= \mathbb{P} \left( \sum_{n \ge 0} W(\tau + n + 1) - W(\tau + n) > \varepsilon \right) \\
&\le \frac{1}{\varepsilon} \sum_{n \ge 0} \mathbb{E} [ |W(\tau + n + 1) - W(\tau + n)| ] \\
&\le \varepsilon^{-1} \sum_{n \ge 0} \mathbb{E} [ |W(\tau + n + 1) - W(\tau + n)|^2 ]^{1/2} \\
&\le C \varepsilon^{-1} e^{-r_{\star}\tau/2} \sum_{n \ge 0} e^{-r_{\star}n/2} \le C \varepsilon^{-1} e^{-r_{\star}\tau/2}.
\end{aligned}
$$

This ends the proof in the case where $\tau$ is deterministic.

Now to prove the result when $\tau$ is some almost surely finite $Z$-adapted stopping time, we just have to do the same computation conditionally on $\mathcal{F}_{\tau}$. Indeed, since $Z(\tau)$ is $\mathcal{F}_{\tau}$-measurable and $Z(\tau + 1) - Z(\tau)$ independent of $\mathcal{F}(\tau)$, (6.1) becomes

$$ \mathbb{E}[(W(\tau + 1) - W(\tau))^2 | \mathcal{F}_{\tau}] \le C e^{-r_{\star}\tau}, $$

and the last computation of the previous case gives

$$ \mathbb{P}(W - W(\tau) > \varepsilon | \mathcal{F}_{\tau}) \le C \varepsilon^{-1} e^{-r_{\star}\tau/2}. $$

The result of the lemma is then proved by taking the expectation in the inequality above.

Finally let us treat the case where $\tau$ is not necessary almost surely finite. We write

$$ 
\begin{aligned}
\mathbb{P}(W(\tau) \le \varepsilon; W > 0) &= \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) + \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau = \infty) \\
&\le \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) + \mathbb{P}(0 < W \le \varepsilon).
\end{aligned}
\quad (6.2)
$$

Note that, on the event $\{\tau < \infty\}$, we have the almost sure convergence

$$ \mathbf{1}_{\{W(\tau \wedge n) \le \varepsilon; W > 0\}} \underset{n \to \infty}{\longrightarrow} \mathbf{1}_{\{W(\tau) \le \varepsilon; W > 0\}}. $$

Hence, by Fatou's lemma,

$$ 
\begin{aligned}
\mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) &\le \liminf_n \mathbb{P}(W(\tau \wedge n) \le \varepsilon; W > 0; \tau < \infty) \\
&\le \liminf_n \mathbb{P}(W(\tau \wedge n) \le \varepsilon; W > 0).
\end{aligned}
$$
where, conditionally on $Z(\tau)$, the $N_i$ are i.i.d. with the same distribution as $Z(1)$. This implies that

$$ 
\begin{aligned}
\mathbb{E}[(W(\tau + 1) - W(\tau))^2] &= e^{-2r_{\star}\tau} \mathbb{E} \left[ \mathrm{Var} \left[ \sum_{i=1}^{Z(\tau)} (N_i e^{-r_{\star}} - 1) |Z(\tau)| \right] \right] \\
&= e^{-2r_{\star}\tau} \mathbb{E}[Z(\tau)] \mathrm{Var} [N_i e^{-r_{\star}} - 1] \leq C e^{-r_{\star}\tau}.
\end{aligned}
\quad (6.1) $$

On the other hand, we write

$$ 
\begin{aligned}
\mathbb{P}(W(\tau) \le \varepsilon; W > 0) \le \mathbb{P}(W \le 2\varepsilon; W > 0) + \mathbb{P}(W(\tau) \le \varepsilon; W > 2\varepsilon) \\
\le \mathbb{P}(W \le 2\varepsilon; W > 0) + \mathbb{P}(W - W(\tau) > \varepsilon).
\end{aligned}
$$

Recalling that, on the event $\{W > 0\}$, $W$ follows an exponential distribution, we can bound the first term of the sum above by $1 - e^{-C\varepsilon} \le C\varepsilon$.

The second term of the sum can be handled as follows:

$$ 
\begin{aligned}
\mathbb{P}(W - W(\tau) > \varepsilon) &= \mathbb{P} \left( \sum_{n \ge 0} W(\tau + n + 1) - W(\tau + n) > \varepsilon \right) \\
&\le \frac{1}{\varepsilon} \sum_{n \ge 0} \mathbb{E} [ |W(\tau + n + 1) - W(\tau + n)| ] \\
&\le \varepsilon^{-1} \sum_{n \ge 0} \mathbb{E} [ |W(\tau + n + 1) - W(\tau + n)|^2 ]^{1/2} \\
&\le C \varepsilon^{-1} e^{-r_{\star} \tau / 2} \sum_{n \ge 0} e^{-r_{\star} n / 2} \le C \varepsilon^{-1} e^{-r_{\star} \tau / 2}.
\end{aligned}
$$

This ends the proof in the case where $\tau$ is deterministic.

Now to prove the result when $\tau$ is some almost surely finite $Z$-adapted stopping time, we just have to do the same computation conditionally on $\mathcal{F}_{\tau}$. Indeed, since $Z(\tau)$ is $\mathcal{F}_{\tau}$-measurable and $Z(\tau + 1) - Z(\tau)$ independent of $\mathcal{F}(\tau)$, (6.1) becomes

$$ \mathbb{E} \left[ (W(\tau + 1) - W(\tau))^2 \mid \mathcal{F}_{\tau} \right] \le C e^{-r_{\star}\tau}, $$

and the last computation of the previous case gives

$$ \mathbb{P}(W - W(\tau) > \varepsilon \mid \mathcal{F}_{\tau}) \le C\varepsilon^{-1} e^{-r_{\star}\tau/2}. $$

The result of the lemma is then proved by taking the expectation in the inequality above.

Finally let us treat the case where $\tau$ is not necessary almost surely finite. We write

$$ 
\begin{aligned}
\mathbb{P}(W(\tau) \le \varepsilon; W > 0) &= \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) + \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau = \infty) \\
&\le \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) + \mathbb{P}(0 < W \le \varepsilon).
\end{aligned}
\quad (6.2) $$

Note that, on the event $\{\tau < \infty\}$, we have the almost sure convergence

$$ \mathbf{1}_{\{W(\tau \wedge n) \le \varepsilon; W > 0\}} \underset{n \to \infty}{\longrightarrow} \mathbf{1}_{\{W(\tau) \le \varepsilon; W > 0\}}. $$

Hence, by Fatou's lemma,

$$ 
\begin{aligned}
\mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) &\le \liminf_n \mathbb{P}(W(\tau \wedge n) \le \varepsilon; W > 0; \tau < \infty) \\
&\le \liminf_n \mathbb{P}(W(\tau \wedge n) \le \varepsilon; W > 0).
\end{aligned}
$$
Then, applying the result of the lemma with the almost surely finite stopping time $\tau \wedge n$, we obtain

$$ \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) \le C \left( \varepsilon + \varepsilon^{-1} \liminf_n \mathbb{E} \left[ e^{-r_* (\tau \wedge n)/2} \right] \right). $$

Noticing that

$$ e^{-r_* (\tau \wedge n)/2} \le \mathbf{1}_{\{\tau < \infty\}} e^{-r_* \tau/2} + e^{-r_* n/2}, $$

we have

$$ \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) \le C \left( \varepsilon + \varepsilon^{-1} \mathbb{E} \left[ \mathbf{1}_{\{\tau < \infty\}} e^{-r_* \tau/2} \right] \right). $$

In addition, to control the second term of the sum in (6.2), we recall that the distribution of $W$ is

$$ \frac{d_*}{b_*} \delta_0 + \frac{r_*}{b_*} \mathcal{E} \left( \frac{r_*}{b_*} \right). $$

Hence

$$ \mathbb{P}(0 < W \le \varepsilon) = \frac{r_*}{b_*} \left(1 - e^{-\frac{r_*}{b_*} \varepsilon}\right) \le \left(\frac{r_*}{b_*}\right)^2 \varepsilon. $$

Recalling (6.2), the result of the lemma is proved. □

**Lemma 6.2.** With the same notation as in Lemma 6.1, for any $0 < \varepsilon < 1$,

$$ \mathbb{P} \left( \inf_{t \ge 0} W(t) \le \varepsilon; W > 0 \right) \le C \varepsilon^{1/4}, $$

for some positive constant $C$ independent of $\varepsilon$.

*Proof.* Let us introduce

$$ s_{\varepsilon} := \frac{1}{r_*} \log (1/\varepsilon). $$

On the event $\{W > 0\}$, we have $Z(t) \ge 1$ for all $t \ge 0$ and further, for $t < s_{\varepsilon}$, $e^{-r_*t} > \varepsilon$. As a consequence

$$ \left\{ \inf_{t \ge 0} W(t) \le \varepsilon; W > 0 \right\} \subseteq \left\{ \inf_{t \ge s_{\varepsilon}} W(t) \le \varepsilon; W > 0 \right\}. $$

For some $\lambda > 1$ whose value will be fixed later, let us denote

$$ S_{\varepsilon} := \inf \left\{ t \ge s_{\varepsilon} : W(t) \le \varepsilon^{1/\lambda} \right\}. $$

Since $\lambda > 1$ and $0 < \varepsilon < 1$, we have $\varepsilon < \varepsilon^{1/\lambda}$, hence

$$ \left\{ \inf_{t \ge s_{\varepsilon}} W(t) \le \varepsilon \right\} \subseteq \{S_{\varepsilon} < \infty\} \subseteq \{W(S_{\varepsilon}) \le \varepsilon^{1/\lambda}\}. $$

Note that the last event above is trivially satisfied only if $S_{\varepsilon} < \infty$.

Then, thanks to Lemma 6.1,

$$ \mathbb{P} \left( W(S_{\varepsilon}) \le \varepsilon^{1/\lambda}; W > 0 \right) \le C \left( \varepsilon^{1/\lambda} + \varepsilon^{-1/\lambda} \mathbb{E} \left[ \mathbf{1}_{\{S_{\varepsilon} < \infty\}} e^{-r_* S_{\varepsilon}/2} \right] \right). $$

Recall that, by definition of $S_{\varepsilon}$, $S_{\varepsilon} \ge s_{\varepsilon}$, and so

$$ \mathbb{E} \left[ \mathbf{1}_{\{S_{\varepsilon} < \infty\}} e^{-r_* S_{\varepsilon}/2} \right] \le e^{-r_* s_{\varepsilon}/2} = \varepsilon^{1/2}, $$

implying that

$$ \mathbb{P} \left( W(S_{\varepsilon}) \le \varepsilon^{1/\lambda}; W > 0 \right) \le C \left( \varepsilon^{1/\lambda} + \varepsilon^{1/2-1/\lambda} \right). $$

Finally, choosing $\lambda := 4$ to optimize the bound above proves the result. □
Then, applying the result of the lemma with the almost surely finite stopping time $\tau \wedge n$, we obtain

$$ \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) \le C \left( \varepsilon + \varepsilon^{-1} \liminf_n \mathbb{E} \left[ e^{-r_* (\tau \wedge n)/2} \right] \right). $$

Noticing that

$$ e^{-r_* (\tau \wedge n)/2} \le \mathbf{1}_{\{\tau < \infty\}} e^{-r_* \tau/2} + e^{-r_* n/2}, $$

we have

$$ \mathbb{P}(W(\tau) \le \varepsilon; W > 0; \tau < \infty) \le C \left( \varepsilon + \varepsilon^{-1} \mathbb{E} \left[ \mathbf{1}_{\{\tau < \infty\}} e^{-r_* \tau/2} \right] \right). $$

In addition, to control the second term of the sum in (6.2), we recall that the distribution of $W$ is

$$ \frac{d_*}{b_*} \delta_0 + \frac{r_*}{b_*} \mathcal{E}(\frac{r_*}{b_*}). $$

Hence

$$ \mathbb{P}(0 < W \le \varepsilon) = \frac{r_*}{b_*} \left(1 - e^{-\frac{r_*}{b_*} \varepsilon}\right) \le \left(\frac{r_*}{b_*}\right)^2 \varepsilon. $$

Recalling (6.2), the result of the lemma is proved.

<figure><img src="image_6.png" /></figure>

**Lemma 6.2.** With the same notation as in Lemma 6.1, for any $0 < \varepsilon < 1$,

$$ \mathbb{P} \left( \inf_{t \ge 0} W(t) \le \varepsilon; W > 0 \right) \le C \varepsilon^{1/4}, $$

for some positive constant $C$ independent of $\varepsilon$.

*Proof.* Let us introduce

$$ s_{\varepsilon} := \frac{1}{r_*} \log (1/\varepsilon). $$

On the event $\{W > 0\}$, we have $Z(t) \ge 1$ for all $t \ge 0$ and further, for $t < s_{\varepsilon}$, $e^{-r_*t} > \varepsilon$. As a consequence

$$ \left\{ \inf_{t \ge 0} W(t) \le \varepsilon; W > 0 \right\} \subseteq \left\{ \inf_{t \ge s_{\varepsilon}} W(t) \le \varepsilon; W > 0 \right\}. $$

For some $\lambda > 1$ whose value will be fixed later, let us denote

$$ S_{\varepsilon} := \inf \left\{ t \ge s_{\varepsilon} : W(t) \le \varepsilon^{1/\lambda} \right\}. $$

Since $\lambda > 1$ and $0 < \varepsilon < 1$, we have $\varepsilon < \varepsilon^{1/\lambda}$, hence

$$ \left\{ \inf_{t \ge s_{\varepsilon}} W(t) \le \varepsilon \right\} \subseteq \left\{ S_{\varepsilon} < \infty \right\} \subseteq \left\{ W(S_{\varepsilon}) \le \varepsilon^{1/\lambda} \right\}. $$

Note that the last event above is trivially satisfied only if $S_{\varepsilon} < \infty$.

Then, thanks to Lemma 6.1,

$$ \mathbb{P} \left( W(S_{\varepsilon}) \le \varepsilon^{1/\lambda}; W > 0 \right) \le C \left( \varepsilon^{1/\lambda} + \varepsilon^{-1/\lambda} \mathbb{E} \left[ \mathbf{1}_{\{S_{\varepsilon} < \infty\}} e^{-r_* S_{\varepsilon}/2} \right] \right). $$

Recall that, by definition of $S_{\varepsilon}$, $S_{\varepsilon} \ge s_{\varepsilon}$, and so

$$ \mathbb{E} \left[ \mathbf{1}_{\{S_{\varepsilon} < \infty\}} e^{-r_* S_{\varepsilon}/2} \right] \le e^{-r_* s_{\varepsilon}/2} = \varepsilon^{1/2}, $$

implying that

$$ \mathbb{P} \left( W(S_{\varepsilon}) \le \varepsilon^{1/\lambda}; W > 0 \right) \le C \left( \varepsilon^{1/\lambda} + \varepsilon^{1/2-1/\lambda} \right). $$

Finally, choosing $\lambda := 4$ to optimize the bound above proves the result.

<figure><img src="image_7.png" /></figure>
## 6.2. Perturbation of stable dynamical system

We provide a result which is strongly inspired from Lemmas 3.1 and 3.2 in [30]. We consider an exponential stable dynamical system: $y' \le -cy$, with $c > 0$. We add a source term, which acts as a perturbation given by $h$. The following result allows to control the value of the dynamical system in terms of fluctuations of the source term $h$ on a unit time interval.

**Lemma 6.3.** Assume that there are three measurable and locally bounded functions $y, h, \phi$ with $h(0) = 0$ and satisfying for all $t \ge 0$,

$$y(t) = y(0) + h(t) + \int_{0}^{t} y(s)\phi(s)ds.$$

Suppose in addition that $\phi$ is bounded, and upper-bounded by some negative number. Then, for all $t \ge 0$,

$$\sup_{s \le t} |y(s)| \le \Gamma \left( |y(0)| \vee \sup_{s \le t} |h(s) - h(\lfloor s \rfloor)| \right),$$

where $\Gamma = (1 + ||\phi||_{\infty})/(1 - \exp(\sup \phi))$.

*Proof.* To begin with, we introduce

$$\Phi(t, s) = e^{\int_{s}^{t} \phi(r)dr} \le e^{-C(t-s)},$$

for some positive constant $C := -\sup \phi > 0$. Then

$$\begin{gathered} \Phi(t, t) = 1; \quad \Phi(t, s)\Phi(s, u) = \Phi(t, u), \\ \partial_t \Phi(t, s) = \Phi(t, s)\phi(t); \quad \partial_s \Phi(t, s) = -\Phi(t, s)\phi(s). \end{gathered}$$

By a standard constant variation argument, one can write

$$y(t) = \Phi(t, 0)y(0) + h(t) + \int_{0}^{t} \Phi(t, s)h(s)\phi(s)ds.$$

Then it results the following decomposition:

$$\begin{aligned} y(t) = & \Phi(t, 0)y(0) + \sum_{j=1}^{\lfloor t \rfloor} \Phi(t, j) \left( h(j) - h(j-1) + \int_{j-1}^{j} \Phi(j, s)\phi(s) (h(s) - h(j-1)) ds \right) \\ & + h(t) - h(\lfloor t \rfloor) + \int_{\lfloor t \rfloor}^{t} \Phi(t, s)\phi(s) (h(s) - h(\lfloor t \rfloor)) ds. \end{aligned}$$

Using the upperbound of $\phi$, we obtain

$$|y(t)| \le e^{-Ct} |y(0)| + \sum_{j=1}^{\lfloor t \rfloor} e^{-C(t-j)} \left( \sup_{j-1 \le s \le j} |h(s) - h(j-1)| \Gamma' \right) + \sup_{\lfloor t \rfloor \le s \le t} |h(s) - h(\lfloor t \rfloor)| \Gamma,$$

with $\Gamma' := 1 + ||\phi||_{\infty}$. The result is proved by choosing $\Gamma = \Gamma' \sum_{j=0}^{\infty} e^{-Cj} = \Gamma'/(1 - e^{-C})$. $\square$

## Acknowledgements:

* The authors are very grateful to Pierre Collet who has clarified some issue involved in the control of hitting values of the dynamical system started close to the equilibrium. They also thank very warmly the two referees of the first version of this work, whose many comments led to a major improvement of the paper.
## 6.2. Perturbation of stable dynamical system

We provide a result which is strongly inspired from Lemmas 3.1 and 3.2 in [30]. We consider an exponential stable dynamical system : $y' \le -cy$, with $c > 0$. We add a source term, which acts as a perturbation given by $h$. The following result allows to control the value of the dynamical system in terms of fluctuations of the source term $h$ on a unit time interval.

**Lemma 6.3.** Assume that there are three measurable and locally bounded functions $y, h, \phi$ with $h(0) = 0$ and satisfying for all $t \ge 0$,

$$y(t) = y(0) + h(t) + \int_{0}^{t} y(s)\phi(s)ds.$$

Suppose in addition that $\phi$ is bounded, and upper-bounded by some negative number. Then, for all $t \ge 0$,

$$\sup_{s \le t} |y(s)| \le \Gamma \left( |y(0)| \vee \sup_{s \le t} |h(s) - h(\lfloor s \rfloor)| \right),$$

where $\Gamma = (1 + ||\phi||_{\infty})/(1 - \exp(\sup \phi))$.

*Proof.* To begin with, we introduce

$$\Phi(t, s) = e^{\int_{s}^{t} \phi(r)dr} \le e^{-C(t-s)},$$

for some positive constant $C := -\sup \phi > 0$. Then

$$\begin{aligned} \Phi(t, t) &= 1; \quad \Phi(t, s)\Phi(s, u) = \Phi(t, u), \\ \partial_t \Phi(t, s) &= \Phi(t, s)\phi(t); \quad \partial_s \Phi(t, s) = -\Phi(t, s)\phi(s). \end{aligned}$$

By a standard constant variation argument, one can write

$$y(t) = \Phi(t, 0)y(0) + h(t) + \int_{0}^{t} \Phi(t, s)h(s)\phi(s)ds.$$

Then it results the following decomposition:

$$\begin{aligned} y(t) = &\Phi(t, 0)y(0) + \sum_{j=1}^{\lfloor t \rfloor} \Phi(t, j) \left( h(j) - h(j-1) + \int_{j-1}^{j} \Phi(j, s)\phi(s) (h(s) - h(j-1)) ds \right) \\ &+ h(t) - h(\lfloor t \rfloor) + \int_{\lfloor t \rfloor}^{t} \Phi(t, s)\phi(s) (h(s) - h(\lfloor t \rfloor)) ds. \end{aligned}$$

Using the upperbound of $\phi$, we obtain

$$|y(t)| \le e^{-Ct} |y(0)| + \sum_{j=1}^{\lfloor t \rfloor} e^{-C(t-j)} \left( \sup_{j-1 \le s \le j} |h(s) - h(j-1)| \Gamma' \right) + \sup_{\lfloor t \rfloor \le s \le t} |h(s) - h(\lfloor t \rfloor)| \Gamma,$$

with $\Gamma' := 1 + ||\phi||_{\infty}$. The result is proved by choosing $\Gamma = \Gamma' \sum_{j=0}^{\infty} e^{-Cj} = \Gamma'/(1 - e^{-C})$. $\square$

## Acknowledgements:

* The authors are very grateful to Pierre Collet who has clarified some issue involved in the control of hitting values of the dynamical system started close to the equilibrium. They also thank very warmly the two referees of the first version of this work, whose many comments led to a major improvement of the paper.
* This work has been supported by the Chair "Modélisation Mathématique et Biodiversité of Veolia Environnement-Ecole Polytechnique-Museum national d'Histoire naturelle-Fondation X and by ITMO Cancer of Aviesan within the framework of the 2021-2023 Cancer Control Strategy, on funds administrated by Inserm.

* Funded by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

## References

[1] K. B. Athreya, P. E. Ney. *Branching processes*. Springer-Verlag Berlin Heidelberg New-York 1972.

[2] M. Baar, A. Bovier, N. Champagnat. From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step. *Ann. Appl. Probab.* 27 (2017), no. 2.

[3] H. Bahouri, J. Chemin and R. Danchin. *Fourier Analysis and Nonlinear Partial Differential Equations* Springer, Grundlehren der mathematischen Wissenschaften, 2011.

[4] F. Ball, P. Donnelly. Strong approximations for epidemic models. *Stoc. Proc. Appl.* Vol. 55, 1995, Pages 1-21.

[5] A.D. Barbour, S. Utev. Approximating the Reed-Frost epidemic process. *Stoc. Proc. Appl.* Vol. 113 (2004) p 173-197.

[6] A. D. Barbour, G. Reinert. Approximating the epidemic curve. *Electron. J. Probab.* 18 (2013), no. 54, 1-30.

[7] A. D. Barbour, K. Hamza, H. Kaspi and F. C. Klebaner. Escape from the boundary in Markov population processes. *Adv. Appl. Probab.* (2015), Vol. 47, No. 4, pp. 1190-1211.

[8] S. Billiard, R. Ferrière, S. Méléard, V.C. Tran. Stochastic dynamics of adaptive trait and neutral marker driven by eco-evolutionary feedbacks. *J. Math. Biol.* 71 (2015), no. 5.

[9] S. Billiard, C. Smadi. The interplay of two mutations in a population of varying size: a stochastic eco-evolutionary model for clonal interference. *Stochastic Process. Appl.* 127 (2017), no. 3, 701-748.

[10] J. Blath, T. Paul and A. Tóbiás. A Stochastic Adaptive Dynamics Model for Bacterial Populations with Mutation, Dormancy and Transfer arXiv:2105.09228 (2021).

[11] C. Bonnet, P. Gou, S. Girel, V. Bansaye, C. Lacout, K. Bailly, M.H. Schlagetter, E. Lauret, S. Méléard, S. Giraudier. Multistage hematopoietic stem cell regulation in the mouse: a combined biological and mathematical approach. *iScience*, 2021 Nov 6; 24 (12):103399. Open access.

[12] C. Bonnet, S. Méléard. Large fluctuations in multi-scale modeling for rest erythropoiesis. *J. Math. Biol.* 82 (2021), no. 6, Paper No. 58.

[13] A. Bovier, L. Coquille, R. Neukirch. The recovery of a recessive allele in a Mendelian diploid model. *J. Math. Biol.* 77 (2018), no. 4.

[14] A. Bovier, L. Coquille, C. Smadi. Crossing a fitness valley as a metastable transition in a stochastic population model. *Ann. Appl. Probab.* 29 no 6 (2019), 3541-3589.

[15] N. Champagnat. A microscopic interpretation for adaptive dynamics trait substitution sequence models. *Stochastic Processes and their Applications*, 116 (2006), 1127-1160.

[16] N. Champagnat and S. Méléard. Polymorphic evolution sequence and evolutionary branching. *Probability Theory and Related Fields*, 151 no 1-2 (2011), 45-94.

[17] N. Champagnat, S. Méléard, S. Mirrahimi, V.C. Tran. Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer. *Ann. Appl. Probab.* 31 (2021), no.4, p. 1820-1867.

[18] L. Coquille, A. Kraut, C. Smadi. Stochastic individual-based models with power law mutation rate on a general finite trait space. *Electron. J. Probab.* 26(123), 37 p., 2021.

[19] O. Diekmann, P.-E. Jabin, S. Mischler, and B. Perthame. The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. *Theoretical Population Biology*, 67 (2005), 257-271.
* This work has been supported by the Chair "Modélisation Mathématique et Biodiversité of Veolia Environnement-Ecole Polytechnique-Museum national d'Histoire naturelle-Fondation X and by ITMO Cancer of Aviesan within the framework of the 2021-2023 Cancer Control Strategy, on funds administrated by Inserm.

* Funded by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

## References

[1] K. B. Athreya, P. E. Ney. *Branching processes*. Springer-Verlag Berlin Heidelberg New-York 1972.

[2] M. Baar, A. Bovier, N. Champagnat. From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step. *Ann. Appl. Probab.* 27 (2017), no. 2.

[3] H. Bahouri, J. Chemin and R. Danchin. *Fourier Analysis and Nonlinear Partial Differential Equations* Springer, Grundlehren der mathematischen Wissenschaften, 2011.

[4] F. Ball, P. Donnelly. Strong approximations for epidemic models. *Stoc. Proc. Appl.* Vol. 55, 1995, Pages 1-21.

[5] A.D. Barbour, S. Utev. Approximating the Reed-Frost epidemic process. *Stoc. Proc. Appl.* Vol. 113 (2004) p 173-197.

[6] A. D. Barbour, G. Reinert. Approximating the epidemic curve. *Electron. J. Probab.* 18 (2013), no. 54, 1-30.

[7] A. D. Barbour, K. Hamza, H. Kaspi and F. C. Klebaner. Escape from the boundary in Markov population processes. *Adv. Appl. Probab.* (2015), Vol. 47, No. 4, pp. 1190-1211.

[8] S. Billiard, R. Ferrière, S. Méléard, V.C. Tran. Stochastic dynamics of adaptive trait and neutral marker driven by eco-evolutionary feedbacks. *J. Math. Biol.* 71 (2015), no. 5.

[9] S. Billiard, C. Smadi. The interplay of two mutations in a population of varying size: a stochastic eco-evolutionary model for clonal interference. *Stochastic Process. Appl.* 127 (2017), no. 3, 701-748.

[10] J. Blath, T. Paul and A. Tóbiás. A Stochastic Adaptive Dynamics Model for Bacterial Populations with Mutation, Dormancy and Transfer arXiv:2105.09228 (2021).

[11] C. Bonnet, P. Gou, S. Girel, V. Bansaye, C. Lacout, K. Bailly, M.H. Schlagetter, E. Lauret, S. Méléard, S. Giraudier. Multistage hematopoietic stem cell regulation in the mouse: a combined biological and mathematical approach. *iScience*, 2021 Nov 6; 24 (12):103399. Open access.

[12] C. Bonnet, S. Méléard. Large fluctuations in multi-scale modeling for rest erythropoiesis. *J. Math. Biol.* 82 (2021), no. 6, Paper No. 58.

[13] A. Bovier, L. Coquille, R. Neukirch. The recovery of a recessive allele in a Mendelian diploid model. *J. Math. Biol.* 77 (2018), no. 4.

[14] A. Bovier, L. Coquille, C. Smadi. Crossing a fitness valley as a metastable transition in a stochastic population model. *Ann. Appl. Probab.* 29 no 6 (2019), 3541-3589.

[15] N. Champagnat. A microscopic interpretation for adaptive dynamics trait substitution sequence models. *Stochastic Processes and their Applications*, 116 (2006), 1127-1160.

[16] N. Champagnat and S. Méléard. Polymorphic evolution sequence and evolutionary branching. *Probability Theory and Related Fields*, 151 no 1-2 (2011), 45-94.

[17] N. Champagnat, S. Méléard, S. Mirrahimi, V.C. Tran. Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer. *Ann. Appl. Probab.* 31 (2021), no.4, p. 1820-1867.

[18] L. Coquille, A. Kraut, C. Smadi. Stochastic individual-based models with power law mutation rate on a general finite trait space. *Electron. J. Probab.* 26(123), 37 p., 2021.

[19] O. Diekmann, P.-E. Jabin, S. Mischler, and B. Perthame. The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. *Theoretical Population Biology*, 67 (2005), 257-271.
[20] R. Durrett. Branching process models of Cancer. Mathematical Biosciences Institute Lecture Series 1.1, Stochastics in Biological Systems, Springer, 2015.

[21] R.Durrett, J. Mayberry. Traveling waves of selective sweeps. *Ann. Appl. Probab.* 21 (2011), no. 2, p. 699–744.

[22] S.N. Ethier, T.G. Kurtz. Markov processes, Characterization and Convergence. John Wiley & Sons, New York, 1986.

[23] J. Hofbauer and K. Sigmund. *Evolutionary Games and Population Dynamics*. Cambridge University Press 1998.

[24] J. Jacod and A.N. Shiryaev. *Limit Theorems for Stochastic Processes*. Springer-Verlag, Berlin, 1987.

[25] A. Jakubowski. On the Skorokhod topology. *Annales de l'Institut Henri Poincaré*, 22 No 3, 263-285, 1986.

[26] A. Joffe and M. Métivier. Weak convergence of sequences of semimartingales with applications to multitype branching processes. *Advances in Applied Probability*, 18 (1986), 20–65.

[27] T. G. Kurtz. Solutions of ordinary differential equations as limits of pure jump Markov processes. *J. Appl. Prob.* 7,49-58, 1970.

[28] T. G. Kurtz. Strong approximation theorems. *Stoch. Proc. Appl.* 6, 223-240, 1978.

[29] J.A.J. Metz, S.A.H. Geritz, G. Meszéna, F.A.J. Jacobs, and J.S. Van Heerwaarden. Adaptative dynamics, a geometrical study of the consequences of nearly faithful reproduction. *S.J. Van Strien & S.M. Verduyn Lunel (ed.)*, *Stochastic and Spatial Structures of Dynamical Systems*, 45, 183–231, 1996.

[30] A. Prodhomme. Strong Gaussian approximation of metastable density-dependent Markov chains on large time scales. arXiv:2010.06861 (2020).

[31] F. Takens. Partially hyperbolic fixed points. *Topology*, Vol 10, 133-147, section 4. (1971)
[20] R. Durrett. Branching process models of Cancer. Mathematical Biosciences Institute Lecture Series 1.1, Stochastics in Biological Systems, Springer, 2015.

[21] R.Durrett, J. Mayberry. Traveling waves of selective sweeps. *Ann. Appl. Probab.* 21 (2011), no. 2, p. 699–744.

[22] S.N. Ethier, T.G. Kurtz. Markov processes, Characterization and Convergence. John Wiley & Sons, New York, 1986.

[23] J. Hofbauer and K. Sigmund. *Evolutionary Games and Population Dynamics*. Cambridge University Press 1998.

[24] J. Jacod and A.N. Shiryaev. *Limit Theorems for Stochastic Processes*. Springer-Verlag, Berlin, 1987.

[25] A. Jakubowski. On the Skorokhod topology. *Annales de l'Institut Henri Poincaré*, 22 No 3, 263-285, 1986.

[26] A. Joffe and M. Métivier. Weak convergence of sequences of semimartingales with applications to multitype branching processes. *Advances in Applied Probability*, 18 (1986), 20–65.

[27] T. G. Kurtz. Solutions of ordinary differential equations as limits of pure jump Markov processes. *J. Appl. Prob.* 7,49-58, 1970.

[28] T. G. Kurtz. Strong approximation theorems. *Stoch. Proc. Appl.* 6, 223-240, 1978.

[29] J.A.J. Metz, S.A.H. Geritz, G. Meszéna, F.A.J. Jacobs, and J.S. Van Heerwaarden. Adaptative dynamics, a geometrical study of the consequences of nearly faithful reproduction. *S.J. Van Strien & S.M. Verduyn Lunel (ed.)*, *Stochastic and Spatial Structures of Dynamical Systems*, 45, 183–231, 1996.

[30] A. Prodhomme. Strong Gaussian approximation of metastable density-dependent Markov chains on large time scales. arXiv:2010.06861 (2020).

[31] F. Takens. Partially hyperbolic fixed points. *Topology*, Vol 10, 133-147, section 4. (1971)