# Controlling the Mean Time to Extinction in Populations of Bacteria

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## Abstract

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## 1. Introduction

## 2. Model

**Random switching of the environment.**For a randomly switching environment, B switches between ${B}_{+}$ and ${B}_{-}$ such that ${B}_{-}<{B}_{+}$ according to ADMN, which is expressed as:

**Deterministic changes in the environment.**Deterministic changes are realized as either abrupt periodic rectangular switches or continuous sinusoidal changes. For a periodic (index p) (possibly asymmetric rectangular and sinusoidal) environmental change, the birth rate varies deterministically, and the master equation reduces to

**Abrupt periodic rectangular switches.**For a periodic rectangular wave, $B\left(t\right)$ switches periodically between ${B}_{+}$ and ${B}_{-}$, again such that ${B}_{-}<{B}_{+}$. Again, $B\left(t\right)$ is given by Equation (2), but with ${\zeta}_{r}$ replaced by ${\zeta}_{p}$, which enters the rectangular wave of period $T=(1/{\nu}_{+})+(1/{\nu}_{-})$, defined as

**Continuous sinusoidal changes.**When $B\left(t\right)$ changes continuously according to a periodic sinusoidal wave, it is given by $B\left(t\right)={B}_{0}+\u03f5sin\left(\omega t\right)$, where $\u03f5$ is the amplitude, and $\omega $ is the angular frequency of the perturbation. If we write the parameters of the sinusoidal perturbation in terms of the parameters of ADMN or a rectangular wave, we obtain $\omega =\pi \nu $, ${B}_{0}=({B}_{+}+{B}_{-})/2$, and $\u03f5=({B}_{+}-{B}_{-})/2$, but $\u03f5$ will vary independently of ${B}_{+}$ and ${B}_{-}$. Figure 1 shows an example of each type of variation in $B\left(t\right)$ and the corresponding sample trajectories of the total population size N until extinction. The different modulations of the environment are visualized in Figure 1 together with the corresponding time evolution of the population size.

**Mean-field level.**Before we look further into the dynamics of these master equations, we should state what these equations yield on the mean-field level in the absence of any noise, as their fixed-point structure is needed later. In the mean-field description, the dynamics of the average number of normals $\overline{n}\left(t\right)$ and persisters $\overline{m}\left(t\right)$ are governed by the rate equations [8]

## 3. Quasi-Stationary Distribution

## 4. Numerical Simulations

#### 4.1. Numerical Simulations for Sinusoidal Perturbations

#### 4.2. Impact of the Switching Rates α and β on the MTE’s Frequency Dependence for the Sinusoidal Change

#### 4.3. Impact of the Minimum Value of the Birth Rates

**The case of ${\mathit{B}}_{-}>\mathbf{1}$.**Let us first consider the case where B switches randomly between ${B}_{+}=1.15$ and ${B}_{-}=1.05$ at the same rate, that is, ${\nu}_{+}={\nu}_{-}=\nu $. The results are shown as connected blue squares in Figure 7a.

**The case of ${\mathit{B}}_{-}<\mathbf{1}$.**So far, the minimum birth rate was chosen to be larger than 1, because in the deterministic limit, the fixed point ${F}_{M}$ loses its stability at $B={B}_{-}=1$. Therefore, as soon as ${B}_{-}\le 1$, the fate of the population is extinction without the need for a large stochastic fluctuation that drives it to extinction via an instanton. In the context of an antimicrobial treatment, the unfavorable environment for the microbial population would correspond to the introduction of biostatic drugs into the environment. Biostatic drugs prevent microbes from growing. When the biostatic drug only reduces the birth/growth rate of the microbes, as discussed in the case above, it is imperfect. Ideally, the biostatic drug should completely stop the growth of the microbes. Therefore, in our simulations, we also studied the case of the birth rate of normals completely going to zero under unfavorable environmental conditions, that is, ${B}_{-}=0$, mimicking the effect of a perfect biostatic drug.

#### 4.4. Deterministic versus Stochastic Changes in the Environment for Square Waves and SDMN

#### 4.5. Duty Cycles with Asymmetric Switching in Competition with Amplitudes

## 5. WKB Approach for the MTE

#### 5.1. Real-Space WKB

#### 5.2. Momentum-Space WKB

## 6. Effects of a Changing Environment: Analytical Approaches

#### 6.1. Sinusoidal Changes in the Environment for Weak Perturbations

#### 6.2. Sinusoidal Changes of the Environment for High Frequencies

## 7. Discussion and Summary of the Results

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Numerical Methods

#### Appendix A.1. Stochastic Simulations

#### Appendix A.1.1. Gillespie Algorithm

#### Time-Independent Rates

- Initialize the algorithm by setting the initial number of normals (n) and persisters (m) and setting $t=0$.
- Calculate the propensity function for each reaction. In the absence of any environmental perturbation, we have four stochastic reactions $i\in \{1,...,4\}$ (birth of normals, death of normals, switching from normals to persisters, and switching from persisters to normals), each with a propensity function ${a}_{i}\in \{Bn(1-n/K),n,\alpha n,\beta m\}$, respectively. In the presence of ADMN, there are two additional stochastic reactions corresponding to environmental switching ${\xi}_{r}\to -{\xi}_{r}$.
- Set ${a}_{0}={\sum}_{i=1}^{M}{a}_{i}$, where $M=4$ for the unperturbed system and $M=6$ for ADMN.
- Generate two random numbers ${r}_{1}$ and ${r}_{2}$ from a uniform distribution $U(0,1)$.
- Find the time until the next reaction should take place, that is, $\Delta t=1/{a}_{0}\mathrm{ln}(1/{r}_{1})$.
- Find the reaction $\mu \in [1,\dots ,M]$ that takes place such that$$\sum _{i=1}^{\mu -1}{a}_{i}<{r}_{2}{a}_{0}\le \sum _{i=1}^{\mu}{a}_{i}.$$
- Set $t=t+\Delta t$ and update the number of normals (n) and persisters (m) according to the reaction $\mu $.
- Return to step 2 or quit.

#### Time-Dependent Rates: Modified Gillespie Algorithm

#### Appendix A.1.2. Modified Next-Reaction Method

- Initialize the algorithm by setting the initial number of normals (n) and persisters (m) and setting $t=0$. For each i, set ${T}_{i}=0$.
- Generate M random numbers ${r}_{i}$ from a uniform distribution ${r}_{i}\in U(0,1)$ and set ${P}_{i}=\mathrm{ln}(1/{r}_{i})$ for each i.
- Calculate $\Delta {t}_{i}$ by solving ${\int}_{t}^{t+\Delta {t}_{i}}{a}_{i}(n\left(t\right),m\left(t\right),s)ds={P}_{i}-{T}_{i}$ for $\Delta {t}_{i}$.
- Set $\Delta t={\mathrm{min}}_{i}\{\Delta {t}_{i}\}$ and let $\Delta {t}_{\mu}$ be the time for which the minimum is realized, that is, let $\Delta {t}_{i}$ be the minimum for the reaction $i=\mu $.
- Increase the time by an increment of $t=t+\Delta t$ and update the number of normals and persisters according to the reaction $\mu $.
- For each i, set ${T}_{i}={T}_{i}+{\int}_{t}^{t+\Delta {t}_{i}}{a}_{i}(n\left(t\right),m\left(t\right),s)ds$.
- For the reaction $\mu $, let r be a uniform random number $r\in U(0,1)$, and set ${P}_{\mu}={P}_{\mu}+\mathrm{ln}(1/r)$.
- Return to step 3 or quit.

#### Appendix A.2. The Adapted Chernykh-Stepanov Iteration Method

## Appendix B. The Kapitsa Method for High Frequencies

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**Figure 1.**Time series of the birth rate $B\left(t\right)$ (

**upper panel**) and sample realization of the total population size N (

**lower panel**) for (

**a**) asymmetric random, (

**b**) periodic rectangular, and (

**c**) sinusoidal perturbations. The parameters are ${B}_{+}=1.5$, ${B}_{-}=1.1$, $\alpha =0.02$, and $\beta =0.02$. For (

**a**,

**b**), $(K,{\nu}_{+},{\nu}_{-})=(500,0.00375,0.00225)$, and for (

**c**), $(K,{B}_{0},\omega ,\u03f5)=(300,1.3,0.007,0.2)$. The initial number of persisters is taken to be 0.3 K/2, and that of normals is 0.7 K/2.

**Figure 2.**The quasi-stationary distribution for SDMN (blue color) and sinusoidal perturbations (orange color) for three values of the environmental switching frequency $\nu =\{0.001,0.1,0.5\}$ (

**a**–

**c**). N is the sum of normals and persisters, measured from 250 sample trajectories for the distribution. Other parameters are ${B}_{+}=1.5$, ${B}_{-}=1.1$, $K=1000$, $\alpha =0.02$, and $\beta =0.02$.

**Figure 3.**Impact of more favorable (

**a**) and more adverse environmental conditions (

**b**) on the quasi-stationary distribution for an ADMN (blue color) and rectangular wave perturbations (orange color): (

**a**) $\nu =0.5$, ${\nu}_{+}=0.25$, and ${\nu}_{-}=0.75$; (

**b**) $\nu =0.5$, ${\nu}_{+}=0.75$, and ${\nu}_{-}=0.25$. N is the sum of normals and persisters, measured from 1200 sample trajectories for the distribution. Otherwise, the conditions are the same as in Figure 2.

**Figure 4.**Leakage of a quasi-stationary distribution of normals and persisters, whose sum is N. In time windows of $t\in \left[100\u2013250\right]$ (

**a**), $t\in \left[350\u2013500\right]$ (

**b**), $t\in \left[500\u2013750\right]$ (

**c**), and $t\in \left[2300\u20132500\right]$ (

**d**), we measure the fraction of 250 sample trajectories, which takes a value N as the sum of normals and persisters that have survived the respective time interval. The environment is modeled as SDMN for $\nu =0.05$. Other parameters are ${B}_{+}=1.2$, ${B}_{-}=1.1$, $K=500$, $\alpha =0.02$, and $\beta =0.02$.

**Figure 5.**(

**a**) Typical non-monotonic dependence of the MTE on the switching rate $\nu $ (or frequency $\omega $) obtained with Gillespie simulations (red dots) for fixed $\alpha $ and $\beta $. The black dashed line shows the value of the MTE for the unperturbed system. The amplitude of perturbation is $\u03f5=0.05$. The dashed green line shows the time $1/\nu $. For MTEs larger than that, the system on average has time to see more than one period of environmental change. (

**b**) Variation in the MTE with the external frequency $\nu $ ($\omega $) for three values of $\u03f5=0.025(\mathrm{black}\mathrm{squares}),0.05(\mathrm{red}\mathrm{stars})$, and $0.075(\mathrm{blue}\mathrm{circles})$. The parameters are $\alpha =\beta =0.02$, $K=500$, $B=1.1$, and 2400 sample trajectories.

**Figure 6.**Gillespie simulations for the MTE as a function of the frequency for (

**a**) different values of $\beta $ and $\alpha =0.02$ and (

**b**) different values of $\alpha $ and $\beta =0.02$. Other parameters are $K=500$, $B=1.1$, $\u03f5=0.05$, and 2400 sample trajectories.

**Figure 7.**(

**a**) MTE as a function of $\nu =\omega /\pi $ when B switches randomly according to SDMN between ${B}_{+}=1.15$ and ${B}_{-}\in \{0,1.05\}$. (

**b**) MTE as a function of $\nu $ when B switches randomly according to SDMN between ${B}_{+}=\{1.2,1.3,1.4,1.5\}$ and ${B}_{-}=0$. Other parameters are $\alpha =0.02$, $\beta =0.02$, and $K=500$ in (

**a**) and $K=1000$ in (

**b**). In the inset, we zoom into $\nu <0.05$.

**Figure 8.**Histogram of escape times (

**a**) for SDMN and (

**b**) for square-wave perturbations. For square waves, all start with the zero phase, for SDMN, half start with +1 and the other half with −1. Parameters are ${B}_{+}=1.15$, ${B}_{-}1.05$, $\alpha =0.02$, $\beta =0.02$, and $K=500$.

**Figure 9.**Variation in the MTE with $\nu $ for a few cases with different duty cycles $\gamma $ and birth rates: (

**a**) ${B}_{+}=1.15,{B}_{-}=0.5$; (

**b**) ${B}_{-}=0.95,{B}_{av}=1.055$. The switching rates are $({\nu}_{+},{\nu}_{-})=(2\nu [1-\gamma ],2\nu \gamma )$. Other parameters are $\alpha =0.02$, $\beta =0.02$, and $K=500$.

**Figure 10.**Variation in the MTE as a function of the frequency of the perturbation predicted by the linear theory for $K=500$, $\alpha =\beta =0.02$, $B=1.1$, and $\u03f5=0.05$. The maximum reduction in the MTE due to the perturbation in the birth rate is observed at small frequencies. For high frequencies ($\omega >>\delta $), the system sees an average of the environmental perturbation, which equals the unperturbed case, and thus, the MTE approaches the unperturbed value.

**Figure 11.**(

**a**) Correction to the action $\Delta S$, rescaled by the unperturbed action ${S}_{0}$ for three values of $\alpha $, while $\beta =0.02$. As $\alpha $ increases, the correction to the action approaches zero more rapidly; that is, the range of frequencies for which the environmental perturbation has a significant effect on the system decreases as $\alpha $ increases. (

**b**) The same as in (

**a**), but for three values of $\beta $ for $\alpha =0.02$. Here, as $\beta $ increases, the range of frequencies for which the environmental perturbation has a significant effect on the system increases together with $\beta $.

**Figure 12.**(

**a**) Logarithmic susceptibility ${\chi}_{S}$ as a function of $\beta $ for four values of $\omega $ with a single peak in ${\chi}_{S}$ for $\alpha =0.02$. (

**b**) The same as (

**a**), but as a function of $\alpha $ for three values of $\omega $ with a single peak in ${\chi}_{S}$ for $\beta =0.02$.

**Figure 13.**MTE as obtained from the Kapitsa approximation (red dots) as a function of the frequency in comparison to (

**a**) the unperturbed case (blue squares) and (

**b**) the linear approximation (blue squares). Parameters are $K=500$, $\alpha =\beta =0.02$, $B=1.1$, and $\u03f5=0.05$.

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Thakur, B.; Meyer-Ortmanns, H.
Controlling the Mean Time to Extinction in Populations of Bacteria. *Entropy* **2023**, *25*, 755.
https://doi.org/10.3390/e25050755

**AMA Style**

Thakur B, Meyer-Ortmanns H.
Controlling the Mean Time to Extinction in Populations of Bacteria. *Entropy*. 2023; 25(5):755.
https://doi.org/10.3390/e25050755

**Chicago/Turabian Style**

Thakur, Bhumika, and Hildegard Meyer-Ortmanns.
2023. "Controlling the Mean Time to Extinction in Populations of Bacteria" *Entropy* 25, no. 5: 755.
https://doi.org/10.3390/e25050755