Stochastic Modeling of Bacterial Population Growth with Antimicrobial Resistance

In this paper we consider a stochastic model of bacterial population growth with antimicrobial resistance under the influence of random fluctuations. We analyze the model for the problem of persistence and extinction of bacterial cells. This analysis shows asymptotic extinction and conditional persistence for growing population. Moreover, we perform computer simulations in order to illustrate the model behavior. The model results have important implications for the eradication of bacterial cells and the emergence of resistance.

• We proposed this stochastic model based on the deterministic within-host model [1] of antibiotic resistance, which consists of the two/three equations for the strains of bacteria. • This simple model is able to capture the stochastic dynamics of bacterial population with antimicrobial resistance. Also, of that mentioned above, it is different from that for instance in [4] where the Markovian birth-and-death model has been used to demonstrate the stochastic nature of eradicating bacteria with antibiotics using the master equation. • Here, we studied the longtime behavior to obtain a threshold for extinction and persistence of bacterial cells.
The advantages of this model are (i) it demonstrate the stochastic effect of eradicating bacteria with antibiotics and (ii) it has significant implications for the prediction of treatment outcomes.
In practice, the results obtained enable us to understand the stochastic dynamics of the bacterial cells and effects of the anti-microbial treatments, which also gives us insights on how to eliminate bacterial cells. As a result, for instance we obtain a threshold which depends on the intrinsic growth rate, the antimicrobial rate and the noise intensity rate. Regarding the threshold, we find that decreasing the intrinsic growth rate of bacterial cells or increasing the noise intensity rate or the antimicrobial rate lead to the elimination of bacterial cells.
The rest of this paper is organized as follows. In Sect. 2, we introduce the stochastic differential equation model. In Sect. 3, we analyze the steady probability distribution. In Sect. 4, we analyze the extinction and persistence of bacterial cells. In Sect. 5 we verify and illustrate the model behavior by computer simulations. Section 6 contains a conclusion.

The Stochastic Differential Equation Model
In this section we introduce a stochastic model for bacterial growth with antibiotics. Different deterministic differential equation models for bacterial growth have been proposed and studied. Commonly used are logistic type growth models where x denotes the bacterial population density at time t, α is the per capita maximum fertility rate of population, β denotes the strength of intra-competition of population. During antimicrobial resistance the bacterial growth kinetics is perturbed by the antibiotic drug, which can be described by a loss term −γ x, where γ is the antimicrobial rate. Taking into account the effect of antimicrobial resistance leads to the model To study the bacterial evolution in stochastic environment, we assume that random fluctuations, such as rapid environmental changes, affect the system through external parameters.
In this case, we suppose that, as random perturbations, the intrinsic growth parameter α in model (2) is regarded as a random variable in the form where α is the mean intrinsic growth rate, ξ is Gaussian white noise, and σ is the intensity of the noise. Then, equation (2) is replaced by stochastic differential equation for the random process X in the form where dW (t) = ξ(t)dt. Here, W is defined on complete probability space ( , F , P) equipped with the natural filtration (F t ) t≥0 associated to the Wiener process. It is assumed that the variable x is dimensionless, i.e., the equation describes changes in relative bacterial population size.

Analysis for Steady Probability Distribution
In this section we analyze the steady state probability density of bacterial cells. First, we observe that the deterministic model can be written where V is the potential function defined by where η = α − γ , see Fig.1. The potential function has a local minimum corresponding to the stable equilibrium and a local maximum at x = 0, which is an unstable equilibrium if α − γ > 0. For α − γ > 0, the bacterial population converges to the stable equilibrium x s = (α − γ )/β. We observe that the population x(t) tends to 0, the state of extinction, namely there is not bacterial cells, if and only if α/γ ≤ 1; while in the stable state α/γ > 1, namely the bacterial cells exist and stay at a certain level. Thus, the system is bistable and the bistable feature of system depends on the parameter γ . Now, the transitional probability density P(x, t) satisfies the corresponding Fokker-Planck equation Then, the steady state probability P s (x) density can be obtained from (5) as in [24] and can be written as where N s is a normalizing constant. Note that the probability density (6) is normalized for η => σ 2 /2. At σ 2 /2 < η < σ 2 the probability density is divergent at x = 0, and we have a nose induced phase transition at the point η = σ 2 , see Fig. 2. Next, we simulate the steady-state probability P s (x) density of bacterial cells for different noise intensities σ 2 and antimicrobial parameter γ . In Fig. 2, we plot the steady-state proba- bility P s (x) density for given parameters α, β, γ and different noise intensities σ 2 . In Fig. 3, we plot the steady-state probability P s (x) density varying with antimicrobial parameter and fixed σ 2 . Clearly, the extinction of bacterial cells increases with increase of γ and is the more probable for large σ 2 .

Analysis for Extinction and Persistence
In this section we analyze longtime behavior using methods in [25] and present analytical results concerning the extinction and persistence for the model Eq. (4).

Extinction
In this subsection we will discus the extinction of the system (4).
namely, X (t) tends to zero exponentially almost surely.
Proof From the Ito formula, we have where f : R → R is defined by However, under condition (7), we have for X (s) ∈ (0, α). It then follows from (9) that This implies lim sup But by the larger number theorem for martingales, we have lim sup We therefore obtain the desired assertion (8) from (12).
then for any given initial values X (0) = X 0 ∈ (0, α), the solution of the model (2) obeys lim sup namely, X (t) tends to zero exponentially almost surely.
Proof From the Ito formula, we have It then follows that In the same way as in the proof of Theorem 1, this implies that lim sup Hence, the proof is complete.

Persistence
In this subsection we will discus the persistence of the model system (4).

Theorem 3 If
then for any given initial values X (0) = X 0 ∈ (0, α), the solution of the model (2) obeys where which is the unique root in (0, α) of That is, X (t) will rise to or above the level ξ infinitely often with probability one.
Proof We begin to prove assertion (19). If it is not true, then there is a sufficiently small ∈ (0, 1) such that It therefore follows from (24) that Now, fix any ω ∈ 1 ∩ 2 . It then follows from the Ito formula and (25) that, for t ≥ T (ω), This yields lim inf But this contradicts (24). We therefore must have the desired assertion (19).

Computer Simulations
In this section we use the Euler-Maruyama method [26] with the time step 10 −2 and present computer simulations in order to illustrate the model behavior particularly for the extinction and persistence of the bacterial cells. According to the conditions in Theorems 1, 2 and 3, the extinction and persistence of bacterial cells rely on the parameter space σ and γ . Figure 4 shows the extinction of bacterial cells according to the conditions in Theorem 1 for the parameters α = 0.05, β = 0.004, γ = 0.035 and σ = 0.2. Figure 5 shows the same result according to the conditions in Theorem 2, with keeping the parameters the same but let σ = 0.4. To illustrate the result in Theorem 3, we keep the same parameter values except σ is reduced to 0.15 from 0.2. The computer simulation in Fig. 6 shows this result, showing clearly fluctuation around the level ζ = 0.9375. To further illustrate the effect of the noise intensity, we keep all the parameter values unchanged but reduce σ to σ = 0.04. In Fig. 7, we show the resulting computation simulation, illustrating clearly the increase of the level ζ = 3.55 and persistence. Figure 8 shows a computer simulation of the distribution of the solution X (t) in the persistent case for higher and lower σ . In this figure we plot histograms, showing the distribution of X (t) in the case of σ = 0.15, 0.1, 0.05 and 0.025.

Conclusion
In this paper, based on the deterministic model equations in [1], we have considered a stochastic model for the dynamics of bacterial population with antimicrobial resistance under the influence of random fluctuations. We have used the technique of parameter perturbation and investigated the effect of multiplicative noise on the evolutionary dynamics. We first have evaluated the steady state probability density of bacterial cells for different noise intensities and antimicrobial intensities. Then, we have studied the longtime behavior and obtained necessary conditions (a threshold) for extinction and persistence of bacterial cells. Further, the model behaviors ware illustrated by computer simulations. The obtained results demonstrate the growth dynamics of the bacterial population which was controlled by the antimicrobial rate and the noise strength rate. Regarding the results of the threshold, it can be used to analyze the drug resistance observed in evolving and variable bacterial cell population.
Overall, the principal theoretical implication of this study is that the stochastic model is capable to define the macroscale properties of the dynamics of bacteria with antimicrobial resistance, capturing stochastic growth, and identifying the specific response to antibiotic treatment. This may be important for the use of therapeutic purposes.
Author Contributions I contribute this work.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). This research did not receive any funding.
Data Availability Not applicable.

Conflict of interest
The author declares no conflict of interest concerning the publication of this manuscript.

Ethical Approval Not applicable.
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