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A comparative study of three mathematical approaches applied to the reversal of AMR

Sebastian Builes, Jhoana P Romero-Leiton, Leon A. Valencia

Abstract

In this work, we study the qualitative properties of a simple mathematical model inspired by antimicrobial resistance (AMR), focusing on the reversal of resistance. In particular, we analyze the model from three perspectives: ordinary differential equations (ODEs), stochastic differential equations (SDEs) driven by Brownian motion, and fractional differential equations (FDEs) with Caputo temporal derivatives. Finally, we perform numerical experiments using data from Escherichia coli exposed to colistin to assess the validity of the qualitative properties of the model.

A comparative study of three mathematical approaches applied to the reversal of AMR

Abstract

In this work, we study the qualitative properties of a simple mathematical model inspired by antimicrobial resistance (AMR), focusing on the reversal of resistance. In particular, we analyze the model from three perspectives: ordinary differential equations (ODEs), stochastic differential equations (SDEs) driven by Brownian motion, and fractional differential equations (FDEs) with Caputo temporal derivatives. Finally, we perform numerical experiments using data from Escherichia coli exposed to colistin to assess the validity of the qualitative properties of the model.

Paper Structure

This paper contains 11 sections, 13 theorems, 53 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical model formulation
  3. Theoretical results
  4. Defining thresholds
  5. The deterministic approach
  6. The stochastic approach
  7. The fractionary approach
  8. Numerical experiments
  9. Discussion
  10. Appendix
  11. Stability

Key Result

Theorem 3.1

For any $R_0 \in (0,N)$, there exists a unique global solution of (model-det) invariant in $(0,N).$

Figures (5)

  • Figure 1: Extinction scenery for the three approaches and different values of $\gamma=$1, 1.25, 1.5, 1.75, 2. For the stochastic case the perturbation parameter was $\sigma=1e-6$. For the fractional case the derivative order was $\alpha=0.7$. The initial condition $R(0)=N-1$.
  • Figure 2: Persistence scenery for the three approaches and different values of $\gamma=$0, 0.1, 0.2, 0.3, 0.4. For the stochastic case the perturbation parameter was $\sigma=1e-6$. For the fractional case the derivative order was $\alpha=0.7$. The initial condition $R(0)=1$.
  • Figure 3: Deterministic and stochastic comparison for different values of $\sigma$=1e-6, 2e-6, 3e-6, 4e-6, 5e-6. Here $\gamma=2$ and $R(0)=N-1$ for extinction, and $\gamma= 0$ and $R(0)=1$ for persistence.
  • Figure 4: Fractional approach for different values of the order of the derivative $\alpha$=0.5, 0.6, 0.7, 0.8. Here $\gamma=1.5$ and $R(0)=N-1$ for extinction, and $\gamma= 0.2$ and $R(0)=1$ for persistence.
  • Figure 5: Stationary mean (histograms) of the stochastic model for different values of $\gamma$ associated with the persistence of resistant bacteria. The parameter values used are those from Table \ref{['table1']}. Additionally, $\sigma=1e-7$ and the initial condition $R(0)=N-1$.

Theorems & Definitions (32)

  • Theorem 3.1: Invariance
  • proof
  • Theorem 3.2: Asymptotic stability
  • proof
  • Theorem 3.3: Extinction
  • proof
  • Theorem 3.4: Persistence
  • proof
  • Theorem 3.5: Invariance
  • proof
  • ...and 22 more