Ponente
Descripción
Antibiotic resistance is currently recognized as one of the most important global health challenges. Antibiotic resistance occurs when bacteria develop defenses that allow them to survive despite exposure to antibiotics, rendering standard treatments ineffective.
Understanding the exact bacterial response to antibiotic treatment remains a fundamental challenge due to the emergence of phenotypic heterogeneity and persistence, which lead to non-exponential and history-dependent killing dynamics. An experimental setting aimed at understanding Escherichia coli persistence under antibiotic exposure was developed at the Escuela Nacional de Ciencias Biológicas of the Instituto Politécnico Nacional of México. This setting provides qualitative and quantitative information on the influence of antibiotic treatment on E.coli, including the emergence of biphasic killing curves, a hallmark of persistence phenomena (Ponce de León-Rosales, S., PUCRA Report, 21-23, 2025).
Fractional-order mathematical models aim to describe complex phenomena that exhibit memory effects that traditional integer-order differential equation models cannot capture (R. L. Magin, Comput. Math. Appl. 59, 1586–1593, 2010). In these models, memory is quantified through the order of the fractional derivative $\alpha$, which determines the strength of the system’s dependence on its past states (M. Du, Z. Wang, H. Hu, Sci. Rep. 3, 3431, 2013). Therefore, in an effort to describe past influence of bacteria dynamics under the influence of antibiotics, we establish a fractional-order McKendrick–von Foerster model to describe the evolution of E. coli population density under antibiotic treatment. The proposed population density evolves according to a transport equation in which a Caputo fractional-order derivative arises from history-dependent cellular responses to antibiotic stress. We start by considering a non-local mortality interaction in the McKendrick-von Foerster equation:
\begin{equation}
\frac{\partial n(a,t)}{\partial t}+\frac{\partial n(a,t)}{\partial a}=-\int_{-\infty}^{t}\mu(a,\tau)K(t-\tau,\alpha)n(a,\tau)d\tau.
\end{equation}
The model incorporates antibiotic-induced mortality that depends on the entire history of the population, consistent with experimental evidence that antibiotic-induced cellular damage (e.g., DNA damage, protein misfolding or oxidative stress) potentially accumulates over time and leads to delayed cell death (Das T, Nath C, Das P, Ghosh K, Logno TA, Debnath P, et al., PLOS ONE, 1-16, 2023). The kernel $K(.)$ represents how past exposure contributes to present mortality. Considering a power-law kernel $K(t)\propto t^{-\alpha}$ and under convenient assumptions, equation (1) can be established as a fractional-order differential equation in which the spatial extent of the population and the diffusive concentration in the culture medium are also considered. We establish the well-posedness of the model in a suitable functional framework, analyze qualitative properties of solutions, and develop numerical simulations of the model. Importantly, we relate the fractional-order history-dependent model dynamics to the experimental observations. Ultimately, this approach might provide a predictive tool to quantify antimicrobial efficacy beyond traditional static endpoints, offering a scientific basis to optimize dosing regimens and understand the memory-dependent survival strategies of E. coli under antibiotic stress.