[Paper Review] Bifurcation analysis and global dynamics of a mathematical model of antibiotic resistance in hospitals

Xiuli Cen, Zhilan Feng+2 more|arXiv (Cornell University)|Sep 27, 2016

Mathematical and Theoretical Epidemiology and Ecology Models1 references3 citations

TL;DR

This paper presents a rigorous analytical study of a hospital antibiotic resistance transmission model that includes superinfection, introducing the invasion reproduction number ℛₐᵣ and proving a backward bifurcation can occur at ℛₐᵣ = 1 when superinfection is present. The key finding is the existence of a critical threshold ℛₐᵣᶜ < ℛₐᵣ < 1, enabling a bistable phenomenon with two interior equilibria, which complicates control efforts and highlights the importance of targeting specific parameters to prevent resurgence of resistant strains.

ABSTRACT

Antibiotic-resistant bacteria has posed a grave threat to public health by causing a number of nosocomial infections in hospitals. Mathematical models have been used to study the transmission of antibiotic-resistant bacteria within a hospital and the measures to control antibiotic resistance in nosocomial pathogens. Studies presented in \cite{LBL,LB} have shown great value in understanding the transmission of antibiotic-resistant bacteria in a hospital. However, their results are limited to numerical simulations of a few different scenarios without analytical analysis of the models in all biologically feasible parameter regions. Bifurcation analysis and identification of the global stability conditions are necessary to assess the interventions which are proposed to limit nosocomial infection and stem the spread of antibiotic-resistant bacteria. In this paper we study the global dynamics of the mathematical model of antibiotic resistance in hospitals in \cite{LBL,LB}. The invasion reproduction number $\mathcal R_{ar}$ of antibiotic-resistant bacteria is introduced. We give the relationship of $\mathcal R_{ar}$ and two control reproduction numbers of sensitive bacteria and resistant bacteria ($\mathcal R_{sc}$ and $\mathcal R_{rc}$). More importantly, we prove that a backward bifurcation may occur at $\mathcal R_{ar}=1$ when the model includes superinfection which is not mentioned in \cite{LB}. That is, there exists a new threshold $\mathcal R_{ar}^c$, and if $\mathcal R_{ar}^c

Motivation & Objective

• To analytically investigate the global dynamics of a hospital antibiotic resistance model that includes superinfection, extending prior numerical-only studies.
• To identify conditions under which a backward bifurcation occurs, which can lead to multiple endemic equilibria even when ℛₐᵣ < 1.
• To determine the dependence of the critical threshold ℛₐᵣᶜ on key epidemiological and treatment parameters, such as patient admission rates, treatment usage, and transmission rates.
• To provide analytical criteria for effective control interventions that can eliminate or suppress antibiotic-resistant bacteria in hospital settings.

Proposed method

• Formal bifurcation analysis is conducted on a system of ordinary differential equations modeling the transmission dynamics of sensitive and resistant bacterial strains in a hospital.
• The invasion reproduction number ℛₐᵣ is derived to assess the potential for resistant strain spread, and its relationship to control reproduction numbers ℛₛc and ℛᵣc is established.
• Planar reduction of the system is performed by substituting X = 1 - S - R, simplifying the model to two equations for S and R.
• Analytical proofs are used to demonstrate the existence of a backward bifurcation at ℛₐᵣ = 1 when superinfection (σ > 0) is included, leading to multiple interior equilibria in the range ℛₐᵣᶜ < ℛₐᵣ < 1.
• Sensitivity and parameter dependence analyses are performed using numerical simulations to explore how ℛₐᵣᶜ varies with m, μ, τ₁, τ₂, β, σ, and c.
• Latin Hypercube Sampling is applied for sensitivity analysis to rank the impact of parameters on the frequency of resistant colonization.

Experimental results

Research questions

• RQ1 Under what conditions does a backward bifurcation occur in the antibiotic resistance model, and how does superinfection influence this phenomenon?
• RQ2 What is the relationship between the invasion reproduction number ℛₐᵣ and the control reproduction numbers ℛₛc and ℛᵣc?
• RQ3 How does the critical threshold ℛₐᵣᶜ affect the existence of multiple endemic equilibria, and what are the implications for disease control?
• RQ4 Which model parameters most significantly influence the size of the threshold ℛₐᵣᶜ, and how can they be targeted to prevent persistent transmission?

Key findings

• A backward bifurcation occurs at ℛₐᵣ = 1 when superinfection (σ > 0) is included, leading to the possibility of multiple interior equilibria for ℛₐᵣᶜ < ℛₐᵣ < 1.
• The critical threshold ℛₐᵣᶜ increases with the proportion of admitted patients already colonized with sensitive bacteria (m), reducing the window for bistability.
• ℛₐᵣᶜ decreases with increasing primary transmission rate β, but only exists above a threshold β*, indicating a non-monotonic dependence.
• The treatment rate of drug 1 (τ₁) has the strongest positive influence on the frequency of resistant colonization, while drug 2 (τ₂) has a negative correlation.
• Sensitivity analysis shows τ₁ is the most influential parameter on resistant strain frequency, followed by σ and β, highlighting its central role in control strategies.
• The model exhibits significant differences in global dynamics when m = 0 versus m > 0, with the latter enabling richer dynamics including bistability.