A novel mathematical model for predicting the nonlinear spread of antimicrobial resistance in hospital environments using stochastic compartmental dynamics

Introduction

Can the unpredictable nature of bacterial transmission inside a hospital be tamed by mathematics? That question has motivated an expanding body of work at the boundary of applied mathematics and infectious disease epidemiology. Antimicrobial resistance the ability of micro-organisms to withstand drugs that once killed them was linked to an estimated 700,000 deaths per year worldwide according to the O'Neill Review [1] . Hospital settings are particularly vulnerable because patients with weakened immune systems share confined spaces, healthcare workers move between beds, and antibiotic use is intensive. Traditional compartmental models such as the SIR and SIS frameworks have been adapted for hospital-acquired infections by several authors [2,3] . Deterministic versions assume large, wellmixed populations and continuous state variables, assumptions that break down in a single ward of 30 to 50 patients where chance events an ungloved hand contact, a missed cleaning cycle can ignite or extinguish an outbreak. Stochastic extensions address this gap. Lipsitch and colleagues [5] introduced random patient admission and discharge into a colonisation model, while Allen [4] provided a general framework for continuous-time Markov chain epidemic models. Despite these advances, most stochastic hospital models retain linear incidence terms and constant transmission rates. Real transmission, however, is nonlinear: contact rates saturate at high patient densities, and healthcare worker behaviour changes in response to visible outbreaks [6] . And the transmission coefficient itself drifts over time due to staff turnover, seasonal cleaning protocols, and shifting antibiotic prescribing habits. This research bridges that gap by constructing a fivecompartment stochastic differential equation (SDE) model SCIRD-AMR with a saturating incidence function and a timedependent transmission rate perturbed by Brownian motion. We prove the existence and uniqueness of a positive global solution, derive the stochastic basic reproduction number, and calibrate the model with ward-level surveillance data from a tertiary hospital in New Delhi. Monte Carlo simulations then compare five plausible intervention strategies. The aim is to give hospital infection-control teams a practical, mathematically grounded tool for forecasting AMR dynamics under uncertainty.

Theoretical Background

The model rests on three mathematical pillars. First, compartmental dynamics partition the hospital patient population into disjoint classes S, C, I, R, D with transition rates governed by ordinary differential equations in the deterministic limit. Second, stochastic perturbation is introduced through an Ito-type noise term appended to the transmission coefficient, so that beta(t) = beta_0 + sigma * dW(t)/dt, where W(t) is a standard Wiener process and sigma controls noise intensity [4] . Third, the incidence function takes the Holling type-II form beta * S * I / (1 + kappa * I) to capture saturation when infected counts are high, a feature absent from standard bilinear incidence [8] . Lyapunov function analysis provides the main stability tool. By constructing a non-negative function V(S,C,I,R) that decreases in expectation along solution trajectories, one can establish almost-sure boundedness and, under tighter conditions, extinction or persistence of the infected compartment [4,12] . The stochastic reproduction number R_s, which accounts for noise variance, replaces the deterministic R_0 and satisfies R_s < R_0 whenever sigma > 0 noise tends to hinder rather than help pathogen persistence [13] .

Material and Methods

Material

Surveillance data were obtained from the infection-control unit of a 350-bed tertiary care hospital affiliated with the University of Delhi, New Delhi, India. The dataset covered the period January 2019 to June 2019 and recorded weekly counts of new AMR-positive isolates from blood cultures, wound swabs, and urine samples across three medical wards.

Patient admission and discharge records for the same wards supplied occupancy and turnover data. Bacterial species included methicillin-resistant Staphylococcus aureus, extended-spectrum beta-lactamase-producing Escherichia coli, and carbapenem-resistant Klebsiella pneumoniae. Ethical clearance was granted by the Institutional Ethics Committee of the University of Delhi (Protocol IEC/2018/MAT-047, approved 12 November 2018). Patient identifiers were removed before the data reached the modelling team; hence individual informed consent was waived.

Methods

The SCIRD-AMR model consists of five coupled stochastic differential equations. Transition rates were estimated from the literature and refined by maximum-likelihood fitting to the observed weekly case counts using the Euler-Maruyama discretisation scheme with a time step of 0.01 days. The saturating incidence parameter kappa was estimated by grid search over [0, 0.5]. Model fitting used data from weeks 1 through 8 (training set); weeks 9 through 12 served as the validation set. Monte Carlo simulations were performed in MATLAB R2019b on a desktop workstation at the Department of Mathematics, University of Delhi. Each scenario was run for 10,000 independent realisations, and summary statistics (mean trajectory, 95 percent prediction intervals) were computed from the ensemble. Five scenarios were tested: (i) baseline no additional intervention; (ii) enhanced hand hygiene, reducing beta_0 by 30 percent [7] ; (iii) patient isolation, reducing contact rate by 50 percent for detected carriers; (iv) combined hand hygiene and isolation; and (v) antibiotic stewardship, reducing colonisation probability alpha by 20 percent. The effective reproduction number R_eff was calculated at each time step from the next-generation matrix adapted for the stochastic setting [14] . Goodness-of-fit was assessed by root-mean-square error and the coefficient of determination R-squared.

Simulation Parameters

All simulations began from the initial condition S(0) = 320, C(0) = 18, I(0) = 7, R(0) = 5, consistent with the average ward census and baseline prevalence on 7 January 2019. The Wiener process was discretised at the same 0.01-day time step. Random number generation used the Mersenne Twister algorithm seeded independently for each realisation to ensure reproducibility. The parameter set is given in Table 1. Sensitivity of peak case counts to sigma was assessed by varying the noise intensity from 0.005 to 0.05 in steps of 0.005. [3] Colonisation rate α 0.024 per day Lipsitch et al. [5] Recovery rate γ 0.071 per day Bonten et al. [7] Discharge rate (C) μ 0.143 per day Pelupessy et al. [9] Discharge rate (I) δ 0.091 per day Austin et al. [10] Discharge rate (R) ν 0.167 per day Cooper et al. [11] Noise intensity σ 0.015 dimensionless Estimated Baseline pop. (N) N 350 patients Hospital data *HH = hand hygiene compliance rate

Results

The calibrated model tracked the observed outbreak trajectory closely. Over the eight-week training window, the mean simulated case count deviated from the observed count by at most 1.8 cases in any single week, and the root-mean-square error across all training weeks was 1.14 cases. On the fourweek validation set the RMSE rose modestly to 1.37, and the R-squared value was 0.94, indicating that the model captured 94 percent of the week-to-week variance in AMR incidence. Table 2 summarises the outcomes of the five intervention scenarios. Under baseline conditions, the outbreak peaked at week 6 with a mean of 31 infected patients and an effective reproduction number of 2.73. Enhanced hand hygiene alone reduced the peak to 19 cases and shortened the outbreak by two weeks. Patient isolation was more effective, bringing the peak to 14 cases. But only the combined strategy drove R_eff below one (0.91), extinguishing the outbreak within seven weeks. Antibiotic stewardship lowered total cases by roughly 18 percent relative to baseline but did not bring R_eff below unity. the line shows the mean predicted trajectory from the stochastic model.

Comprehensive Interpretation

Sensitivity analysis revealed that peak case count rose by roughly 11 percent when sigma increased from 0.005 to 0.05, confirming that environmental stochasticity amplifies outbreak severity. The 95 percent prediction interval widened from plus-or-minus 3.2 cases at sigma = 0.005 to plus-or-minus 8.7 cases at sigma = 0.05. The saturating incidence parameter kappa was estimated at 0.12 (95 percent CI: 0.08 to 0.17), indicating moderate density dependence in contact patterns.

Discussion

The SCIRD-AMR framework offers several advantages over existing hospital AMR models. First, the saturating incidence function accounts for behavioural feedbacks that limit transmission at high prevalence a phenomenon documented by Bootsma and colleagues [6] but rarely formalised in stochastic models. Second, the Brownian-motion perturbation of beta provides a parsimonious way to represent the many small, unmeasured factors that cause day-to-day variation in transmission risk.

Our finding that single interventions failed to push R_eff below one aligns with the conclusions of Robotham and colleagues [15] , who used a deterministic model to argue for bundled infection-control strategies. The stochastic perspective adds nuance: even under the combined strategy, 4.3 percent of realisations produced secondary peaks, suggesting that post-intervention surveillance must remain vigilant.

Several limitations deserve mention. The model treats the hospital as a single well-mixed unit, ignoring ward-to-ward transfer. Data from a single hospital in New Delhi may not generalise to other settings with different staffing ratios or resistance profiles. And the six-month calibration window limits our ability to capture seasonal trends. Future extensions could incorporate spatial structure through a network of coupled wards and integrate genomic data to distinguish imported from locally acquired resistance.

Conclusion

This research introduced a stochastic compartmental model SCIRD-AMR for predicting nonlinear AMR spread in hospitals. The model incorporates time-varying transmission, saturating incidence, and Brownian-motion perturbations, features that together yield a realistic picture of outbreak uncertainty. Calibration against real surveillance data from a New Delhi hospital produced strong predictive accuracy (Rsquared = 0.94). Intervention simulations demonstrated that hand hygiene improvements and patient isolation must be combined to bring the effective reproduction number below one. Antibiotic stewardship helps but is insufficient alone. Hospital infection-control programmes should adopt bundled strategies and maintain post-intervention monitoring, as residual stochastic risk can reignite outbreaks even after apparent control. The SCIRD-AMR framework is freely available and can be adapted to other pathogens and hospital configurations, making it a practical decision-support tool for clinicians and public-health officials.

Acknowledgement

AcknowledgementsWe gratefully acknowledge the support provided by the authorities for this research.

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