A model of antibiotic-resistant bacterial epidemics in hospitals

Abstract

The emergence of drug-resistant strains of bacteria is an increasing threat to society, especially in hospital settings. Many antibiotics that were formerly effective in combating bacterial infections in hospital patients are no longer effective because of the evolution of resistant strains, which compromises medical care worldwide. In this article, we formulate a two-level population model to quantify key elements in nosocomial (hospital-acquired) infections. At the bacteria level, patients infected with these strains generate both nonresistant and resistant bacteria. At the patient level, susceptible patients are infected by infected patients at rates proportional to the total bacteria load of each strain present in the hospital. The objectives of this paper are to analyze the dynamic elements of nonresistant and resistant bacteria strains in epidemic populations in hospital environments and to provide understanding of measures to avoid the endemicity of resistant antibiotic strains.

Fig. 1.

Flow diagram of bacteria populations in an infected patient. (A) Plasmid-free bacteria in a single infected host before antibiotic treatment. Here, log(2)/β_F is the doubling time, and β_Fκ_F is the carrying capacity of the bacteria population. (B) Plasmid-free and plasmid-bearing bacteria in a single infected host before antibiotic treatment. Here log(2)/β_- is the doubling time of the plasmid-free strain, log(2)/β₊ is the doubling time of the plasmid-bearing strain, τ is the recombination rate, and γ is the reversion rate.

Fig. 2.

Simulation of Eq. 1. Plasmid-free bacteria population in a single infected host before and during antibiotic treatment. Here β_F = 12.0 log(2) (the doubling time is 2 h before treatment), β_F = -2.0 after treatment, and κ_F = 10¹⁰. Treatment starts at day 5 and lasts 15 days. Treatment is successful in eliminating the nonresistant bacterial strain.

Fig. 3.

Simulation of system 2 with β_- = 8.0 × log(2), β₊ = 4.0× log(2), γ = 0.00001, τ = 0.001, κ_F = 10¹⁰, and σ = -2.77 < 0. The plasmid-free [V^-(a)] population has lim_a→∞ V^- (a) = 5.55 × 10¹⁰, and the plasmid-bearing [V⁺(a)] population has lim_a→∞ V⁺(a) = 0.

Fig. 4.

Simulation of system 2 with β_- = 8.0 × log(2), β₊ = 9.0 × log(2), γ = 0.00001, τ = 0.001, κ_F = 10¹⁰ and σ = 0.69 > 0. The plasmid-free [V^-(a)] population has lim_a→∞ V^-(a) = 1.27 × 10⁶, and the plasmid-bearing [V⁺(a)] population has lim_a→∞ V⁺(a) = 6.23 × 10¹⁰.

Fig. 5.

Flow diagram of epidemic populations in the hospital. Susceptible patients acquire nonresistant or resistant bacteria at infection age 0 at time t.

Fig. 6.

Parametric determination of epidemics in the hospital. The admission rate λ, the average length of stay of uninfected patients 1/ν, the infection rate parameter η, and the average patient bacterial loads H_F = T_F/L_N (nonresistant strain) and H_R = T_V+/L_R (resistant strain) determine three possible epidemic outcomes.

Fig. 7.

Simulation of system 3 when parameters lie in Region I. (Upper) Treatment starts on day 3 and lasts 21 days for patients undergoing treatment. β_F = 12.0 × log(2) before treatment and -2.0 during treatment for patients infected only with the nonresistant strain. β_- = 8.0 × log(2) before treatment and -2.0 during treatment and β₊ = 4.0 × log(2) before and during treatment for patients infected with the resistant strain. γ = 0.00001, τ = 0.001, κ_F = 10¹⁰. Treatment eliminates the nonresistant strain in patients with only this strain but not in patients with the resistant strain. (Lower) λ = 5.0, ν = 0.05, η = 3.0 × 10^-14, μ_N = 0.05, μ_R = 0.04, T_F = 1.31 × 10¹¹ < T_V- = 2.84 × 10¹¹, T_V+ = 4.44 × 10¹¹, and R₀ = 0.85 < 1. (S(t), I_N(t), I_R(t)) converges to the steady state (98,0,0).

Fig. 8.

Simulation of system 3 when parameters lie in Region II. (Upper) Treatment starts on day 5 and lasts 14 days for patients undergoing treatment.β_F = 12.0 × log(2) before treatment and -2.0 during treatment for patients infected with the nonresistant strain. β_- = 8.0 × log(2) before treatment and -2.0 during treatment, β₊ = 4.0 × log(2) before and during treatment for patients infected with the resistant strain. γ = 0.00001, τ = 0.001, κ = 10¹⁰_F. The nonresistant strain is present in patients with the resistant strain because of the reversion of plasmid-bearing to plasmid-free bacteria. (Lower) λ = 5.0, ν = 0.05, η = 5.0 × 10^-14, μ_N = 0.05, μ_R = 0.04, T_F = 2.62 × 10¹¹> T_V + = 1.76 × 10¹¹, T_V - = 6.30 × 10¹¹, and R₀ = 1.31 > 1. The nonresistant strain becomes endemic, and the resistant strain is eliminated. (S(t), I_N(t), I_R(t)) converges to the steady state (76,24,0).

Fig. 9.

Simulation of system 3 when parameters lie in Region III. (Upper) Treatment starts on day 1 and lasts 7 days for patients undergoing treatment. β_F = 12.0 × log(2) before treatment and -0.8 during treatment for patients infected with the nonresistant strain. β_- = 3.0 × log(2) before treatment and -2.0 during treatment, β₊ = 3.0 × log(2) before and during treatment for patients infected with the resistant strain. γ = 0.00001, τ = 0.001, κ = 10¹⁰_F. (Lower) λ = 5.0, ν = 0.05, η = 5.0 × 10^-14, μ_N = 0.05, μ_R = 0.05, T_F = 5.62 × 10⁶< T_V+ = 2.48 × 10¹¹, T_V- = 5.48 × 10⁷, and R₀ = 1.24 > 1. The population of patients infected only with the nonresistant strain is effectively eliminated, and patients infected with the resistant strain completely dominate. (S(t), I_N(t), I_R(t)) converges to the steady state (83,0,17).