The evolution of antibiotic resistance in a structured host population

The evolution of antibiotic resistance in opportunistic pathogens such as Streptococcus pneumoniae, Escherichia coli or Staphylococcus aureus is a major public health problem, as infection with resistant strains leads to prolonged hospital stay and increased risk of death. Here, we develop a new model of the evolution of antibiotic resistance in a commensal bacterial population adapting to a heterogeneous host population composed of untreated and treated hosts, and structured in different host classes with different antibiotic use. Examples of host classes include age groups and geographic locations. Explicitly modelling the antibiotic treatment reveals that the emergence of a resistant strain is favoured by more frequent but shorter antibiotic courses, and by higher transmission rates. In addition, in a structured host population, localized transmission in host classes promotes both local adaptation of the bacterial population and the global maintenance of coexistence between sensitive and resistant strains. When transmission rates are heterogeneous across host classes, resistant strains evolve more readily in core groups of transmission. These findings have implications for the better management of antibiotic resistance: reducing the rate at which individuals receive antibiotics is more effective to reduce resistance than reducing the duration of treatment. Reducing the rate of treatment in a targeted class of the host population allows greater reduction in resistance, but determining which class to target is difficult in practice.

where ( $ * and 1 $ * are the equilibrium density of uncolonised, untreated hosts when the sensitive strain (resident) is at equilibrium in the population, and ( & * and 1 & * are the equilibrium densities of uncolonised, treated hosts when the sensitive strain is at equilibrium in the population. The matrix describing initial growth of the sensitive strain, 5 , is analogous. Coexistence occurs if and only if 5 > 0 and " > 0, where 5 and " are the dominant eigenvalues of the matrices 5 and " . We first derive an expression for the eigenvalues as a function of the equilibrium density of uncolonised 9 $ * and 9 & * when the resistant strain is alone (for 5 ) and when the sensitive strain is alone (for " ). Then we derive expressions for these equilibria ( 9 $ * and 9 & * ) to finally obtain closed-form expressions for 5 and " . Calculations are detailed in a companion Mathematica notebook.

Equilibrium expressions for the densities of uncolonised hosts at the resident strain equilibrium
Equation (3)  and to the first order, the ratios 9 $ * / $ * appearing in equation (3) in the antibiotic clearance term are simply approximated as 9 . The subscript "WT" denotes the resident wild type. The density of uncolonised untreated hosts depends on Γ :,;& = N > O P ?@ Q > F C >,?@ E > A >,?@ →> F C >,?@ F N > C >,?@ Q > RC >,?@ Q > P ?@ R E > A >,?@ →> R Q > , a small term representing the impact of antibiotic treatment. The density of uncolonised treated hosts depends on , a small positive term.

Impact of reducing treatment rate or reducing treatment duration on resistance
When both resistant and sensitive strains coexist in the population, the equilibrium frequency of resistance is well predicted by the ratio of invasion fitnesses of the resistant and the sensitive strain (sup. fig. 1). To investigate how a change in treatment rates ( 9 and 9 0 ) or treatment duration ( 9 ) would affect the frequency of resistance, therefore, we study how a change in these parameters impact the invasion fitnesses. We do so in the 'full inter-class transmission' scenario and we expect results to be similar in the 'no inter-class transmission' scenario.
From the derivatives in Supplementary Table 1 In these equations, the terms Δ / (resp. Δω/ω) represent the magnitude of the intervention, and the remainder of the equation represents the impact of the intervention. Reducing the treatment rate directly reduces the resistant strain fitness and increases the sensitive strain fitness (term in 0 ), an effect that reducing treatment duration does not have. Thus, the impact of reducing the treatment rate on resistance is always greater than the impact of reducing treatment duration, " 5 * + A Y A X W > " 5 * . The beneficial impact of reducing the treatment rate on the sensitive strain is also always greater than the impact of reducing treatment duration when 9 0 = 9 , and numerical investigation suggest this is also likely to be true even when 9 0 > 9 for plausible parameter values.
Supplementary Figure 1: the frequency of resistance correlates well with the invasion fitnesses. logit is shown as a function of log " − log 5 across 228 random simulations where coexistence between the sensitive and the resistant strain occurred. Parameters are: three classes of size 9 drawn independently in each class in a uniform[0,1] then rescaled to ( + 1 + c = 1. Clearance rates 9,5 = 9," drawn independently in each class in a uniform[0.5, 2] month -1 ; treatment rates 9 drawn independently in each class in a uniform[0,0.1] month -1 and 9 0 = 0 9 with the factor 0 drawn in a uniform[1, 1.5]; treatment cessation 9 drawn independently in each class in a uniform [3,5]