Optimising Antibiotic Usage to Treat Bacterial Infections

The increase in antibiotic resistant bacteria poses a threat to the continued use of antibiotics to treat bacterial infections. The overuse and misuse of antibiotics has been identified as a significant driver in the emergence of resistance. Finding optimal treatment regimens is therefore critical in ensuring the prolonged effectiveness of these antibiotics. This study uses mathematical modelling to analyse the effect traditional treatment regimens have on the dynamics of a bacterial infection. Using a novel approach, a genetic algorithm, the study then identifies improved treatment regimens. Using a single antibiotic the genetic algorithm identifies regimens which minimise the amount of antibiotic used while maximising bacterial eradication. Although exact treatments are highly dependent on parameter values and initial bacterial load, a significant common trend is identified throughout the results. A treatment regimen consisting of a high initial dose followed by an extended tapering of doses is found to optimise the use of antibiotics. This consistently improves the success of eradicating infections, uses less antibiotic than traditional regimens and reduces the time to eradication. The use of genetic algorithms to optimise treatment regimens enables an extensive search of possible regimens, with previous regimens directing the search into regions of better performance.

Supplementary Table S1 -Results from 1% initial resistant population Table S1: Comparison of traditional dosage vectors (runs A, B, C and D), dosage vectors produced by the GA with deterministic modelling (runs E, F, G and K) and dosage vectors produced by the GA with stochastic modelling (runs H, I and J) for an infection with a resistant population of 1% of the total bacterial population.

Supplementary Equations -Analytical Analysis of Antibiotic Free System
Using stability analysis the steady states of the system, in the absence of antibiotic, can be determined. At equilibrium, dS/dt = dR/dt = 0, there are four equilibrium points: Stability of the equilibrium points are found by calculating the Jacobian (Eq. 3) at each of the equilibria and calculating the corresponding eigenvalues.
The Jacobian matrix (Eq. 4) is a diagonal matrix and therefore the eigenvalues can be found on the diagonal. The extinction equilibrium is stable when Eq. 5 and 6 are satisfied.
When the natural death rate is higher than the replication rate, for both the susceptible and resistant strains, the system will tend to extinction.
Eq. 7 is upper triangular and therefore the eigenvalues can be found on the diagonal. For the resistant free equilibrium to be stable it must satisfy Eq. 8 and 9.
θ < r (8) The replication rate must be greater than the death rate otherwise the susceptible population would die out, therefore Eq. 8 must hold true. A lower transmission rate or a higher cost benefits the susceptible population.
Eq. 10 is lower triangular and the eigenvalues can be found on the diagonal. Therefore for the susceptible free equilibrium to be stable it must satisfy Eq. 11 and 12 The net replication rate must be greater than the death rate otherwise the resistant population would die out, therefore Eq. 11 must hold true. A higher transmission rate or a lower cost benefits the resistant population making it possible for the resistance bacteria to invade and out-compete an entirely susceptible population. 4) Analysis of the stability of the co-existence equilibrium is not possible due to the eigenvalues being analytically intractable. If it is hypothesised that stable coexistence is possible then from the previous equilibrium points it can be concluded that co-existence will occur, assuming a positive net growth rate for both bacteria, if: Using the analytical analysis parameter values were chosen such that they satisfy Eq. 8 and 9. Therefore the resistant strain would not out-compete the susceptible strain in the absence of antibiotics.
Supplementary Table S2 -Results from varied weights within the objective function