Mathematical models of drug-resistant tuberculosis lack bacterial heterogeneity: A systematic review

Drug-resistant tuberculosis (DR-TB) threatens progress in the control of TB. Mathematical models are increasingly being used to guide public health decisions on managing both antimicrobial resistance (AMR) and TB. It is important to consider bacterial heterogeneity in models as it can have consequences for predictions of resistance prevalence, which may affect decision-making. We conducted a systematic review of published mathematical models to determine the modelling landscape and to explore methods for including bacterial heterogeneity. Our first objective was to identify and analyse the general characteristics of mathematical models of DR-mycobacteria, including M. tuberculosis. The second objective was to analyse methods of including bacterial heterogeneity in these models. We had different definitions of heterogeneity depending on the model level. For between-host models of mycobacterium, heterogeneity was defined as any model where bacteria of the same resistance level were further differentiated. For bacterial population models, heterogeneity was defined as having multiple distinct resistant populations. The search was conducted following PRISMA guidelines in five databases, with studies included if they were mechanistic or simulation models of DR-mycobacteria. We identified 195 studies modelling DR-mycobacteria, with most being dynamic transmission models of non-treatment intervention impact in M. tuberculosis (n = 58). Studies were set in a limited number of specific countries, and 44% of models (n = 85) included only a single level of "multidrug-resistance (MDR)". Only 23 models (8 between-host) included any bacterial heterogeneity. Most of these also captured multiple antibiotic-resistant classes (n = 17), but six models included heterogeneity in bacterial populations resistant to a single antibiotic. Heterogeneity was usually represented by different fitness values for bacteria resistant to the same antibiotic (61%, n = 14). A large and growing body of mathematical models of DR-mycobacterium is being used to explore intervention impact to support policy as well as theoretical explorations of resistance dynamics. However, the majority lack bacterial heterogeneity, suggesting that important evolutionary effects may be missed.

Combining mutation and horizontal gene transfer in a within-host model of antibiotic resistance

Antibiotics are used extensively to control infections in humans and animals, usually by injection or a course of oral tablets. There are several methods by which bacteria can develop antimicrobial resistance (AMR), including mutation during DNA replication and plasmid mediated horizontal gene transfer (HGT). We present a model for the development of AMR within a single host animal. We derive criteria for a resistant mutant strain to replace the existing wild-type bacteria, and for co-existence of the wild-type and mutant. Where resistance develops through HGT via conjugation we derive criteria for the resistant strain to be excluded or co-exist with the wild-type. Our results are presented as bifurcation diagrams with thresholds determined by the relative fitness of the bacteria strains, expressed in terms of reproduction numbers. The results show that it is possible that applying and then relaxing antibiotic control may lead to the bacterial load returning to pre-control levels, but with an altered structure with regard to the variants that comprise the population. Removing antimicrobial selection pressure will not necessarily reduce AMR and, at a population level, other approaches to infection prevention and control are required, particularly when AMR is driven by both mutation and mobile genetic elements.

A MODEL ON BACTERIAL RESISTANCE CONSIDERING A GENERALIZED LAW OF MASS ACTION FOR PLASMID REPLICATION

Bacterial plasmids play a fundamental role in antibiotic resistance. However, a lack of knowledge about their biology is an obstacle in fully understanding the mechanisms and properties of plasmid-mediated resistance. This has motivated investigations of real systems in vitro to analyze the transfer and replication of plasmids. In this work, we address this issue with mathematical modeling. We formulate and perform a qualitative analysis of a nonlinear system of ordinary differential equations describing the competition dynamics between plasmids and sensitive and resistant bacteria. In addition, we estimated parameter values from empirical data. Our model predicts scenarios consistent with biological phenomena. The elimination or spread of infection depends on factors associated with bacterial reproduction and the transfer and replication of plasmids. From the estimated parameters, three bacterial growth experiments were analyzed in vitro. We determined the experiment with the highest bacterial growth rate and the highest rate of plasmid transfer. Moreover, numerical simulations were performed to predict bacterial growth.

Mathematical modelling to study the horizontal transfer of antimicrobial resistance genes in bacteria: current state of the field and recommendations

Antimicrobial resistance (AMR) is one of the greatest public health challenges we are currently facing. To develop effective interventions against this, it is essential to understand the processes behind the spread of AMR. These are partly dependent on the dynamics of horizontal transfer of resistance genes between bacteria, which can occur by conjugation (direct contact), transformation (uptake from the environment) or transduction (mediated by bacteriophages). Mathematical modelling is a powerful tool to investigate the dynamics of AMR; however, the extent of its use to study the horizontal transfer of AMR genes is currently unclear. In this systematic review, we searched for mathematical modelling studies that focused on horizontal transfer of AMR genes. We compared their aims and methods using a list of predetermined criteria and used our results to assess the current state of this research field. Of the 43 studies we identified, most focused on the transfer of single genes by conjugation in Escherichia coli in culture and its impact on the bacterial evolutionary dynamics. Our findings highlight the existence of an important research gap in the dynamics of transformation and transduction and the overall public health implications of horizontal transfer of AMR genes. To further develop this field and improve our ability to control AMR, it is essential that we clarify the structural complexity required to study the dynamics of horizontal gene transfer, which will require cooperation between microbiologists and modellers.

Dynamic analysis of conversion from a drug-sensitivity strain to a drug-resistant strain

In this paper, a mathematical model of conversion from a drug-sensitivity strain to a drug-resistant strain is given to investigate how antibiotic usage may be optimized to preserve or restore antibiotic effectiveness. This novel theoretical framework could result in an optimal criterion on how to reduce the drug resistance to a reasonable range by using the antibiotic dressing strategy. The sufficient conditions of existence of order-1 periodic solution are obtained in view of the geometrical theory of the semi-continuous dynamical system and the qualitative properties of the corresponding continuous system. The stability of the order-1 periodic solution is proved by means of [H. J. Guo, L. S. Chen and X. Y. Song, Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property, Appl. Math. Comput. 271 (2015) 905–922]. Finally, our results are confirmed by means of numerical simulations.

Send more data: a systematic review of mathematical models of antimicrobial resistance

Background

Antimicrobial resistance is a global health problem that demands all possible means to control it. Mathematical modelling is a valuable tool for understanding the mechanisms of AMR development and spread, and can help us to investigate and propose novel control strategies. However, it is of vital importance that mathematical models have a broad utility, which can be assured if good modelling practice is followed.

Objective

The objective of this study was to provide a comprehensive systematic review of published models of AMR development and spread. Furthermore, the study aimed to identify gaps in the knowledge required to develop useful models.

Methods

The review comprised a comprehensive literature search with 38 selected studies. Information was extracted from the selected papers using an adaptation of previously published frameworks, and was evaluated using the TRACE good modelling practice guidelines.

Results

None of the selected papers fulfilled the TRACE guidelines. We recommend that future mathematical models should: a) model the biological processes mechanistically, b) incorporate uncertainty and variability in the system using stochastic modelling, c) include a sensitivity analysis and model external and internal validation.

Conclusion

Many mathematical models of AMR development and spread exist. There is still a lack of knowledge about antimicrobial resistance, which restricts the development of useful mathematical models.

Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma

In this work we formulate a model for the population dynamics of Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB). Our main interest is to assess the impact of the competition among bacteria on the infection prevalence. For this end, we assume that Mtb population has two types of growth. The first one is due to bacteria produced in the interior of each infected macrophage, and it is assumed that is proportional to the number of infected macrophages. The second one is of logistic type due to the competition among free bacteria released by the same infected macrophages. The qualitative analysis and numerical results suggests the existence of forward, backward and S-shaped bifurcations when the associated reproduction number R0 of the Mtb is less unity. In addition, qualitative analysis of the model shows that there may be up to three bacteria-present equilibria, two locally asymptotically stable, and one unstable.