Modeling the Spread of Antibiotic Resistance

Transcription

1 Modeling the Spread of Antibiotic Resistance J. S. Hallinan Department of Computer Science and Electrical Engineering, The University of Queensland, Brisbane, QLD 4072 Australia Abstract: This paper describes a stochastic implementation of Austin et al. s (1999) model of the spread of antibiotic resistance in a population of fixed size under varying conditions of antibiotic use. The population is divided into sub-groups: individuals colonized by commensal bacteria and an uncolonized group. The colonized group is further divided according to whether the commensal bacteria are sensitive or resistant to antibiotics. This study uses Monte Carlo techniques to model the dynamics of the evolution of the antibiotic resistant population, a study that cannot be done in the original model. The Monte Carlo approach allows the investigation of the transient dynamics of the spread of resistance, the effects of finite (especially small) populations and the interaction of model parameters. 1 Introduction The development of antibiotic therapy following Sir Alexander Fleming s serendipitous discovery of penicillin in 1928 had a major impact upon public health. For the first time in human history diseases of bacterial origin, such as typhoid, cholera, and opportunistic infections were reduced from major causes of mortality to relatively minor inconveniences. In the last twenty years, however, it has become apparent that the ability of bacterial pathogens to develop significant levels of resistance to antibiotics has the potential to outstrip the rate of development of novel antibiotic compounds. In order to avoid a return to the era of untreatable infections, a detailed understanding of the relationship between antibiotic use and the spread of antibiotic resistance is essential. Several groups have developed models of the spread of antibiotic resistance (Blower, Small & Hopewell, 1996; Bonhoeffer, Lipsitch & Levin, 1997; Castillo-Chavez & Feng, 1997; Levin et al., 1997; Stewart et al., 1998). A particularly interesting approach is that of Austin, Kristinsson & Anderson (1999), who propose a model based upon a population genetics approach. This model uses coupled ordinary differential equations (ODEs) to J. Wiles Department of Computer Science and Electrical Engineering, and School of Psychology The University of Queensland, Brisbane, QLD 4072 Australia janetw@csee.uq.edu.au determine the level of carriage of resistant bacteria in a population of fixed size with respect to time. All of the models mentioned above are mathematically based. In order for such a model to be tractable it must be very simple and concerns have been expresssed that such simplification may reduce the ability of the system to model the complex processes involved (Todd, 1996). An alternative approach is that of evolutionary computation (EC), in which individuals, rather than trends, are modeled. Such an approach has not previously been explored in this problem domain, to the best of our knowledge. An EC approach is likely to be fruitful for the investigation of the spread of antibiotic resistance because it permits an examination of the dynamics of a population as it evolves, leading to a greater understanding of the relative importance of interactions between model parameters. An EC model also incorporates a stochastic element, enabling analysis of the relative probabilities of different long-term outcomes from the same initial conditions. Finally, an EC model is easily extended to incorporate conditions that may be too complex to be computationally tractable in a model based on ODEs. This paper describes the implementation and performance of an EC model based on that described by Austin et al. (1999). 2 Implementation of the Model In the model proposed by Austin et al. (1999) members of the population may carry commensal bacteria, organisms which live within the host without either harming or benefiting it. These commensals in turn may carry genes coding for antibiotic resistance. Since the commensal bacteria cause no significant morbidity to the host, antibiotics are assumed to be prescribed for conditions unrelated to the status of the host with respect to infection with commensals. The chance of an individual receiving antibiotic treatment is thus independant of its infection status. The population therefore consists of five categories of individuals: Uncolonized individuals;

2 Individuals colonized with antibiotic sensitive bacteria; Individuals colonized with antiobiotic resistant bacteria; Uncolonized individuals who are undergoing antibiotic treatment; ly colonized individuals who are undergoing antibiotic treatment. The size of the population is constant, and the proportion of the population in each category at any given time is dependant upon several parameters. 2.1 Design of the Model Since the current model is a quantified version of the continuous Austin et al. model, time must be explicitly modelled. The timestep chosen was a 10 day period, equal to that of the average duration of treatment in the casestudies of Austin et al. (1999). This is the minimum time period in which any change occurs in the model; using smaller timesteps would add complexity without improving functionality, while the use of larger timesteps incurs the risk of missing potentially important behaviour. Because of this equivalence in the model, any individual undergoing treatment in one time step may be assumed to have finished treatment by the end of the timestep. The fivestate model of Austin et al. (1999) thus becomes a threestate model (Figure 1). colonization RESISTANT (z 0 ) Status unchanged Status unchanged UNCOLONIZED (x 0 ) Reversion Superinfection Figure 1. The basic model colonization SENSITIVE (y 0 ) Status unchanged In any timestep there is a defined probability of transition from any given state to another. The model therefore becomes, in effect, a random Monte Carlo process. The transition probabilities depend upon the model parameterization. 2.2 Transition Probabilities Transition between uncolonized and colonized states is based upon the reproductive number of the bacterium,. is defined as the average number of secondary cases of colonized hosts generated when the primary case is introduced in a naïve population (Austin et al., 1999, p. 1153). depends on the effective contact rate, β, the rate of exit of individuals from the population, µ, and the natural clearance rate of the bacterium, f: = β / (µ + f). The values of µ and f are assumed to be constant; β may vary depending upon whether the commensal is carrying a resistant or sensitive gene, with a sensitive bacterium enjoying a fitness advantage over resistance-bearing commensals. The assumption that resistance incurs a fitness cost is based on the observation that in an antibiotic-free environment, resistance tends to be present at very low levels. This assumption is subject to debate (e.g. Schrag & Perrat, 1996), and is a candidate for further computational experimentation. Experimentation with MC models enables us to study the effects of differential bacterial fitness on the equilibrium levels of carriage of the organism in the population. The differential infectivity of resistant and sensitive bacteria is implemented as a separate β z (infectivity of resistant bacteria) and β y (sensitive bacteria). β y and β z are calculated from the transmission fitness parameter, defined by Austin et al. as transmission fitness = /, where is the reproductive number of the resistant bacterium. The transmission fitness value given by these authors is greater than one for both sets of data examined; a value which suggests that resistant bacteria have a fitness advantage over sensitive bacteria, since the reversion rates from resistant to uncolonized and from sensitive to uncolonized are the same. Since this parameterization contradicts the assumption that resistance incurs a fitness cost, transmission fitness was implemented in this model as /. In this model, the probability of transition from the uncolonized state, x, to the resistantly colonized state, y, at timestep t is: y0 ( t) y ( t) P( x y β y0( t) ) 1 1 x0( t) β = = and the probability of transition from uncolonized to sensitively colonized state, z, is

3 z β ( t) P( x z z0 ( t ) 1 1 x0 ( 0) β = = Note that the transition probabilities depend on the proportion of the population in each state at each timestep. The probability of reversion from a resistantly colonized to an uncolonized state at any timestep is: P(zÆ x) = (µ + f) A transition from resistantly colonized to sensitively colonized status, or vice versa, is referred to as superinfection. If sensitively colonized bacteria have a fitness advantage over resistantly colonized bacteria, as discussed above, some proportion of contacts between individuals in the two states will result in a transition from resistant to sensitive status. This superinfection fitness, φ, modifies the contact probability, β y (t): P y ( z y) = β ( t) φ. The selection pressure in favour of resistance is provided by the level of antibiotic treatment. The probability of antibiotic treatment is assumed to be independent of the colonization status of the individual; colonization by a commensal, regardless of its resistance status, causes negligable morbidity. Treatment is assumed to be prescribed for an independantly occurring condition. Treatment level, α, is dependant upon the level of prescribing in the community, γ, and the compliance level, g: γ α =. g The status of an uncolonized individual who is treated will be unchanged, as will that of a resistantly colonized individual (resistance is assumed to be complete). A sensitively colonized individual will revert to uncolonized status. The probability of reversion for a sensitively colonized individual is thus: P(y Æ x) = (µ + f) + α 2.3 Parameterization The parameterization for our Monte Carlo model was taken from the values used by Austin et al. (1999), modified as appropriate for a discrete system using 10-day timesteps. The parameterization is shown in Table 1. z0 ( t) Table 1. Parameterization of the Model Parameter Symbol Value Reproductive number (sensitive bacteria) 2.0 Reproductive number (resistant bacteria) * Contact probability β /Maximum age µ /Duration of carriage f Superinfection fitness φ 0.43 Initial proportion uncolonized Initial proportion sensitive Initial proportion resistant Probability of treatment 3 Dynamics of the Model x 0 (0) y 0 (0) z 0 (0) α Behaviour in the Absence of Selection Pressure For the simulations described below the model consisted of a population of 1,000 individuals. Population size remained constant throughout the course of a run. The initial proportions of individuals in each of the states described in Figure 1 were set to those determined by Austin et al. for their Finland data; that is 50% sensitively colonized, 0.2% resistantly colonized and 49.8% uncolonized. For each of the simulations reported in Figures 2 to 6 indicative results are reported, showing the results of one typical run of the model. In a population of fixed size, a rise in the proportion of individuals in one state must be accompanied by a fall in the proportion of individuals in one or both of the other states. In addition, since the probabilities of transition between states are dependant upon the proportions of various states, there are dynamic interactions between individuals which evolve over time. According to Austin et al. the numbers of an organism with reproductive number, undergoing constant selection pressure, should increase in a sigmoidal manner, finally persisting in the population at a proportion of 1 (1/ ). This behaviour can be demonstrated using the MC model, but is influenced by many of the model parameters, * The value of depends upon the transmission fitness parameter

4 particularly the average duration of carriage of the resistant organism, f. The course of evolution of the resistant percentage of individuals was initially studied in a population in which was 2.0, the superinfection fitness was 0.0, transmission fitness was 1.0, and there was no antibiotic being prescribed (see Figure 2). The model was initialized as described above, and then allowed to run until the proportions of the three types of individuals appeared to have stabilized. The duration of carriage (average persistence of the bacterium in an infected individual, measured in months) was varied and the effect upon the population observed. Proportion of population Figure 2. Evolution of resistance in populations with no selection and = 2.0, testing different time courses for average persistence of infection by resistant bacteria. From Figure 2 it can be seen that as the average duration of carriage of the bacterium increases, the proportion of the population carrying the resistant commensal also increases, and the rate at which this asymptote is reached slows. This observation is counterintuitive it would appear that increasing the average persistence time of the bacterium should increase, rather than decrease, the rate at which it spreads through the population. The explanation for the observed behaviour lies in the interactions between the resistantly colonized hosts and the rest of the population. bacteria can only spread to uncolonized hosts. However, uncolonized hosts are also the target of sensitive bacteria, and there is a much higher initial proportion of sensitively colonized hosts. The sensitive bacteria also benefit from the increased duration of carriage and these two factors high initial numbers and increased duration of carriage give the sensitive bacteria an initial advantage over the resistant bacteria in the competition for uncolonized hosts. Over time this advantage is counteracted by the selection pressure of antibiotic use, sensitive numbers fall, and resistant numbers rise, but the longer the duration of carriage the longer this initial advantage lasts. The expected level of carriage of 0.5 is only observed with a low average duration of carriage (0.5 months). The levels of persistence of the resistant bacterium in the population evident from these simulations are much higher than those observed in real populations. A closer examination of the model parameters reveals two parameters whose values critically affect the spread of the resistant bacterium. These parameters are the superinfection fitness, φ, and the transmission fitness, /. The superinfection fitness represents the probability that a resistantly infected individual, upon contact with a sensitively colonized individual, will be converted to sensitively colonized status. A positive value for this parameter indicates selection pressure against the spread of the resistant strain. The transmission fitness is a measure of the differential fitness of the sensitive and resistant bacterial strains. A value of 1.0 indicates that the resistant bacterium is at no fitness disadvantage, while values greater than 1.0 impose increasingly greater fitness advantage to the sensitive bacterium. For example, a transmission fitness of 1.24, as used by Austin et al., leads to a of 1.6 if is 2.0. is used to calculate the effective contact rate, β z, of resistant bacteria, while is used to calculate β y, the effective contact rate of sensitive bacteria. β z will therefore be lower than β y, and sensitive bacteria will spread more readily through the population. With the model parameterized as described in Table 1, and no antibiotic prescription, resistant bacteria do not persist in the simulation, as long as there are sensitively colonized bacteria present in the population. Reducing either the transmission fitness or the superinfection fitness leads to the persistence of resistant bacteria. The effects of varying these parameters are graphed in the next two figures. In Figure 3 the proportion of resistantly colonized hosts is plotted for varying values of φ, with transmission fitness held constant at the value determined by Austin et al., while Figure 4 shows the opposite condition varying values of transmission fitness, with φ held constant.

5 Proportion of pop. resistant Proportion of pop. resistant Figure 3. Spread of resistant bacteria with superinfection fitness = 0.43, no selection, and varying values of transmission fitness Figure 4. Spread of resistant bacteria with transmission fitness = 1.29, no selection, and varying values of φ. for further experiments we used a transmission fitness of 1.0 (i.e. no fitness advantage to sensitive bacteria) and a superinfection fitness of 0.2. With these parameter settings and no selection pressure, the equilibrium levels of resistant and uncolonized bacteria are both around 25%, with approximately 50% of the population sensitively colonized. 3.2 Behaviour in the Presence of Selection Pressure Parameters such as transmission fitness and superinfection fitness are very difficult to establish reliably. Austin et al. developed their model from a theoretical basis, and then determined the appropriate values for model parameters using empirical data from two epidemiological studies. Values for parameters such as the amount and duration of antibiotic prescription are, however, considerably easier to establish. Using the values determined by Austin et al. (average treatment rate of 1.42 courses of antibiotic treatment per head of population per year, and an average duration of treatment of 10 days) we observed the pattern of evolution shown in Figure 5. For this simulation the population size was increased to 10,000 in order to obtain a clear trend. With a population size of 1,000 the variability inherent in the simulation obscured the difference between the equilibrium levels of uncolonized and resistant individuals. Proportion of pop. resistant Uncolonized Although the course of evolution is affected in different ways by these two parameters, the overall effect is the same as the parameter value increases the equilibrium level of resistant carriage decreases from the theoretically expected level (0.5 in this case, since is 2.0), declining eventually to 0.0. The existence of a fitness penalty associated with antibiotic resistance is a subject of debate amongst biologists. Figure 3 shows that this postulated phenomenon can have a dramatic effect upon the spread of resistance. Since the existence of a fitness penalty associated with antibiotic resistance is not an essential part of the model, Figure 5. Evolution of a population under antibiotic selection pressure Under this antibiotic regime the final equilibrium level of resistance is slightly higher than it is without selection pressure, and the equilibrium reached amongst the three states appears to be stable. There is a characteristic pattern of evolution. The population is originally antibiotic-naïve. When antibiotic treatment commences there is an initial sharp rise in the number of sensitively colonized

6 individuals, since they have a fitness advantage over resistantly colonized hosts. The number of uncolonized individuals correspondingly drops. In the course of several time steps the effects of antibiotic treatment of sensitive individuals leads to a gradual drop in their numbers. Because of the differential infectivity of resistant and sensitive bacteria, resistant bacteria need uncolonized hosts to which to spread. As more uncolonized hosts are colonized by resistant (and sensitive) bacteria, the number of uncolonized individuals falls, and the proportion of resistantly colonized hosts rises to an equilibrium level. Since sensitive bacteria have a fitness advantage over resistant bacteria, they predominate in the population at equilibrium at this level of antibiotic use. The level of antibiotic treatment used in this simulation is relatively low each individual has a 0.4% chance of receiving antibiotic treatment in any timestep. The effect of this parameter, while evident, is small compared with the effects of, φ, and the transmission fitness, as discussed above. In Figure 6 the effect upon the evolution of resistance of increasing numbers of courses of treatment is plotted. Proportion of pop. resistant No treatment 2 courses/yr 5 courses/yr 10 courses/yr Figure 6. Effect of increasing number of antibiotic courses per year on the evolution of resistance. From Figure 6 it can be seen that increasing the number of courses of antibiotic per person per year increases both the level of antibiotic resistance reached in the population, and the speed with which that level is reached, as would be predicted. 3.3 The Effect of Population Size In all of the simulations discussed so far, the output of a single, typical, run of the model have been reported. A Monte Carlo model, however, is highly stochastic, and the element of chance is particularly important in determining the fate of small populations. Modeling using a system of coupled ordinary differential equations, as has typically been done previously, implicitly assumes that the population of interest is of infinite size. A stochastic model permits an examination of the range of behaviours that may arise by chance in a given situation, and allows an estimate to be made of the relative probabilities of each outcome. Table 2 shows the results of multiple runs of the resistance model, using the parameterization discussed above. In order to allow the investigation of small population sizes, the initial proportion of resistantly colonized hosts was set to 2% instead of 0.2%, and the proportion of uncolonized individuals to 48.0% instead of 49.8%. The model was run 20 times for each population size, and the final proportions of individuals of each status were recorded after 100 timesteps. Table 2. Effects of population size on course of evolution final proportion of population. Population Size Statistic ,000 10,000 Average Uncolonized Std Dev Uncolonized Average Std Dev. Average Std Dev. Range Uncolonized Range Range Table 2 illustrates the importance of population size to the course of evolution. In the population of size 50, resistance was lost completely in 18 out of 20 runs, a phenomenon which occurred ocasionally in a population of 100, and never occurred in larger populations. In the absence of mutation the gene for resistance is never recovered. As population size increases, the standard deviation of the distributions of the outcomes decreases as the range of outcomes decreases. Small populations are much more

7 likely to have aberrant equilibrium values, and inspection of the course of evolution indicates that small populations are much less likely to reach an equilibrium state. This observation has implications for the management of antbiotic resistance in restricted populations such as nursing homes and hospitals, which will be discussed in more detail below. 4 Discussion The Monte Carlo model described in this paper uses a simple paradigm: a population is modelled as a collection of individuals each of which is in one of three clearly defined states at any given time. Individuals have the opportunity to change state at each time step, with each state transition occuring with a specified probability. This simplicity has the advantage of making the behaviour of the model relatively easy to understand. In Austen et al. s model based on ODEs, closed form solutions of the steady state behaviours of the population groups are possible. In the Monte Carlo model, additional aspects of the spread of antibiotic resistence can be studied. Firstly, in addition to the steady state conditions, transient behaviours can be studied. In particular, two observations are of interest: In simulations of the model without selection pressure, the rate of increase of the resistent individuals was slower with longer average persistence, a counter-intuitive result (Figure 2). A single parameter governed persistence in both the sensitive and resistently colonised individuals and we hypothesized that the increased persistence in the sensitive bacteria, together with an initial numerical advantage, slowed the rate of infection by the resistent bacteria at a proportionately greater rate than infection by the resistent individuals. Further investigation is required to test this hypothesis; an examination of the effect of introducing separate parameters for the two persistence levels might yield interesting insights into this behaviour. A second aspect of the transient behaviour of the model that is interesting is the ability to quantify the speed with which antibiotic resistance rises in a population under a given set of conditions (see Figure 6). Isolation (i.e., quarantine) of infected individuals or groups is the reccomended method for in halting the spread of infection (CDC, 1995). The slower the rate of increase of infection in a population, the more time there is to quarantine infected individuals, regardless of the predicted steady state levels of infection. The issue of isolation relates to the second major benefit of the MC model, the ability to simulate the effects of population size. Austin et al. argue that a constant population size is characteristic of some of the environments of most interest to epidemiologists, such as hospitals, nursing homes, and day care centres. Their model predicts the expected steady state levels of infection if such populations remain isolated. The MC model allows us to extend their results to study the diversity of behaviours exhibited by the model under a variety of population sizes. The experimental results reported in Table 2 indicate that a small constant population may behave in a manner very different from a large population with the same parameters. Most nursing homes have populations of less than 100 residents, while large hospitals may have several thousand patients. In a very small population the prospect of eliminating the resistant bacteria by chance is reasonably high; but when resistance does become established, it can reach higher levels in the population than is the case in a larger group. The most relevant result here is for the smallest population size tested (see population size 50 in Table 2), in which resistance was eliminated in 18/20 runs. This result is consistent with intuition small populations will generally be resistance free, and will remain so unless infected from outside. The MC model could also be extended to study other aspects of finite populations. The most obvious restriction of the current formulation is the constant population size. Few natural populations remain exactly the same in size; individuals enter and leave most populations, carrying with them new pathogens. Individuals carrying resistant commensals may be quarantined when discovered. The MC model could be extended to simulate a variable population with a set rate of immigration and emigration to study such conditions. As with any computational model, it is important to consider what has been omitted, as well as what has been included. The series of simulations reported in this paper show the diversity of behaviours accompanies changes in fundamental parameters of the model, reflecting different assumptions about the environment in which bacteria compete to infect their hosts. There are many further extensions that could be made to this model. For example, one limitation of the model is the grouping of superinfected individuals with those who are sensitively colonized, which was done for consistency with Austin et al. s original formulation of the problem. With their published parameters, these two groups behave identically, and are collapsed into one state. However, the original model by Austin et al. (1999) did incorporate a parameter, ρ, which represented the probability that a sensitively colonized individual receiving antibiotic treatment becomes resistantly colonized. This transition would be due to either de novo mutation or to the elimination of sensitive commensals in a superinfected individual resulting in a resistant infection. This parameter was not used in their case study. If a realistic value for this parameter could be estimated, it could be implemented in

8 the MC model by representing sensitively infected and superinfected individuals seperately; making the model a four-state instead of a three-state system. The model of the spread on antibiotic resistance described in this paper is reasonably simple, and is constrained by the limitations imposed by this simplicity. Despite these limitations, we believe that an EC modeling approach has much to offer in a problem domain such as this. It provides the ability to examine the dynamic behaviour of a complex system, and the way in which these dynamics are affected by parameters such as population size, in a way which cannot be done using most other approaches. 9. Todd, P. M. (1996). The causes and effects of evolutionary simulations in the behavioural sciences. In Adaptive Individuals in Evolving Populations: Models and Algorithms. Proceedings Volume XXVI of the Santa Fe Institute for Studies in the Sciences of Complexity. pp Bibliography 1. Austin, D. J., Kristinsson, K. G., & Anderson, R. M. (1999). The relationship between the volume of antimicrobial consumption in human communities and the frequency of resistance. Proceedings of the National Academy of Sciences of the USA, 96, Blower, S. M., Small, P. M. & Hopewell, P. C. (1996). Control strategies for tuberculosis epidemics: new models for old problems. Science 273, Bonhoeffer, S., Lipsitch, M. & Levin, B. R. (1997). Evaluating treatment protocols to prevent antibiotic resistance. Proceedings of the National Academy of Sciences USA 94(22), Castillo-Chavez, C. & Feng, Z. (1997). To treat or not to treat: the case of tuberculosis. Journal of Mathematics in Biology, 35(6), CEC (1995). Recommendations for preventing the spread of Vancomycin resistance: Recommendations of the Hospital Infection Control Practices Advisory Committee (HICPAC). Centres for Disease Control MMWR 44(RR12): htm. Downloaded 21/1/ Levin, B. R., Lipsitch, M., Perrot, V., Scrag, S., Antia, R., Simonsen, L., Moore-Walker, N. & Stewart, F. M. (1997). The population genetics of antibiotic resistance. Clinical Infectious Diseases, 24, S9 S Schrag, S. J. & Perrot, V. (1996). Reducing antibiotic resistance (Letter). Nature 381(6578), Stewart, F. M., Antia, R. Levin, B. R., Lipsitch, M. & Mittler, J. E. (1998). The population genetics of antibiotic resistance. II: Analytic theory for sustained populations of bacteria in a community of hosts Theoretical Population Biology, 53(2),