Identification of mathematical model of bacteria population under the antibiotic influence
Serovajsky S., Nurseitov D., Azimov A.
Introduction
As is known, the efficiency of action of the antibiotics on pathogenic bacteria decreases quite rapidly. Therefore, considerable facilities are distinguished for development of new types of antibiotics. However, their action is temporary.
As a result, the problem of antibiotic resistance is one of the most general in modern medicine (see, for example, [1, 2, 3]).
One of the most important directions in the analysis of the resistance process is mathematical modeling. The different models are known for description of this phenomenon (see, for example, [4, 5, 6, 7, 8, 9]). However, that effective use of these models is possible only after their identification. These models have parameters that are not determine by the direct experiment. They can be found only as a result of solving inverse problems on the base of known information about the state of the system [10]. We determine the parameters of the mathematical model of the population of sensitive and resistant bacteria under the action of bactericidal and bacteriostatic antibiotics [8, 9] on the basis of available experimental data [11]. The partial case of this inverse problem on the basis of Verhulst equation is considered before [12].
The mathematical model of the considered process is based on the following suppositions:
1. The population of bacteria is under the influence of antibiotic.
2. An antibiotic can possess both bactericidal and bacteriostatic effects.
3. A population is heterogeneous and consists of bacteria sensible and resistant to the action to the antibiotic.
4. The sensitive bacteria are more viable than resistant bacteria in the absence of an antibiotic.
5. The sensible bacteria prevail initially in a population.
6. The habitat of the population is bounded.
7. Transitions from sensitive bacteria to resistant ones are possible, and vice versa by mutations and transfer of plasmids carrying a gene of resistance to the action of antibiotic.
At the done assumptions a process is described by the following system of differential equations [8, 9]
˙ x1=
a1
1 +s(t)(x1)θs −b1(x1+x2)
x1+a12x2−c(t)(x1)θc,
˙
x2= [a2−b2(x1+x2)]x2+ a21
1 +s(t)(x1)θsx1,
with initial conditions
x1(0) =x10, x2(0) =x20.
The functionsx1andx2characterize here the quantity of bacteria, respectively, sensitive and resistant to the action of the antibiotic, and x10 and x20 are the initial values of these quantities. The parameters a1 and a2 describe the in- creases of quantity of bacteria of both types,b1andb2are their sensitiveness to the boundedness of the habitat,a12anda21are intensity of transitions from one type of bacteria to other by mutations or transfer of plasmids carrying the resis- tant gene, the parametersθc andθscharacteristics of activities for bactericidal and bacteriostatic antibiotics, and the value c(t) and s(t) have some positive values at the stage of action of the antibiotic and equal zero in its absence.
Substantial predominance of sensible bacteria in initial moment of time cor- responds to inequality
x10> x20.
The increased viability of sensitive bacteria in comparison with resistant bacteria is characterized by inequality
a1
b1 > a2
b2.
It should be noted that all the parameters included in the mathematical model under consideration cannot be directly determined experimentally. They can be found exceptionally in the process of solving the corresponding inverse problems. In this case, in view of the considerable number of unknown parame- ters and the small volume of the information being measured, it is not possible to find all the unknowns at once by solving a single inverse problem. However, the process under study has a natural multistage character. Therefore, the re- quired parameters can be determined by solving a series of essentially simpler inverse problems.
1 The initial stage of infection
The investigated process begins with the stage of infection. There is still no antibiotic that the following equalities hold c(t) = 0, s(t) = 0. In view of substantial predominance of sensible bacteria on the initial stage of process a population can be considered homogeneous, i.e. we have unique state function, characterizing the quantity of population. Under these conditions, transitions from one type of bacteria to another do not make sense. Hence, the mathemat- ical model of the process is substantially simplified and takes the form:
˙
x= (a−bx)x, x(0) =x0.
Thus, the system is described by the Verhulst equation with unknown values of the parametersaandb.
We can find the unknown parameters with using the measurable values of the function x at certain instants of time, which corresponds to the conditions
x(tj) =yj, j= 1, . . . , M.
Thus, Problem 1 reduces to minimizing the value J =J(a, b) =
M
X
j=1
[x(tj)−yj]2.
This problem is solved by three methods. In the first case, the standard gradient method is used [9]. In this case, the function appearing in the gradient of the functional is the solution of the adjoint equation including delta functions. This is approximated directly in the computer experiment. The second algorithm also applies the gradient method. However, here the straight-line interpolation of measureable data is preliminary conducted. As a result, the adjoint system no contains a delta function. In the third case, the problem is solved using a genetic algorithm [13]. To test the efficiency of algorithms, calculations were first carried out for the model case, when the exact values of the unknown parametersaand bare assumed to be known. The results of computer experiment are presented in Table 1. We have the quantity of iterationsk, the found values of the parameters ak andbk, the values of the absolute errors ∆a and ∆b and the relative errorsδa and δb for the explicit gradient method (method 1), the gradient method with interpolation (method 2), and the genetic method (method 3).
Table 1: The results of calculations for Problem 1 for valuesa= 3 andb= 2.
method k ak ∆a δa bk ∆b δb
1 18532 2.998 0.002 8·10−4 1.992 0.002 9·10−4 2 11187 2.976 0.024 0.008 1.983 0.017 0.009 3 71 2.999 2·10−4 6·10−5 1.999 10−4 10−4 As can be seen from the obtained results, all the algorithms turn out to be quite effective, which is explained by the simplicity of the problem being solved. Note that the genetic algorithm requires significantly fewer iterations (by several orders of magnitude) with almost the same accuracy of computations.
In turn, due to application of preliminary interpolation of the known data it is succeeded to obtain reduction approximately in one and a half times of number of iterations, but with the decline of exactness of calculation approximately on an order at the same initial approaching and criterion of precipice of iteration process. It is explained by the fact that in the case of interpolation, we actually assume that at intermediate points we know information about the system, which in reality does not correspond to reality. Missing information at each point in time we artificially add ourselves, by interpolating.
In practice, experimental data are set with a considerable error. In this connection verification of efficiency of algorithms was conducted also for a case, when information is set with a 10% noise (see Table 2).
Table 2: The results of calculations for problem 1 fora= 3 andb= 2 with 10%
noise.
method k ak ∆a δa bk ∆b δb
1 18021 2.484 0.516 0.172 1.577 0.423 0.211 2 10623 2.502 0.498 0.166 1.592 0.408 0.204 3 999 3.326 0.326 0.109 2.259 0.259 0.130
Comparing the information from Table 1 and Table 2, we note that the ac- curacy of the solution of the problem in the presence of noise is substantially reduced. This is quite natural. However, we note that the difference in the quantity of iterations between the genetic algorithm (method 3) and the gra- dient method (methods 1 and 2) has significantly decreased (up to one order).
But the use of interpolation (method 2) allows at the same relations between the quantities of iteration, what in previous case to obtain the substantial increase of exactness of results by comparison to a method 1. This can be explained by the fact of smoothing of information by the interpolation. Taking into account that the exact solution of the Verhulst equation is a fairly regular function, the smoothing of noisy information has a positive effect on the computational pro- cess. On the basis of the obtained results, it can be concluded that interpolation of the initial data is effective in the case of high noise in conditions of sufficient smoothness of the system state function. As for the genetic algorithm, in gen- eral, it is inferior to the gradient method for the accuracy of calculations, but it exceeds it in the sense of the number of iterations.
Further investigation of Problem 1 is based on laboratory experiments, de- livered at the Scientific Center of Anti-infectious Drugs (Almaty) [11]. Thus examined a bacteria E.coli. Measuring of quantity of bacteria was conducted through every hour. In this case, the optical density of the medium with bacteria was directly measured, that is proportional to the value of quantity of bacteria, was directly measured thus. In order to avoid the use of extraordinarily large numbers we conducted an analysis directly on the values of optical density. The relevant information is given in Table 3.
Table 3: Results of the experiment.
tj, hour 0 1 2 3 4
yj, optical density 63 83 136 217 313
The results of calculations based on experimental data are given in Table 4.
Table 4: Results of calculations for the problem 1 on the basis of experimental data.
method k ak bk
1 225 0.406 0
2 552 0.407 0
3 999 0.419 10−4
It is noteworthy that all three methods give practically identical values of the required parameters. The necessary amount of iterations is here different, but has the same order. In this case, the greatest quantity of iterations requires a genetic algorithm, and the least quantity requires a gradient method without interpolation. We also note that according to the calculation, the coefficientb, which characterizes the degree of influence of the boundedness of the habitat, turns to the zero. Actually, this is not surprising, since we are considering the initial stage of the process of infection, when the number of bacteria is still relatively small, and the limited life space has not yet affected.
2 Complete stage of infection
Consider now the system at the full stage of infection. In this case, we already have a heterogeneous population of bacteria and take into account the tran- sitions from one type of bacteria to other. However, the treatment is not yet conducted, due to the influence of the antibiotic is not taken into account. Thus, the system is described by the equations
˙
x1= [a1−b1(x1+x2)]x1+a12x2,
˙
x2= [a2−b2(x1+x2)]x2+a21x1,
with corresponding initial conditions. In this case, the values of both functions of the state are measured in certain moments of time that corresponds to the equalities
xi(tj) =yij, j= 1, . . . , M, i= 1,2.
Thus, Problem 2 reduces to searching for of parameters a1, a2, b1, b2, a12, a21
from the condition of the minimum of functional J =J(a1, a2, b1, b2, a12, a21) =
2
X
i=1 M
X
j=1
[xi(tj)−yij]2.
We use Nelder–Mead method [14] for solving this problem. The results with model values of the parameters are given in Table 5. The corresponding solutions of equations by comparison to the results of measuring are brought around to Fig. 1.
Table 5: The results of calculations for Problem 2.
a1 a2 b1 b2 a12 a21
exact solution 3 2 0.008 0.009 10−4 2·10−4
approximate solution 2.999 2.001 0.008 0.009 10−4 6·10−4 absolute error 7·10−4 6·10−4 2·10−6 3·10−6 5·10−6 3·10−4
relative error 2.5·10−4 3·10−4 2.8·10−4 3.3·10−4 0.05 1.5 By Table 5, Nelder–Mead method provides high enough exactness of deter-
mination of coefficients for the state equation. Consider the dynamics of the process. We observe the growth of the bacterial population at the initial stage.
However, in the future, in view of the boundedness of the habitat, the gradual replacement by more viable sensitive bacteria of resistant bacteria is observed.
This is typical for the Volterra equations of competition [15]. It should be noted that the model examined in this case differs from classic equations of competition the presence of elements, characterizing transitions from one type of microorganisms to other. Therefore, weaker bacteria are not completely re- placed by stronger bacteria. However, the final value of the number of weaker resistant bacteria turns out to be sufficiently small because of the smallness of the corresponding coefficients (these transitions are relatively rare). The calcu- lations were also carried out with the case when the measurements were given with some noise (see Table 6).
Calculations have shown that the presence of noise does not change the situation, but increases the error of calculations.
t
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x1,x2
0 50 100 150 200 250 300 350 400
y1j
y2j
x1
x2
Figure 1: The solution of the state equations for Problem 2 and the results of measuring.
Table 6: Results of calculations for problem 2 with 10% noise.
a1 a2 b1 b2 a12 a21
exact solution 3 2 0.008 0.009 10−4 2·10−4
approximate solution 2.999 2.001 0.008 0.009 10−4 6·10−4 absolute error 7·10−4 6·10−4 2·10−6 3·10−6 5·10−6 3·10−4 relative error 0.084 0.072 0.059 0.0042 0.321 4.67
3 Infection and treatment by a potent bacteri- cidal antibiotic
We consider the completely process with the stages of infection and treatment by a potent antibiotic. In this case, resistant bacteria, present initially in a antibiotic small amount, will not have time to multiply. Thus, they can be neglected, i.e. there is a homogeneous population of bacteria that is sensible to the action of antibiotic. Thus, it is not needed naturally to take into account transitions from one type of bacteria to other. We consider here the case of bactericidal antibiotic that kills directly bacteria. Then the system is described by equation
˙
x= (a−bx)x−c(t)xθ,
where the value c(t) is equal to zero before the time of beginning of treatment and takes a constant valueγon the stage of treatment.
It is required to determine the value of coefficients a, b, γ, θ by measuring of the state of the system, both on the stage of infection and on the stage of
treatment. Thus, Problem 3 reduces to minimizing the functional J =J(a, b, γ, θ) =
M
X
j=1
[x(tj)−yj]2.
Laboratory experiment was also done an in the Scientific Center of Anti- infectious Drugs (Almaty) for the bacteria of E.coli [11]. The ampicillin is chosen as the antibiotic. The experiment is set every hour at the infection stage (see Table 7), and the optical density of the medium is measured. The antibiotic is entered in 6 hours 20 minutes from the beginning of experiment. Thus, on the stage of treatment experiment is done a more frequently (see Table 8, where time is measured in minutes, and the origin is beginning treatment).
Table 7: Results of the experiment for Problem 3 at the stage of infection.
tj, hour 0 1 2 3 4 5 6 6.33
yj, optical density 57 146 186 224 270 348 446 489
Table 8: Results of the experiment for Problem 3 at the stage of treatment.
tj, hour 0 15 30 45 60 80 100
yj, optical density 489 311 330 343 339 317 300
The problem was solved in two stages. First, at the stage of infection ac- cording to the results of the experiment from Table 7 the parametersaandbare determined. Then the parametersγandθare found on the basis of information from Table 8 with using the already found values ofaandb. The problem was solved by the method of Nelder–Mead. As a result, the values of the parameters area= 0.34,b= 3·10−4,γ= 3·10−6 andθ= 3.08. The corresponding values of the bacterial population in comparison with the experimental data are shown in Fig. 2. There is an increase population of bacteria at the stage of infection and a fairly rapid drop in their quantity at the stage of treatment.
4 Treatment by a bactericidal antibiotic for a heterogeneous population
Problem 3 corresponds to the case when the antibiotic is so strong that the resistance of bacteria does not show up. However, a more typical situation is the heterogeneous population. We consideration only the stage of treatment.
The antibiotic is still bactericidal. Then the system is described by the equations
˙
x1= [a1−b1(x1+x2)]x1+a12x2−cxθ1,
˙
x2= [a2−b2(x1+x2)]x2+a21x1,
Suppose all coefficients of equations except for characteristics of antibiotic are already determined by the analysis of the system on the stage of infection.
Thus, it is required to determine the values of the parametersc andθfrom the
t
0 1 2 3 4 5 6 7 8 9 10
x
50 100 150 200 250 300 350 400 450 500
yj x
Figure 2: Solution of the state equation for Problem 3 and results of measuring.
results of measurement of both state functions. Problem 4 reduces to minimizing the functional
J =J(c, θ) =
2
X
i=1 M
X
j=1
[xi(tj)−yij]2.
The problem is solved by the method of Nelder–Mead. The results of cal- culations are given in Table 9. They differ sufficiently high degree of accuracy.
The corresponding solutions of the equations of state are shown in Fig. 3. The quantity of sensitive bacteria is sharply reduced because of the action of antibi- otic. However, there is a gradual increase of the quantity of resistant bacteria because the antibiotic has no effect.
Table 9: Results of calculations for the Problem 4.
c θ
exact solution 0.1 2.1
approximate solution 0.10001 2.099 absolute error 9·10−6 3·10−5
relative error 9·10−5 1.4·10−6
In Table 10 shows the results of calculations in the case of the measuring with noise. Naturally, the error of the solution decreases in comparison with the previous case because of the noise.
Table 10: Results of calculations for Problem 2 with 10% noise.
c θ
exact solution 0.1 2.1 approximate solution 0.102 2.001
absolute error 0.002 0.091 relative error 0.02 0.04
t
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x
0 50 100 150 200 250
y1j
y2j
x1
x2
Figure 3: Solution of the state equation for Problem 3 and results of measuring.
5 Treatment by a bacteriostatic antibiotic
Consider now the process of treatment by a bacteriostatic antibiotic that sup- presses the birth rate of bacteria. In this case, we do not take into account the transitions from one type of bacteria to another, which have small effect on the system. In addition, we neglect the first term (unit) in the denominator of the first term of the first equation, which is much smaller than the second term. As a result, we have the system
˙ x1=
χa1
xθ1 −b1(x1+x2)
x1,
˙
x2= [a2−b2(x1+x2)]x2, whereχ=s−1.
The problem is to determine the parameterχ, characterizing the properties of the antibiotic, based on the measurement results of both state functions.
Thus, Problem 5 reduces to minimizing the functional J =J(χ) =
2
X
i=1 M
X
j=1
[xi(tj)−yij]2.
A gradient method is used for solving this problem. The results are given in Table 11. The calculations are carried out with different values of the exact solution of the problem. The exact solutionχ, the quantity of iterations k, the approximate solution χk, the absolute error ∆χ, and the relative errorδχ are indicated in this table.
The calculations show a sufficiently high efficiency of the algorithm.
Table 11: Calculation results for Problem 5 for different values ofχ.
χ k χk ∆χ δχ
0.1 47 0.1 10−7 10−6
1 33 1 2·10−5 2·10−5 10 84 10.0001 5·10−4 5·10−5 100 48 99.999 0.0013 10−5
6 Treatment a bacteriostatic antibiotic at mea- suring of total quantity of bacteria
A practical research of a system with a heterogeneous population of bacteria is complicated because the available results of measuring of the optical density are not able to distinguish between resistant bacteria and sensitive bacteria.
Therefore, the solution of inverse problems for the case of total measuring of quantity of bacteria only is of interest. Thus, the measurement results have the form
x1(tj) +x2(tj) =yj, j= 1, . . . , M.
We will consider of process of treatment a bacteriostatic antibiotic, when the system is described as well as for Problem 5. Thus, problem 6 reduces to minimizing the functional
J =J(χ) =
M
X
j=1
[x1(tj) +x2(tj)−yj]2.
This problem is also solved by the gradient method. The results of calcula- tions are given in Table 12.
Table 12: Calculation results for Problem 6 for different valuesχ.
χ k χk ∆χ δχ
0.1 77 0.0999 10−4 0.001 1 23 1.002 0.002 0.002 10 43 10.046 0.0456 0.0046 100 21 100.29 0.29 0.003
Comparing the results of solving Problems 5 and 6, we note that the accuracy of the solution in the case of measuring only the total quantity of bacteria has decreased, which is quite natural, since in this case the volume of the measured information was halved. Nevertheless, Problem 6 still can be analyzed. Thus using the measurement of only the total quantity of bacteria leads to quite meaningful results. The results obtained make it possible to better understand the phenomenon of antibiotic resistance and can serve as a stage on the path of further research, in particular, the study of the phenomenon of reversion, when, under the action of specific preparations, resistant bacteria become sensitive to the action of antibiotics.
Gratitude
We would like to thank M. V. Lankina and S. Kassymbekova of the Scientific Center of Anti-infectious Drugs for the natural experiment and discussion of these results.
This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan under the grant number 1746/GF4 and by the Ministry for Investments and Development under the project number 006/156.
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