Comparison of stochastic and random models for bacterial resistance
- Mehmet Merdan^{1},
- Zafer Bekiryazici^{2}Email authorView ORCID ID profile,
- Tulay Kesemen^{3} and
- Tahir Khaniyev^{4}
DOI: 10.1186/s13662-017-1191-5
© The Author(s) 2017
Received: 3 November 2016
Accepted: 25 April 2017
Published: 9 May 2017
Abstract
In this study, a mathematical model of bacterial resistance considering the immune system response and antibiotic therapy is examined under random conditions. A random model consisting of random differential equations is obtained by using the existing deterministic model. Similarly, stochastic effect terms are added to the deterministic model to form a stochastic model consisting of stochastic differential equations. The results from the random and stochastic models are also compared with the results of the deterministic model to investigate the behavior of the model components under random conditions.
Keywords
stochastic differential equation random differential equation Milstein scheme Euler-Maruyama scheme antibiotic resistanceMSC
34F05 92D301 Introduction
Epidemic diseases, the effects of drugs and many other phenomena in the fields of health, medicine, biology etc. have been widely analyzed through the use of mathematical models in the last century. The majority of the models used in epidemiology include systems of deterministic differential equations. However, it is well known that some of the deterministic quantities used in modeling, e.g. epidemiological, biological events are acquired through statistical analysis of the real-life data. Many of the numerical values of the parameters used in modeling studies, especially for newly emerging diseases, the modification of an existing model for a new disease or for trial of new drugs/treatments, are determined from the statistical investigation of the limited number of data available on the event. Thus, the uncertainty of these parameters is neglected in deterministic models. This uncertainty can be modeled in the equation systems using random variables or stochastic processes. By implementing random components into the deterministic equation system and analyzing the statistical properties of the results, we aim to obtain information on the properties of the randomness of these parameters. The use of random and stochastic terms is more effective in this sense, compared with other tools of deterministic analysis or fractional calculus. In this context, we will be using random effect and stochastic noise terms on a mathematical model for antibiotic resistance to analyze the randomness of results.
The World Health Organization (WHO) describes ‘Antimicrobial Resistance (AMR)’ as follows: Resistance of a microorganism to an antimicrobial drug that was originally effective for treatment of infections caused by it. WHO reports that without effective anti-infective treatment, many standard medical treatments will fail or turn into very high risk procedures [1]. Antibiotic therapy is the most common method for battling bacterial infections worldwide. Most of the classes of antibiotics used today have been introduced in the ‘golden era of antibiotics’ 1940s-1960s [2]. However, various factors have caused a decline in research aimed at the discovery of novel antibacterial agents [3, 4]. Only five new significant classes of antibiotics were discovered in the last 40 years [5]. A novel treatment for bacterial infections is necessary, since the introduction of every antibiotic is followed by the occurrence of new bacteria resistant to that class [6]. Resistance is acquired either through mutations in the chromosome (vertical evolution) or through conjugation and transduction which can take place between the same or different bacteria that may also cause multiple drug resistance (horizontal evolution) [5, 7, 8]. The public threat of antibiotic resistance is also affected by inappropriate use of anti-infective medicine for human and animal food production, together with inadequate measures to control the spread of diseases. Increasing resistance of bacteria has social and economic implications like growing morbidity and mortality from infections and the rise of treatment costs [1, 8].
There are many other mathematical models which analyze antibiotic resistance from different perspectives: Dasbasi and Ozturk investigated a model of bacterial resistance to multiple drugs and immune system response [9]; Ternent et al. used a model of combined antibiotic and anti-virulent treatment to monitor the population dynamics of bacteria [5]; Ibarguen-Mondragon et al. evaluated a model on bacterial resistance to multiple antibiotics caused by spontaneous mutations [10]; D’Agata et al. studied a model of antibiotic resistance in hospitals [8]; Austin and Anderson investigated antibiotic resistance within patient, hospital and the community using a simple mathematical model [11]. Most of the models on bacterial resistance are compartmental models, investigating the course of the event through changes in the compartments of the model using deterministic differential equation systems. There are also some mathematical models that study fractional derivation on several phenomena like river blindness disease [12]; lassa hemorrhagic fever [13]; rubella disease [14]; immunegenic tumors [15] and heat transfer [16]. Since we concentrate on the random nature of the parameters of the antibiotic resistance model, we will be implementing random effects into the deterministic system rather than fractional derivation. Some modeling studies which compare stochastic and deterministic models in this regard can be given as follows: Imran et al. compared the models for analyzing Hepatitis C [17]; Lahrouz et al. used a SIRS epidemic model for comparison [18] and Bekiryazici et al. compared the results of models for Dengue disease [19].
In this study, we will be using a deterministic model of immune system response and bacterial resistance with antibiotic therapy to form random and stochastic models of the event [9]. The random and stochastic models will be obtained by adding random effect and stochastic noise terms into the deterministic model to analyze the uncertainty of the antibiotic resistance. The motivation of this work is the previous studies of the authors where the random dynamics of an avian-human influenza model [20] and a biochemical reaction model [21] were analyzed. Similarly, the random behavior of the solutions of our model will be analyzed from the simulations and solutions of the random and stochastic models. First, the deterministic solution of the model will be given along with the phase portraits to investigate the deterministic behavior of the model components. Using the deterministic model, a random model of antibiotic resistance will be obtained by adding random effect terms to the parameters of the equation system. The random parameters will represent the uncertainty in the nature of the parameters and, thus, we will be able to analyze the randomness of the system though the solutions of the random system. Also, a stochastic model of antibiotic resistance will be obtained by adding stochastic noise terms to the deterministic model. Solutions of all three models will be given in the conclusion, where a comparison of the results will underline the random behavior of the model and the interpretations of variations from the deterministic model. The statistical properties of the random results will provide useful insights for the uncertainty in the behavior of antibiotic resistance which cannot be modeled by using deterministic equations.
2 Deterministic model of antibiotic resistance
Descriptions and values of the parameters
Parameter | Description | Value |
---|---|---|
\(\beta_{S}\) | Birth rate of sensitive bacteria | \(0.8~\mbox{day}^{-1}\) |
T | Carrying capacity of bacterial population | \(10^{9}~\mbox{cells}\) |
c | Fitness cost | 0.5 (dimensionless) |
η̄ | Death rate of sensitive and resistant bacteria | \(0.3~\mbox{day}^{-1}\) |
μ | Sensitive bac. mutation rate by exposure to antibiotic | \(10^{-6}~\mbox{mut} \times \mbox{gen}\) |
σ | Conjugation rate of bacteria | \(10^{-5}~\mbox{day}^{-1}\) |
\(E_{\max}\) | Maximum killing rate of antibiotic | \(26.4~\mbox{day}^{-1}\) |
\(E_{50}\) | Antibiotic concentration for half max. kill rate | \(5~\mu\mbox{g}/\mbox{ml}\) |
\(\beta_{B}\) | Recruitment rate of immune cells | \(3~\mbox{day}^{-1}\) |
Λ | Carrying capacity of immune cells | \(1.8\times10^{5}~\mbox{cells}\) |
λ | Loss rate of immune cells by apoptosis | \(6\times10^{-6}~\mbox{cells}^{-1}\mbox{ days}^{-1}\) |
α | Dose of antibiotic administration | \(5~\mbox{mg}/\mbox{kg}/\mbox{day}\) |
2.1 Deterministic results
Extremum values obtained for the variables
Variable | Maximum | Time | Minimum | Time |
---|---|---|---|---|
S(t) | 6,000 | 0 | 5.824 | 1.09 |
R(t) | 20.16 | 0.09 | 0.004912 | 1.5 |
B(t) | 89.91 | 1.5 | 1 | 0 |
A(t) | 4 | 0 | 0.002193 | 1.5 |
The number of susceptible cells decreases to 5.824 around \(t=1.09\) meaning almost all of the susceptible bacteria either gain resistance or are destroyed through the process. Taking the changes in the number of resistant bacteria into account, it can be said that a part of the susceptible bacteria turn to resistant bacteria in the beginning, but all of these are destroyed afterwards. The number of resistant bacteria slightly increases in the beginning of the process and hits its maximum value 20.16 at \(t=0.09\), after which it starts decreasing. This shows that susceptible bacteria conjugate with resistant bacteria and some of the susceptible bacteria gain resistance to antibiotics. However, both the resistant bacteria and the susceptible bacteria are almost cleared through the end of the process. The size of the immune cell population keeps increasing and hits its maximum value 89.91 at the end of the process, \(t=1.5\), meaning the body keeps sending immune cells to the region of infection until all the bacteria are cleared. The concentration of the antibiotic keeps decreasing throughout the process until its hits its minimum value 0.002193 at the end of the process \(t=1.5\). The equation system used in this study is a well-posed model and the results obtained from the model are compatible with the results of the referred study for the parameters under consideration. The results for the deterministic, random and stochastic models will be investigated for the first 36 hours of the event (\(t\in(0,1.5)\)). Although the random nature of the model components is the main focus of this study, the steady states and other dynamics of the model can provide an in-depth analysis of the deterministic event [24, 25].
2.2 Deterministic behavior of model components
3 Random model of antibiotic resistance
3.1 Random results
Matlab produces random numbers according to the standard normal distribution with the randn command. We simulate the numerical solutions of the random system using the fourth order Runge-Kutta method. We produce more than 10^{5} simulations of the random model. The characteristics of the random systems obtained by using these simulations will be used to interpret the random behavior of the model. Note that although the equations contain random numbers, a system of deterministic equations is obtained for every simulation. The coefficients vary from simulation to simulation, but the equations and therefore the derivatives are deterministic.
3.1.1 Solution curves
Extremum values of variables
Variable | Maximum | Time | Minimum | Time |
---|---|---|---|---|
S(t) | 6,000 | 0 | 0.3081 | 1.5 |
R(t) | 20.31 | 0.15 | 0.01054 | 1.5 |
B(t) | 92.42 | 1.5 | 1 | 0 |
A(t) | 4 | 0 | 0.002193 | 1.5 |
3.1.2 Expected values
Extremum values of the expected values
Variable | Maximum | Time | Minimum | Time |
---|---|---|---|---|
E[S(t)] | 6,000 | 0 | 0.3442 | 1.5 |
E[R(t)] | 20.18 | 0.135 | 0.01376 | 1.5 |
E[B(t)] | 91.59 | 1.5 | 1 | 0 |
E[A(t)] | 4 | 0 | 0.002386 | 1.5 |
Figure 6 shows the expectations of the random variables. These expectations are approximately valid for all possible trials of the event under the assumed random conditions. The expected values are similar to the deterministic results, as expected.
3.1.3 Variances
Extremum values of the variances
Variable | Maximum | Time | Minimum | Time |
---|---|---|---|---|
Var[S(t)] | 18,740 | 0.105 | 0 | 0 |
Var[R(t)] | 1.202 | 0.915 | 0 | 0 |
Var[B(t)] | 447.6 | 1.5 | 0 | 0 |
Var[A(t)] | 0.005509 | 0.195 | 0 | 0 |
Figure 7 shows the variance of the random variables. A high variance of the model components show that some of the variables may produce significantly different behavior from the suggestions of the deterministic results. Once again, the results for the variance are approximately valid for all trials.
3.1.4 Confidence intervals
Extremum values of the confidence intervals
Variable | Maximum | Time | Minimum | Time |
---|---|---|---|---|
CI[S(t)] | 6,000 | 0 | 0.3442 | 1.5 |
CI[R(t)] | 20.5 | 0.18 | 0.01376 | 1.5 |
CI[B(t)] | 155.1 | 1.5 | 1 | 0 |
CI[A(t)] | 4 | 0 | 6.083 × 10^{−5} | 1.275 |
The random behavior of the components of the model can be seen in the graphs. It is seen that the random behavior of the variables are similar to their deterministic behavior. Although there are numerical differences in the solutions of the random system (2), the graphs of the solutions are almost the same. Considering the random results, it can be said that the model produces the same results under random conditions, only at different quantities. It should be noted that the simulations of the model are compatible with the deterministic results and the real-life behavior of the components. The confidence intervals are obtained by using the results for the variance and the expectations of the components and may tend to provide negative results when the expectation of a variable tends to zero. However, these should be neglected since the results for the subpopulations of an epidemic model can only assume positive values [24, 25].
Deterministic results indicate that the population of resistant bacteria will reach its maximum value \(\max(R(t))=20.16\) at \(t=0.09\), which is roughly about 2 hours after the start of the process. Random results suggest that the rise in the number of resistant bacteria is expected to continue a little longer until about 3.5 hours into the process, when \(\max(R(t))=20.31\) will be obtained at \(t=0.15\). On the other hand, deterministic results and random results are quite similar in numerical values besides the behavior for the variables \(S(t)\), \(B(t)\) and \(A(t)\). \(S(t)\), the population of susceptible bacteria, decreases throughout the process in both cases, though slower in the random model, reaching very similar extremum values in both ends. Similarly, \(B(t)\), the population of immune cells, increases throughout the process in both cases, reaching very similar extremum values at both ends. \(A(t)\), the concentration of the antibiotic, also decreases through the process in both cases, reaching very similar extremum values at both ends.
4 Stochastic model of antibiotic resistance
Random outcomes in real life can be modeled in equation systems by the use of stochastic differential equations too. Their difference from random DEs is that stochastic differential equations contain noise terms in terms of Wiener processes. Although these equations are written as differential equations, they are interpreted as integral equations. In this study, we will be interested in the numerical solutions of SDEs.
4.1 Stochastic results
4.1.1 Euler-Maruyama
Extremum values of the realizations
Variable | Maximum | Time | Minimum | Time |
---|---|---|---|---|
S(t) | 6,206 | 0.001 | 0.1078 | 1.5 |
R(t) | 30.13 | 0.39 | 0.002046 | 1.5 |
B(t) | 97.83 | 1.477 | 0.9254 | 0.008 |
A(t) | 4 | 0 | 0.001317 | 1.499 |
4.1.2 Milstein
Extremum values of the realizations
Variable | Maximum | Time | Minimum | Time |
---|---|---|---|---|
S(t) | 6,000 | 0 | 1.229 | 1.5 |
R(t) | 20.78 | 0.024 | 0.01164 | 1.5 |
B(t) | 88.59 | 1.487 | 0.946 | 0.071 |
A(t) | 4.114 | 0.001 | 0.0006289 | 1.475 |
It can be seen that the expectations of the random model and the realizations of the stochastic model produce similar patterns throughout the process. The similarity in the results of these two models will be investigated in the next part.
5 Comparison of random and stochastic results
Extremum values in deterministic, random and stochastic models
Det. Max | Rand. Max | Stoch. Max | Det. Min | Rand. Min | Stoch. Min | |
---|---|---|---|---|---|---|
S | (6,000,0) | (6,000,0) | (6,000,0) | (5.824,1.09) | (0.3442,1.5) | (1.229,1.5) |
R | (20.16,0.09) | (20.18,0.135) | (20.78,0.024) | (0.004912,1.5) | (0.01376,1.5) | (0.01164,1.5) |
B | (89.91,1.5) | (91.59,1.5) | (88.59,1.487) | (1,0) | (1,0) | (0.946,0.071) |
A | (4.0) | (4,0) | (4.114,0.001) | (0.002193,1.5) | (0.002386,1.5) | (0.0006289,1.475) |
Table 9 shows the impact of the random effects and stochastic noise on the extreme values of the variables. While there are some numerical differences in some of the values, together with the graphs, it can be said that the stochastic and random models are meaningful and successful in displaying the randomness of bacterial resistance.
6 Conclusion
In this study, the existing deterministic model for bacterial resistance with immune system response and antibiotic therapy was used to form random and stochastic models of the case. The parameters of the deterministic model are obtained from a statistical evaluation of various bacterial resistance cases from around the world, but the deterministic analysis neglects the randomness of these parameters and regards these quantities as stable values. Thus, by introducing Gaussian distributed random effect terms, we formed a random equation system that models the real-life randomness of bacterial resistance. A random analysis for the dynamics of the antibiotic resistance model is authentic and shows that, even under small random effects, some of the components produce very unlikely results. The comparison between the deterministic and random results shows a correspondence between the behaviors of the models, while the random effects cause an inevitable difference in the random case, as expected. Although the standard deviations of the random effects added to the parameters were 5% of their deterministic values, it can be seen from the results that especially the population of resistant bacteria and the population of immune cells are significantly affected from these variations. The immune cell population size, \(B(t)\), gets its maximum expected value 92.42 at \(t=1.5\). The maximum standard deviation of \(B(t)\) is estimated as 21.16 at \(t=1.5\) using the results of its variance. This value is approximately 22.89% of its expectation, meaning that under the assumed random conditions, the immune cell population could vary at most about 22.89% from its expectation, or similarly from the estimations of the deterministic model, which neglects the randomness of the event. Similarly, the population size of the resistant bacteria, \(R(t)\), gets a maximum standard deviation at \(t=0.915\) with an approximate value of 1.10. On the other hand, \(R(t)\) has an approximate expected value of 6.849 at \(t=0.915\). Thus, the number of resistant bacteria could vary up to 16.06% at the time of its maximum standard deviation. The amount of randomness in these two variables can also be visualized from the confidence interval graphs. Confidence interval graphs of 99% show the intervals within three standard deviations from the expectation and a large interval for these two variables show that neglecting the randomness in the model could result in seriously misleading information. A deviation of 5% in the parameters of the random model causes a maximum of 22.89% deviation for \(B(t)\), 16.06% maximum deviation for \(R(t)\), 6.19% maximum deviation for \(S(t)\) and 4.88% maximum deviation for \(A(t)\). The significant deviations in \(B(t)\) and \(R(t)\) were also seen in the stochastic graphs. The volatility in the realizations of the model under stochastic noise is in correspondence with the confidence intervals of the random variables. For instance, \(R(t)\) gets values between 18.51 and 12.639 in the time interval \([0.7,0.824]\), just as the 99% confidence interval suggests.
The non-negligible deviation and volatility in the random and stochastic models show that the deterministic analysis of bacterial resistance is incapable of modeling the real-life randomness of the event. Our analysis shows that even with small random effects and small diffusion coefficients, the populations of resistant bacteria and immune cells show significant deviations from the deterministic results. The analyses in this study could be continued by using various values for the parameters or various distributions for the random effects. The magnitudes of the random effects or the diffusion coefficients of the stochastic noise could also be altered to monitor the effects of these changes on the results. These suggestions provide a basis for new studies and these options for random models will be evaluated in new papers. Hence, we believe this study will stimulate similar random investigations of models from all areas of science.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- World Health Organization: Antimicrobial resistance report (2015)
- Lewis, K: Platforms for antibiotic discovery. Nat. Rev. Drug Discov. 12(5), 371-387 (2013) View ArticleGoogle Scholar
- Mellbye, B, Schuster, M: The sociomicrobiology of antivirulence drug resistance: a proof of concept. mBio 2(5), e00131-11 (2011) View ArticleGoogle Scholar
- Projan, S, Shlaes, D: Antibacterial drug discovery: is it all down hill from here? Clin. Microbiol. Infect. 10(s4), 18-22 (2004) View ArticleGoogle Scholar
- Ternent, L, Dyson, RJ, Krachler, A-M, Jabbari, S: Bacterial fitness shapes the population dynamics of antibiotic resistant and -susceptible bacteria in a model of combined antibiotic and anti-virulent treatment. J. Theor. Biol. 372, 1-11 (2015) View ArticleMATHGoogle Scholar
- Clatworthy, A, Pierson, E, Hung, D: Targeting virulence: a new paradigm for antimicrobial therapy. Nat. Chem. Biol. 3(9), 541-548 (2007) View ArticleGoogle Scholar
- Dasbasi, B, Ozturk, I: Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response. SpringerPlus 5, 408 (2016) View ArticleGoogle Scholar
- D’Agata, EMC, Magal, P, Olivier, D, Ruan, S, Webb, GF: Modeling antibiotic resistance in hospitals: the impact of minimizing treatment duration. J. Theor. Biol. 249(3), 487-499 (2007) MathSciNetView ArticleGoogle Scholar
- Dasbasi, B, Ozturk, I: Mathematical modelling of immune system response and bacterial resistance with antibiotic therapy. In: Proceedings of the International Conference on Mathematics and Mathematics Education, Elazig/Turkey, 12-14 May 2016, pp. 348-350 (2016) Google Scholar
- Ibargüen-Mondragón, E, Mosquera, S, Cerón, M, Burbano-Rosero, EM, Hidalgo-Bonilla, SP, Esteva, L, Romero-Leitóne, JP: Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations. Biosystems 117, 60-67 (2014) View ArticleGoogle Scholar
- Austin, DJ, Anderson, RM: Studies of antibiotic resistance within the patient, hospitals and the community using simple mathematical models. Philos. Trans. R. Soc. Lond. B 354(1384), 721-738 (1999) View ArticleGoogle Scholar
- Atangana, A, Alqahtani, RT: Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative. Entropy 18(2), 40 (2016) View ArticleGoogle Scholar
- Atangana, A: A novel model for the lassa hemorrhagic fever: deathly disease for pregnant women. Neural Comput. Appl. 26(8), 1895-1903 (2015) View ArticleGoogle Scholar
- Atangana, A, Alkahtani, BST: Modeling the spread of rubella disease using the concept of with local derivative with fractional parameter. Complexity 21(6), 442-451 (2016) MathSciNetView ArticleGoogle Scholar
- Arshad, S, Baleanu, D, Huang, J, Tang, Y, Al Qurashi, MM: Dynamical analysis of fractional order model of immunogenic tumors. Adv. Mech. Eng. 8(7), 1-13 (2016) View ArticleGoogle Scholar
- Atangana, A, Baleanu, D: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763-769 (2016) View ArticleGoogle Scholar
- Imran, M, Hassan, M, Dur-E-Ahmad, M, Khan, A: A comparison of deterministic and stochastic model for hepatitis C with an isolation stage. J. Biol. Dyn. 7(1), 276-301 (2013) View ArticleGoogle Scholar
- Lahrouz, A, Omari, L, Settati, A, Belmaati, A: Comparison of deterministic and stochastic SIRS epidemic model with saturating incidence and immigration. Arab. J. Math. 4 (2), 101-116 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Bekiryazici, Z, Merdan, M, Kesemen, T, Najmuldeen, M: Mathematical modeling of dengue disease under random effects. Math. Sci. Appl. E-Notes 4(2), 58-70 (2016) MathSciNetMATHGoogle Scholar
- Merdan, M, Khaniyev, T: On the behavior of solutions under the influence of stochastic effect of avian-human influenza epidemic model. Int. J. Biotechnol. Biochem. 4(1), 75-100 (2008) Google Scholar
- Bekiryazici, Z, Merdan, M, Kesemen, T, Khaniyev, T: Random modeling of biochemical reactions under Gaussian random effects. Turk. J. Math. Comput. Sci. 5, 8-18 (2016) Google Scholar
- Pugliese, A, Gandolfi, A: A simple model of pathogen-immune dynamics including specific and non-specific immunity. Math. Biosci. 214(1), 73-80 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Coll, P: Drugs with activity against Mycobacterium tuberculosis. Enferm. Infecc. Microbiol. Clín. 21(6), 299-307 (2003) View ArticleGoogle Scholar
- González-Parra, G, Arenas, AJ, Chen-Charpentier, BM: Combination of nonstandard schemes and Richardson’s extrapolation to improve the numerical solution of population models. Math. Comput. Model. 52(7), 1030-1036 (2010) View ArticleMATHGoogle Scholar
- Arenas, AJ, González-Parra, G, Chen-Charpentier, BM: A nonstandard numerical scheme of predictor-corrector type for epidemic models. Comput. Math. Appl. 59(12), 3740-3749 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Soong, TT: Random Differential Equations in Science and Engineering. Academic Press, New York (1973) MATHGoogle Scholar
- Feller, W: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968) MATHGoogle Scholar
- Kloeden, PE, Platen, E: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992) View ArticleMATHGoogle Scholar
- Cyganowski, S, Kloeden, P, Ombach, J: From Elementary Probability to Stochastic Differential Equations with MAPLE. Springer, Berlin (2001) MATHGoogle Scholar