Exploring behaviors of stochastic differential equation models of biological systems using change of measures
 Sumit Kumar Jha^{1}Email author and
 Christopher James Langmead^{2, 3}Email author
https://doi.org/10.1186/1471210513S5S8
© Jha and Langmead; licensee BioMed Central Ltd. 2012
Published: 12 April 2012
Abstract
Stochastic Differential Equations (SDE) are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e.g., oncogenesis) can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a highlevel behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discretestate stochastic models, the analysis of SDE and other continuousstate models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, nonidentically distributed samples, and Girsanov's theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of nonidentically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur.
Background
The dynamics of biological systems are largely driven by stochastic processes and subject to random external perturbations. The consequences of such random processes are often investigated through the development and analysis of stochastic models (e.g., [1–4]). Unfortunately, the validation and analysis of stochastic models can be very challenging [5, 6], especially when the model is intended to investigate rare, but biologically significant behaviors (e.g., oncogenesis). The goal of this paper is to introduce an algorithm for examining such rare behaviors in Stochastic Differential Equation (SDE) models. The algorithm takes as input the SDE model, , and a highlevel description of a dynamical behavior, ϕ (e.g., a formula in temporal logic). It then computes a statistically rigorous bound on the probability that the given model exhibits the stated behavior using a combination of biased sampling and Bayesian Statistical Model Checking [7, 8].
Our method uses a combination of biased sampling and sequential hypothesis testing [9, 10] to estimate the probability that the model satisfies the property. Briefly, the algorithm randomly perturbs the computational model prior to generating each sample in order to expose rare but interesting behaviors. These perturbations cause a change of measure. We note that in the context of SDEs, a change of measure is itself a stochastic process and so importance sampling, which assumes that the change of measure with respect to the biased distribution is known, cannot be used. Our technique does not require us to know the exact magnitudes of the changes of measure, nor the RadonNikodym derivatives [11]. Instead, it ensures that that the geometric average of these derivatives is bounded. This is a much weaker assumption than is required by importance sampling, but we will show that it is sufficient for the purposes of obtaining bounds via sequential hypothesis testing.
Related work
Our method performs statistical model checking using hypothesis testing [12, 13], which has been used previously to analyze in a variety of domains (e.g., [14–19, 19]), including computational biology (e.g., [7, 8, 20–22]). There are two key differences between the method in this paper and existing methods. First, existing methods generate independently and identically distributed (i.i.d.) samples. Our method, in contrast, generates independent but nonidentically distributed (noni.i.d.) samples. It does so to expose rare behaviors. Second, whereas most existing methods for statistical model checking use classical statistics, our method employs Bayesian statistics [23, 24]. We have previously shown that Bayesian statistics confers advantages in the context of statistical verification in terms of efficiency and flexibility [7, 8, 19, 25].
Methods
Our method draws on concepts from several different fields. We begin by briefly surveying the semantics of stochastic differential equations, a language for formally specifying dynamic behaviors, Girsanov's theorem on change of measures, and results on consistency and concentration of Bayesian posteriors.
Stochastic differential equation models
 1.
W_{0} = 0
 2.
W_{ t } is continuous (almost surely).
 3.
W_{ t } has independent normally distributed increments:

W_{ t } W_{ s }and W_{t'} W_{s'}are independent if 0 ≤ s <t <s' <t'.

, where $\mathcal{N}\left(0,\phantom{\rule{2.77695pt}{0ex}}ts\right)$ denotes the normal distribution with mean 0 and variance t  s. Note that the symbol ~ is used to indicate "is distributed as".${W}_{t}{W}_{s}~\mathcal{N}\left(0,\phantom{\rule{2.77695pt}{0ex}}ts\right)$
In what follows, will refer to a system of stochastic differential equations. We note that a system of stochastic differential equations comes equipped with an inherent probability space and a natural probability measure μ. Our algorithm repeatedly and randomly perturbs the probability measure of the Brownian motion in the model which, in turn, changes the underlying measure in an effort to expose rare behaviors. These changes can be characterized using Girsanov's Theorem.
Girsanov's theorem for perturbing stochastic differential equation models
Here, Z_{ t } is the RadonNikodym derivative of $\widehat{P}$ with respect to P for t <T. The Brownian motion ${\u0174}_{t}$ under $\widehat{P}$ is given by: ${\u0174}_{t}={W}_{t}+{\int}_{0}^{t}{\theta}_{u}du$. The nonstochastic component of the stochastic differential equation is not affected by change of measures. Thus, a change of measure for SDEs is a stochastic process (unlike importance sampling for explicit probability distributions).
Specifying dynamic behaviors
Next, we define a formalism for encoding highlevel behavioral specification that our algorithm will test against .
Definition 1 (Adapted Finitely Monitorable). Let σ be a finitelength trace from the stochastic differential equation . A specification ϕ is said to be adapted finitely monitorable (AFM) if it is possible to decide whether σ satisfies ϕ, denoted ϕ ⊨ ϕ.
Certain AFM specifications can be expressed as formulas in Bounded Linear Temporal Logic (BLTL) [30–32]. Informally, BLTL formulas can capture the ordering of events.
Definition 2 (Probabilistic Adapted Finitely Monitorable). A specification ϕ is said to be probabilistic adapted finitely monitorable (PAFM) if it is possible to (deterministically or probabilistically) decide whether satisfies ϕ with probability at least θ, denoted $\mathcal{M}\models {P}_{\ge \theta}\left(\varphi \right)$.
Some common examples of PAFM specifications include Probabilistic Bounded Linear Temporal Logic (PBLTL) (e.g., see [19]) and Continuous Specification Logic. Note that temporal logic is only one means for constructing specifications; other formalisms can also be used, like Statecharts [33, 34].
Semantics of bounded linear temporal logic (BLTL)
 1.
σ^{ k } ⊨ x ~ v if and only if V(σ, k, x) ~ v, where v ∈ ℝ and ~ ∈ {>, <, =}.
 2.
σ^{ k } ⊨ ϕ_{1} ∨ϕ_{2} if and only if σ^{ k } ⊨ ϕ_{1} or σ^{ k } ⊨ ϕ_{2}.
 3.
σ^{ k } ⊨ ϕ_{1} ∧ ϕ_{2} if and only if σ^{ k } ⊨ ϕ_{1} and σ^{ k } ⊨ ϕ_{2};
 4.
σ^{ k } ⊨ ¬ϕ_{1} if and only if σ^{ k } ⊨ ϕ_{1} does not hold.
 5.
σ^{ k }⊨ ϕ_{1}U^{ t }ϕ_{2} if and only if there exists i ∈ ℕ such that: (a) 0 ≤ Σ_{0 ≤ l <i}Δ_{k+l}≤ t; (b) σ^{k+i}⊨ ϕ_{2}; and (c) for each 0 ≤ j < i, σ^{k+j}⊨ ϕ_{1};
Statistical model validation
Our algorithm performs statistical model checking using Bayesian sequential hypothesis testing [23] on noni.i.d. samples. The statistical model checking problem is to decide whether a model satisfies a probabilistic adapted finitely monitorable formula ϕ with probability at least θ. That is, whether $\mathcal{M}\models {P}_{\ge \theta}\left(\varphi \right)$ where θ ∈ (0, 1).
Sequential hypothesis testing
Let ρ be the (unknown but fixed) probability of the model satisfying ϕ. We can now restate the statistical model checking problem as deciding between the two composite hypotheses: H_{0} : ρ ≽ θ and H_{1} : ρ < θ. Here, the null hypothesis H_{0} indicates that satisfies the AFM formula ϕ with probability at least θ, while the alternate hypothesis H_{1} denotes that satisfies the AFM formula ϕ with probability less than θ.
Definition 3 (Type I and II errors). A Type I error is an error where the hypothesis test asserts that the null hypothesis H_{0} is false, when in fact H_{0} is true. Conversely, a Type II error is an error where the hypothesis test asserts that the null hypothesis H_{0} is true, when in fact H_{0} is false.
The basic idea behind any statistical model checking algorithm based on sequential hypothesis testing is to iteratively sample traces from the stochastic process. Each trace is then evaluated with a trace verifier [32], which determines whether the trace satisfies the specification i.e. σ_{ i } ⊨ ϕ. This is always feasible because the specifications used are adapted and finitely monitorable. Two accumulators keep track of the total number of traces sampled, and the number of satisfying traces, respectively. The procedure continues until there is enough information to reject either H_{0} or H_{1}.
Bayesian sequential hypothesis testing
Recall that for any finite trace σ_{ i } of the system and an Adapted Finitely Monitorable (AFM) formula ϕ, we can decide whether σ_{ i } satisfies ϕ. Therefore, we can define a random variable X_{ i } denoting the outcome of σ_{ i } ⊨ ϕ. Thus, X_{ i } is a Bernoulli random variable with probability mass function $f\left({x}_{i}\rho \right)={\rho}^{{x}_{i}}{\left(1\rho \right)}^{1{x}_{i}}$, where x_{ i } = 1 if and only if σ_{ i } ⊨_{ q } ϕ, otherwise x_{ i } = 0.
Bayesian statistics requires that prior probability distributions be specified for the unknown quantity, which is ρ in our case. Thus we will model the unknown quantity as a random variable u with prior density g(u). The prior probability distribution is usually based on our previous experiences and beliefs about the system. Noninformative or objective prior probability distribution [35] can be used when nothing is known about the probability of the system satisfying the AFM formula.
Suppose we have a sequence of independent random variables X_{1},..., X_{ n } defined as above, and let d = (x_{1},..., x_{ n }) denote a sample of those variables.
Definition 4. The Bayes factor of sample d and hypotheses H_{0} and H_{1} is $\mathcal{B}=\frac{P\left(d{H}_{0}\right)}{P\left(d{H}_{1}\right)}$.
We note that the Bayes factor depends on both the data d and on the prior g, so it may be considered a measure of confidence in H_{0} vs. H_{1} provided by the data x_{1},..., x_{ n }, and "weighted" by the prior g.
Noni.i.d. Bayesian sequential hypothesis testing
Traditional methods for hypothesis testing, including those outlined in the previous two subsection*s, assume that the samples are drawn i.i.d.. In this section* we show that noni.i.d. samples can also be used, provided that certain conditions hold. In particular, if one can bound the change in measure associated with the nonidentical sampling, one can can also bound the Type I and Type II errors under a change of measure. Our algorithm bounds the change of measure, and thus the error.
We begin by reviewing some fundamental concepts from Bayesian statistics including KL divergence, KL support, affinity, and δseparation, and then restate an important result on the concentration of Bayesian posteriors [35, 37].
Definition 5 (KullbackLeibler (KL) Divergence). Given a parameterized family of probability distributions {f_{ θ }}, the KullbackLeibler (KL) divergence K(θ_{0}, θ) between the distributions corresponding to two parameters θ and θ_{0} is: $K\phantom{\rule{2.77695pt}{0ex}}\left({\theta}_{0},\phantom{\rule{2.77695pt}{0ex}}\theta \right)={E}_{{\theta}_{0}}\left[{f}_{\theta}/{f}_{{\theta}_{{0}_{}}}\right]$. Note that ${E}_{{\theta}_{0}}$ is the expectation computed under the probability measure ${f}_{{\theta}_{0}}$.
Definition 6 (KL Neighborhood). Given a parameterized family of probability distributions {f_{ θ }}, the KL neighborhood K_{ ε } (θ_{0}) of a parameter value θ_{0} is given by the set {θ : K (θ_{0}, θ) < ε}.
Definition 7 (KL Support). A point θ_{0} is said to be in the KL support of a prior Π if and only if for all ε > 0, Π (K_{ ε } (θ_{0})) > 0.
Definition 8 (Affinity). The affinity Aff(f, g) between any two densities is defined as $\text{A}\text{f}\text{f}\left(f,\phantom{\rule{2.77695pt}{0ex}}g\right)=\int \sqrt{fg}d\mu $.
Definition 9 (Strong δSeparation). Let A ⊂ [0, 1] and δ > 0. The set A and the point θ_{0} are said to be strongly δseparated if and only if for any proper probability distribution v on A, $\text{A}\text{f}\text{f}\left({f}_{{\theta}_{0}},{\int}_{A}{f}_{\theta}\left({x}_{1}\right)v\left(d\theta \right)\right)<\delta $.
Given these definitions, it can be shown that the Bayesian posterior concentrates exponentially under certain technical conditions [35, 37].
Bounding errors under a change of measure
Next, we develop the machinery needed to compute bounds on the TypeI/TypeII errors for a testing strategy based on noni.i.d. samples.
A stochastic differential equation model is naturally associated with a probability measure μ. Our noni.i.d. sampling strategy can be thought of as the assignment of a set of probability measures μ_{1}, μ_{2},... to . Each unique sample σ_{ i } is associated with an implied probability measure μ_{ i } and is generated from under μ_{ i } in an i.i.d. manner. Our proofs require that all the implied probability measures are equivalent. That is, an event is possible (resp. impossible) under a probability measure if and only if it is possible (resp. impossible) under the original probability measure μ.
A sampling algorithm can compute ${p}_{{\mu}_{i}}\left({X}_{i}u\right)$ by drawing independent samples from a stochastic differential equation model under the new "modified" probability measure. We note that it is not easy to compute the change of measure $\frac{{p}_{{\mu}_{i}}\left({X}_{i}u\right)}{{p}_{\mu}\left({X}_{i}u\right)}$ algebraically or numerically. However, our algorithm does not need to compute this quantity explicitly. It simply establishes bounds on it.
Thus, by allowing the sampling algorithm to change measures by at most c, we have changed the posterior probability of observing a behavior by at most c^{2}.
Termination conditions for noni.i.d. sampling
Traditional (i.e., i.i.d.) Bayesian Sequential Hypothesis Testing is guaranteed to terminate. That is, only a finite number of samples are required before the test selects one of the hypotheses. We now consider the conditions under which a Bayesian Sequential Hypothesis Testing based procedure using noni.i.d. samples will terminate. To do this, we first need to show that the posterior probability distribution will concentrate on a particular value as we see more an more samples from the model.
To consider the conditions under which our algorithm will terminate after observing n samples, note that the factor introduced due to the change of measure c^{2n}can outweigh the gain made by the concentration of the probability measure e^{nb}. This is not surprising because our construction thus far does not force the test not to bias against a sample in an intelligent way. That is, a maliciously designed testing procedure could simply avoid the error prone regions of the design. To address this, we define the notion of a fair testing strategy that does not engage in such malicious sampling.
Note that a fair test strategy does not need to sample from the underlying distribution in an i.i.d. manner. However, it must guarantee that the probability of observing the given behavior in a large number of observations is not altered substantially by the noni.i.d. sampling. Intuitively, we want to make sure that we bias for each sample as many times as we bias against it. Our main result shows that such a long term neutrality is sufficient to generate statistical guarantees on an otherwise noni.i.d. testing procedure.
Definition 11. An ηfair test is said to be eventually fair if and only if 1 ≤ η^{4} <e^{ b }, where b is the constant in the exponential posterior concentration theorem.
The notion of a eventually fair test corresponds to a testing strategy that is not malicious or adversarial, and is making an honest attempt to sample from all the events in the long run.
Algorithm
Results and discussion
We applied our algorithm to two SDE models of tumor dynamics from the literature. The first model is a single dimensional stochastic differential equation for the influence of chemotherapy on cancer cells, and the second model is a pair of SDEs that describe an immunogenic tumor.
Lefever and Garay model
Here, x is the amount of tumor cells, A_{0}cos(ωt) denotes the influence of a periodic chemotherapy treatment, r_{0} is the linear per capita birth rate of cancer cells, K is the carrying capacity of the environment, and β represents the influence of the immune cells. Note that W_{ t } is the standard Brownian Motion, and default model parameters were those used in [38]..
We note that there are circumstances when our algorithm may require more samples than one based on i.i.d. sampling. This will happen when the property being tested has relatively high probability. For example, we tested the property that the probability of eradicating the tumor is at most 1%. The i.i.d. algorithm required 1374 samples to deny this possibility while the noni.i.d. algorithm required 1526 samples. Thus, our algorithm is best used to examine rare behaviors.
Nonlinear immunogenic tumor model
The parameters x_{1} an y_{1} denote the stochastic equilibrium point of the model. Briefly, the model assumes that the amount of noise increases with the distance to the equilibrium point.
We also considered the property that the number of tumor cells increases to 4.0 units. We evaluated whether this property is true with probability at least 0.000005 under a Bayes Factor of 100, 000. The i.i.d. sampling algorithm did not produce an answer even after observing 10, 000 samples. The noni.i.d. model validation algorithm answered affirmatively after observing 6, 736 samples. Once again, the real impact of the proposed algorithm lies in uncovering rare behaviors and bounding their probability of occurrence.
Discussion
Our results confirm that noni.i.d. sampling reduces the number of samples required in the context of hypothesis testing  when the property under consideration is rare. Moreover, the benefits of noni.i.d. sampling increase with the rarity of the property, as confirmed by the divergence of the lines in Figures 2 and 3. We note, however, that if the property isn't rare then a noni.i.d. sampling strategy will actually require a larger number of samples than an i.i.d. strategy. Thus, our algorithm is only appropriate for investigating rare behaviors.
Conclusions
We have introduced the first algorithm for verifying properties of stochastic differential equations using sequential hypothesis testing. Our technique combines Bayesian statistical model checking and noni.i.d. sampling and provides guarantees in terms of termination, and the number of samples needed to achieve those bounds. The method is most suitable when the behavior of interest is the exception and not the norm.
The present paper only considers SDEs with independent Brownian noise. We believe that these results can be extended to handle SDEs with certain kinds of correlated noise. Another interesting direction for future work is the extension of these method to stochastic partial differential equations, which are used to model spatially inhomogeneous processes. Such analysis methods could be used, for example, to investigate properties concerning spatial properties of tumors, the propagation of electrical waves in cardiac tissue, or more generally, to the diffusion processes observed in nature.
Declarations
Acknowledgements
The authors acknowledge the feedback received from the anonymous reviewers for the first IEEE Conference on Compuational Advances in Bio and Medical Sciences (ICCABS) 2011.
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 5, 2012: Selected articles from the First IEEE International Conference on Computational Advances in Bio and medical Sciences (ICCABS 2011): Bioinformatics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/13/S5.
Authors’ Affiliations
References
 Faeder JR, Blinov ML, Goldstein B, Hlavacek WS: Rulebased modeling of biochemical networks. Complexity 2005, 10(4):22–41. 10.1002/cplx.20074View ArticleGoogle Scholar
 Haigh J: Stochastic Modelling for Systems Biology by D. J. Wilkinson. Journal Of The Royal Statistical Society Series A 2007, 170: 261–261. [http://ideas.repec.org/a/bla/jorssa/v170y2007i1p261–261.html] 10.1111/j.1467985X.2006.00455_14.xView ArticleGoogle Scholar
 Iosifescu M, Tautu P, Iosifescu M: Stochastic processes and applications in biology and medicine [by] M. Iosifescu [and] P. Tautu. Editura Academiei; SpringerVerlag, Bucuresti, New York; 1973.Google Scholar
 Twycross J, Band L, Bennett MJ, King J, Krasnogor N: Stochastic and deterministic multiscale models for systems biology: an auxintransport case study. BMC Systems Biology 2010., 4(34): [http://www.biomedcentral.com/17520509/4/34/abstract]
 Kwiatkowska M, Norman G, Parker D: Stochastic Model Checking. In Formal Methods for the Design of Computer, Communication and Software Systems: Performance Evaluation (SFM'07). Volume 4486. Edited by: Bernardo M, Hillston J. Springer; 2007:220–270. LNCS (Tutorial Volume) LNCS (Tutorial Volume) 10.1007/9783540725220_6Google Scholar
 Kwiatkowska M, Norman G, Parker D: Advances and Challenges of Probabilistic Model Checking. In Proc 48th Annual Allerton Conference on Communication, Control and Computing. IEEE Press; 2010.Google Scholar
 Langmead C: Generalized Queries and Bayesian Statistical Model Checking in Dynamic Bayesian Networks: Application to Personalized Medicine. Proc of the 8th International Conference on Computational Systems Bioinformatics (CSB) 2009, 201–212.Google Scholar
 Jha SK, Clarke EM, Langmead CJ, Legay A, Platzer A, Zuliani P: A Bayesian Approach to Model Checking Biological Systems. In CMSB Volume 5688 of Lecture Notes in Computer Science. Edited by: Degano P, Gorrieri R. Springer; 2009:218–234.Google Scholar
 Wald A: Sequential Analysis. New York: John Wiley and Son; 1947.Google Scholar
 Lehmann EL, Romano JP: Testing statistical hypotheses. 3rd edition. Springer Texts in Statistics, New York: Springer; 2005.Google Scholar
 Edgar GA: RadonNikodym theorem. Duke Mathematical Journal 1975, 42: 447–450. 10.1215/S0012709475042428View ArticleGoogle Scholar
 Younes HLS, Simmons RG: Probabilistic Verification of Discrete Event Systems Using Acceptance Sampling. In CAV, Volume 2404 of Lecture Notes in Computer Science. Edited by: Brinksma E, Larsen KG. Springer; 2002:223–235.Google Scholar
 Younes HLS, Kwiatkowska MZ, Norman G, Parker D: Numerical vs. Statistical Probabilistic Model Checking: An Empirical Study. TACAS 2004, 46–60.Google Scholar
 Lassaigne R, Peyronnet S: Approximate Verification of Probabilistic Systems. PAPMPROBMIV 2002, 213–214.Google Scholar
 Hérault T, Lassaigne R, Magniette F, Peyronnet S: Approximate Probabilistic Model Checking. In Proc 5th International Conference on Verification, Model Checking and Abstract Interpretation (VMCAI'04), Volume 2937 of LNCS. Springer; 2004.Google Scholar
 Sen K, Viswanathan M, Agha G: Statistical Model Checking of BlackBox Probabilistic Systems. CAV 2004, 202–215.Google Scholar
 Grosu R, Smolka S: Monte Carlo Model Checking. CAV 2005, 271–286.Google Scholar
 Gondi K, Patel Y, Sistla AP: Monitoring the Full Range of omegaRegular Properties of Stochastic Systems. In Verification, Model Checking, and Abstract Interpretation, 10th International Conference, VMCAI 2009, Volume 5403 of LNCS. Springer; 2009:105–119.Google Scholar
 Jha SK, Langmead C, Ramesh S, Mohalik S: When to stop verification? Statistical Tradeoff between Costs and Expected Losses. Proceedings of Design Automation and test In Europe (DATE) 2011, 1309–1314.Google Scholar
 Clarke EM, Faeder JR, Langmead CJ, Harris LA, Jha SK, Legay A: Statistical Model Checking in BioLab: Applications to the Automated Analysis of TCell Receptor Signaling Pathway. CMSB 2008, 231–250.Google Scholar
 Langmead CJ, Jha SK: Predicting Protein Folding Kinetics Via Temporal Logic Model Checking. In WABI, Volume 4645 of Lecture Notes in Computer Science. Edited by: Giancarlo R, Hannenhalli S. Springer; 2007:252–264.Google Scholar
 Jha S, Langmead C: Synthesis and Infeasibility Analysis for Stochastic Models of Biochemical Systems using Statistical Model Checking and Abstraction Refinement.Google Scholar
 Jeffreys H: Theory of Probability. Oxford University Press; 1939.Google Scholar
 Gelman A, Carlin JB, Stern HS, Rubin DB: Bayesian Data Analysis. London: Chapman & Hall; 1995.Google Scholar
 Jha S, Langmead C: Exploring Behaviors of SDE Models of Biological Systems using Change of Measures. Proc of the 1st IEEE International Conference on Computational Advances in Bio and medical Sciences (ICCABS) 2011, 111–117.Google Scholar
 Øksendal B:Stochastic Differential Equations: An Introduction with Applications (Universitext). 6th edition. Springer; 2003. [http://www.amazon.com/exec/obidos/redirect?tag=citeulike0720\&path=ASIN/3540047581]View ArticleGoogle Scholar
 Karatzas I, Shreve SE:Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics). 2nd edition. Springer; 1991. [http://www.amazon.com/exec/obidos/redirect?tag=citeulike07–20\&path=ASIN/0387976558]Google Scholar
 Wang MC, Uhlenbeck GE: On the Theory of the Brownian Motion II. Reviews of Modern Physics 1945, 17(2–3):323. [http://dx.doi.org/10.1103/RevModPhys.17.323] 10.1103/RevModPhys.17.323View ArticleGoogle Scholar
 Girsanov IV: On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures. Theory of Probability and its Applications 1960, 5(3):285–301. 10.1137/1105027View ArticleGoogle Scholar
 Pnueli A: The Temporal Logic of Programs. In FOCS. IEEE; 1977:46–57.Google Scholar
 Owicki SS, Lamport L: Proving Liveness Properties of Concurrent Programs. ACM Trans Program Lang Syst 1982, 4(3):455–495. 10.1145/357172.357178View ArticleGoogle Scholar
 Finkbeiner B, Sipma H: Checking Finite Traces Using Alternating Automata. Formal Methods in System Design 2004, 24(2):101–127.View ArticleGoogle Scholar
 Harel D: Statecharts: A Visual Formalism for Complex Systems. Science of Computer Programming 1987, 8(3):231–274. [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.20.4799] 10.1016/01676423(87)900359View ArticleGoogle Scholar
 Ehrig H, Orejas F, Padberg J: Relevance, Integration and Classification of Specification Formalisms and Formal Specification Techniques.[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.7137]
 Berger J: Statistical Decision Theory and Bayesian Analysis. SpringerVerlag; 1985.View ArticleGoogle Scholar
 Jeffreys H: Theory of probability/by Harold Jeffreys. 3rd edition. Clarendon Press, Oxford; 1961.Google Scholar
 Choi T, Ramamoorthi RV: Remarks on consistency of posterior distributions. ArXiv eprints 2008.Google Scholar
 Lefever R, RGaray S: Biomathematics and Cell Kinetics. Elsevier, NorthHollan biomedical Press; 1978. chap. Local description of immune tumor rejection,:333 chap. Local description of immune tumor rejection,:333Google Scholar
 Horhat R, Horhat R, Opris D: The simulation of a stochastic model for tumourimmune system. Proceedings of the 2nd WSEAS international conference on Biomedical electronics and biomedical informatics BEBI'09, Stevens Point, Wisconsin, USA: World Scientific and Engineering Academy and Society (WSEAS); 2009, 247–252. [http://portal.acm.org/citation.cfm?id=1946539.1946584]Google Scholar
 Kuznetsov V, Makalkin I, Taylor M, Perelson A: Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bulletin of Mathematical Biology 1994, 56: 295–321. 10.1007/BF02460644 [http://dx.doi.org/10.1007/BF02460644] 10.1007/BF02460644View ArticlePubMedGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.