Numerical Integration of the Master Equation in Some Models of Stochastic Epidemiology

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation – up to a desired precision – in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1

Signal Transduction Pathways in the Pentameric Ligand-Gated Ion Channels

The mechanisms of allosteric action within pentameric ligand-gated ion channels (pLGICs) remain to be determined. Using crystallography, site-directed mutagenesis, and two-electrode voltage clamp measurements, we identified two functionally relevant sites in the extracellular (EC) domain of the bacterial pLGIC from Gloeobacter violaceus (GLIC). One site is at the C-loop region, where the NQN mutation (D91N, E177Q, and D178N) eliminated inter-subunit salt bridges in the open-channel GLIC structure and thereby shifted the channel activation to a higher agonist concentration. The other site is below the C-loop, where binding of the anesthetic ketamine inhibited GLIC currents in a concentration dependent manner. To understand how a perturbation signal in the EC domain, either resulting from the NQN mutation or ketamine binding, is transduced to the channel gate, we have used the Perturbation-based Markovian Transmission (PMT) model to determine dynamic responses of the GLIC channel and signaling pathways upon initial perturbations in the EC domain of GLIC. Despite the existence of many possible routes for the initial perturbation signal to reach the channel gate, the PMT model in combination with Yen's algorithm revealed that perturbation signals with the highest probability flow travel either via the β1-β2 loop or through pre-TM1. The β1-β2 loop occurs in either intra- or inter-subunit pathways, while pre-TM1 occurs exclusively in inter-subunit pathways. Residues involved in both types of pathways are well supported by previous experimental data on nAChR. The direct coupling between pre-TM1 and TM2 of the adjacent subunit adds new insight into the allosteric signaling mechanism in pLGICs. © 2013 Mowrey et al

Solving the chemical master equation using sliding windows

<p>Abstract</p> <p>Background</p> <p>The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.</p> <p>Results</p> <p>In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.</p> <p>Conclusions</p> <p>The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.</p

High order structure preserving explicit methods for solving linear-quadratic optimal control problems

[EN] We consider the numerical integration of linear-quadratic optimal control problems. This problem requires the solution of a boundary value problem: a non-autonomous matrix Riccati differential equation (RDE) with final conditions coupled with the state vector equation with initial conditions. The RDE has positive definite matrix solution and to numerically preserve this qualitative property we propose first to integrate this equation backward in time with a sufficiently accurate scheme. Then, this problem turns into an initial value problem, and we analyse splitting and Magnus integrators for the forward time integration which preserve the positive definite matrix solutions for the RDE. Duplicating the time as two new coordinates and using appropriate splitting methods, high order methods preserving the desired property can be obtained. The schemes make sequential computations and do not require the storrage of intermediate results, so the storage requirements are minimal. The proposed methods are also adapted for solving linear-quadratic N-player differential games. The performance of the splitting methods can be considerably improved if the system is a perturbation of an exactly solvable problem and the system is properly split. Some numerical examples illustrate the performance of the proposed methods.The author wishes to thank the University of California San Diego for its hospitality where part of this work was done. He also acknowledges the support of the Ministerio de Ciencia e Innovacion (Spain) under the coordinated project MTM2010-18246-C03. The author also acknowledges the suggestions by the referees to improve the presentation of this work.Blanes Zamora, S. (2015). High order structure preserving explicit methods for solving linear-quadratic optimal control problems. Numerical Algorithms. 69:271-290. https://doi.org/10.1007/s11075-014-9894-0S27129069Abou-Kandil, H., Freiling, G., Ionescy, V., Jank, G.: Matrix Riccati equations in control and systems theory. Basel, Burkhäuser Verlag (2003)Al-Mohy, A.H., Higham, N.J.: Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators. SIAM. J. Sci. Comp. 33, 488–511 (2011)Anderson, B.D.O., Moore, J.B.: Optimal control: linear quadratic methods. Dover, New York (1990)Ascher, U.M., Mattheij, R.M., Russell, R.D.: Numerical solutions of boundary value problems for ordinary differential equations. Prentice-Hall, Englewood Cliffs (1988)Bader, P., Blanes, S., Ponsoda, E.: Structure preserving integrators for solving linear quadratic optimal control problems with applications to describe the flight of a quadrotor. J. Comput. Appl. Math. 262, 223–233 (2014)Basar, T., Olsder, G.J.: Dynamic non cooperative game theory, 2nd Ed, SIAM, Philadelphhia (1999)Blanes, S., Casas, F.: On the necessity of negative coefficients for operator splitting schemes of order higher than two. Appl. Num. Math. 54, 23–37 (2005)Blanes, S., Casas, F., Farrés, A., Laskar, J., Makazaga, J., Murua, A.: New families of symplectic splitting methods for numerical integration in dynamical astronomy. Appl. Numer. Math. 68, 58–72 (2013)Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009)Blanes, S., Casas, F., Ros, J.: High order optimized geometric integrators for linear differential equations. BIT 42, 262–284 (2002)Blanes, S., Diele, F., Marangi, C., Ragni, S.: Splitting and composition methods for explicit time dependence in separable dynamical systems. J. Comput. Appl. Math. 235, 646–659 (2010)Blanes, S., Moan, P.C.: Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nystrm methods. J. Comput. Appl. Math. 142, 313–330 (2002)Blanes, S., Ponsoda, E.: Magnus integrators for solving linear-quadratic differential games. J. Comput. Appl. Math. 236, 3394–3408 (2012)Brif, C., Chakrabarti, R., Rabitz, H.: Control of quantum phenomena: past, present and future. New J. Phys. 12, 075008(68pp) (2010)Cruz, J.B., Chen, C.I.: Series Nash solution of two person non zero sum linear quadratic games. J. Optim. Theory Appl. 7, 240–257 (1971)Dieci, L., Eirola, T.: Positive definitness in the numerical solution of Riccati differential quations. Numer. Math. 67, 303–313 (1994)Engwerda, J.: LQ dynamic optimization and differential games. Wiley (2005)Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations (2nd edition). Springer Series in Computational Mathematics, 31. Springer-Verlag (2006)Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286 (2010)Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, New York (1985)Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie group methods. Acta Numerica 9, 215–365 (2000)Iserles, A., Nørsett, S.P.: On the solution of linear differential equations in Lie groups. Phil. Trans. R. Soc. Lond. A 357, 983–1019 (1999)Jódar, L., Ponsoda, E.: Non-autonomous Riccati-type matrix differential equations: existence interval, construction of continuous numerical solutions and error bounds. IMA. J. Num. Anal. 15, 61–74 (1995)Jódar, L., Ponsoda, E., Company, R.: Solutions of coupled Riccati equations arising in differential games. Control. Cybern. 24, 117–128 (1995)Kaitala, V, Pohjola, M. In: Carraro, Filar (eds.) : Sustainable international agreement on greenhouse warming. A game theory study. Control and Game Theoretic Models of the Environment, pp 67–87. Birkhauser, Boston (1995)Keller, H.B.: Numerical solution of two point boundary value problems. In: CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 24. SIAM, Philadelphia (1976)McLachlan, R.I.: Composition methods in the presence of small parameters. BIT 35, 258–268 (1995)McLachlan, R.I., Quispel, R.: Splitting Methods. Acta Numer. 11, 341–434 (2002)Moler, C.B., Van Loan, C.F.: Nineteen Dubious Ways to Compute the Exponential of a Matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)Na, T.Y.: Computational methods in engineering boundary value problems. In: Mathematics in Science and Engineering, Vol. 145. Accademic Press, New York (1979)Palao, J.P., Kosloff, R.: Quantum computing by an optimal control algorithm for unitry transformations. Phys. Rev. Lett. 28 (2002)Peirce, A.P., Dahleh, M.A., Rabitz, H.: Optimal control of quantum-mechanical systems: existence, numerical approximation, and applications. Phys. Rev. A 37, 4950–4967 (1988)Reid, W.T.: Riccati Differential Equations. Academic, New York (1972)Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)Sidje, R.B.: Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Software 24, 130–156 (1998)Speyer, J.L., Jacobson, D.H.: Primer on optimal control theory. SIAM, Philadelphia (2010)Starr, A.W., Ho, Y.C.: Non-zero sum differential games. J. Optim. Theory and Appl 3, 179–197 (1969)Zhu, W., Rabitz, H.: A rapid monotonically convergent iteration algorithm for quantum optimal control ever the expectation value of a positive definite operator. J. Chem. Phys. 109, 385–391 (1998

Estimating a Markovian epidemic model using household serial interval data from the early phase of an epidemic

The clinical serial interval of an infectious disease is the time between date of symptom onset in an index case and the date of symptom onset in one of its secondary cases. It is a quantity which is commonly collected during a pandemic and is of fundamental importance to public health policy and mathematical modelling. In this paper we present a novel method for calculating the serial interval distribution for a Markovian model of household transmission dynamics. This allows the use of Bayesian MCMC methods, with explicit evaluation of the likelihood, to fit to serial interval data and infer parameters of the underlying model. We use simulated and real data to verify the accuracy of our methodology and illustrate the importance of accounting for household size. The output of our approach can be used to produce posterior distributions of population level epidemic characteristics.Andrew J. Black, Joshua V. Ros

Efficient tomography of a quantum many-body system

Quantum state tomography (QST) is the gold standard technique for obtaining an estimate for the state of small quantum systems in the laboratory [1]. Its application to systems with more than a few constituents (e.g. particles) soon becomes impractical as the e ff ort required grows exponentially with the number of constituents. Developing more e ffi cient techniques is particularly pressing as precisely-controllable quantum systems that are well beyond the reach of QST are emerging in laboratories. Motivated by this, there is a considerable ongoing e ff ort to develop new state characterisation tools for quantum many-body systems [2–11]. Here we demonstrate Matrix Product State (MPS) tomography [2], which is theoretically proven to allow the states of a broad class of quantum systems to be accurately estimated with an e ff ort that increases e ffi ciently with constituent number. We use the technique to reconstruct the dynamical state of a trapped-ion quantum simulator comprising up to 14 entangled and individually-controlled spins (qubits): a size far beyond the practical limits of QST. Our results reveal the dynamical growth of entanglement and description complexity as correlations spread out during a quench: a necessary condition for future beyond-classical performance. MPS tomography should therefore find widespread use to study large quantum many-body systems and to benchmark and verify quantum simulators and computers

A numerical study of large sparse matrix exponentials arising in Markov chains

Krylov subspace techniques have been shown to yield robust methods for the numerical computation of large sparse matrix exponentials and especially the transient solutions of Markov Chains. The attractiveness of these methods results from the fact that they allow us to compute the action of a matrix exponential operator on an operand vector without having to compute, explicitly, the matrix exponential in isolation. In this paper we compare a Krylov-based method with some of the current approaches used for computing transient solutions of Markov chains. After a brief synthesis of the features of the methods used, wide-ranging numerical comparisons are performed on a power challenge array supercomputer on three different models. (C) 1999 Elsevier Science B.V. All rights reserved.AMS Classification: 65F99; 65L05; 65U05