Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

[EN] This paper presents a full probabilistic description of the solution of random SI-type epidemiological models which are based on nonlinear differential equations. This description consists of determining: the first probability density function of the solution in terms of the density functions of the diffusion coefficient and the initial condition, which are assumed to be independent random variables; the expectation and variance functions of the solution as well as confidence intervals and, finally, the distribution of time until a given proportion of susceptibles remains in the population. The obtained formulas are general since they are valid regardless the probability distributions assigned to the random inputs. We also present a pair of illustrative examples including in one of them the application of the theoretical results to model the diffusion of a technology using real data.This work has been partially supported by the Ministerio de Economia y Competitividad Grants MTM2013-41765-P and TRA2012-36932.Casabán Bartual, MC.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation. 24(1):86-97. https://doi.org/10.1016/j.cnsns.2014.12.016S869724

Mean square solution of Bessel differential equation with uncertainties

[EN] This paper deals with the study of a Bessel-type differential equation where input parameters (coefficient and initial conditions) are assumed to be random variables. Using the so-called Lp-random calculus and assuming moment conditions on the random variables in the equation, a mean square convergent generalized power series solution is constructed. As a result of this convergence, the sequences of the mean and standard deviation obtained from the truncated power series solution are convergent as well. The results obtained in the random framework extend their deterministic counterpart. The theory is illustrated in two examples in which several distributions on the random inputs are assumed. Finally, we show through examples that the proposed method is computationally faster than Monte Carlo method.This work has been partially supported by the Spanish Ministerio de Economía y Competitividad grant MTM2013-41765-P and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement No. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and Mexican Conacyt.Cortés, J.; Jódar Sánchez, LA.; Villafuerte, L. (2017). Mean square solution of Bessel differential equation with uncertainties. Journal of Computational and Applied Mathematics. 309:383-395. https://doi.org/10.1016/j.cam.2016.01.034S38339530

A comprehensive probabilistic solution of random SIS-type epidemiological models using the Random Variable Transformation technique

[EN] This paper provides a complete probabilistic description of SIS-type epidemiological models where all the input parameters (contagion rate, recovery rate and initial conditions) are assumed to be random variables. By applying the Random Variable Transformation technique, the first probability density function, the mean and the variance functions as well as confidence intervals associated to the solution of SIS-type epidemiological models are determined under the general hypothesis that the random inputs have any joint probability density function. The distributions to describe the time until a given proportion of the population remains susceptible and infected are also determined. Lastly, a probabilistic description of the so-called basic reproductive number is included. The theoretical results are applied to an illustrative example showing good fitting.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P and TRA2012-36932. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Casabán, M.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation. 32:199-210. https://doi.org/10.1016/j.cnsns.2015.08.009S1992103

Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques

[EN] This paper deals with the determination of the first probability density function of the solution stochastic process to the homogeneous Riccati differential equation taking advantage of both linearization and Random Variable Transformation techniques. The study is split in all possible casuistries regarding the deterministic/random character of the involved input parameters. An illustrative example is provided for each one of the considered cases.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P and TRA2012-36932.Casabán, MC.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2016). Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques. Journal of Computational and Applied Mathematics. 291:20-35. https://doi.org/10.1016/j.cam.2014.11.028S203529

Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function

[EN] This paper deals with the approximate computation of the first probability density function of the solution stochastic process to second-order linear differential equations with random analytic coefficients about ordinary points under very general hypotheses. Our approach is based on considering approximations of the solution stochastic process via truncated power series solution obtained from the random regular power series method together with the so-called Random Variable Transformation technique. The validity of the proposed method is shown through several illustrative examples.This work has been partially supported by the Ministerio de Econom ia y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation. 331:33-45. https://doi.org/10.1016/j.amc.2018.02.051S334533

Solving random homogeneous linear second-order differential equations: a full probabilistic description

[EN] In this paper a complete probabilistic description for the solution of random homogeneous linear second-order differential equations via the computation of its two first probability density functions is given. As a consequence, all unidimensional and two-dimensional statistical moments can be straightforwardly determined, in particular, mean, variance and covariance functions, as well as the first-order conditional law. With the aim of providing more generality, in a first step, all involved input parameters are assumed to be statistically dependent random variables having an arbitrary joint probability density function. Second, the particular case that just initial conditions are random variables is also analysed. Both problems have common and distinctive feature which are highlighted in our analysis. The study is based on random variable transformation method. As a consequence of our study, the well-known deterministic results are nicely generalized. Several illustrative examples are included.This work has been partially supported by the Spanish M. C. Y. T. Grant MTM2013-41765-P.Casabán, M.; Cortés, J.; Romero, J.; Roselló, M. (2016). Solving random homogeneous linear second-order differential equations: a full probabilistic description. Mediterranean Journal of Mathematics. 13(6):3817-3836. https://doi.org/10.1007/s00009-016-0716-6S38173836136Øksendal B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2007)Soong T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Neckel, T., Rupp, F.: Random Differential Equations in Scientific Computing. Versita, London (2013)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 1–18 (2014). doi: 10.1007/s00009-014-0452-8Villafuerte, L., Chen-Charpentier, B.M.: A random differential transform method: theory and applications. Appl. Math. Lett. 25(10), 1490–1494 (2012). doi: 10.1016/j.aml.2011.12.033Licea, J.A., Villafuerte, L., Chen-Charpentier, B.M.: Analytic and numerical solutions of a Riccati differential equation with random coefficients. J. Comput. Appl. Math. 239, 208–219 (2013). doi: 10.1016/j.cam.2012.09.040Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Probabilistic solution of random homogeneous linear second-order difference equations. Appl. Math. Lett. 34, 27–32 (2014). doi: 10.1016/j.aml.2014.03.010Santos, L.T., Dorini, F.A., Cunha, M.C.C.: The probability density function to the random linear transport equation. Appl. Math. Comput. 216 (5), 1524–1530 (2010). doi: 10.16/j.amc.2010.03.001El-Tawil, M., El-Tahan, W., Hussein, A.: Using FEM-RVT technique for solving a randomly excited ordinary differential equation with a random operator. Appl. Math. Comput. 187(2), 856–867 (2007). doi: 10.1016/j.amc.2006.08.164Hussein, A., Selim, M.M.: Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Appl. Math. Comput. 218(13), 7193–7203 (2012). doi: 10.1016/j.amc.2011.12.088Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Probabilistic solution of random SI-type epidemiological models using the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul. 24(1–3), 86–97 (2015). doi: 10.1016/j.cnsns.2014.12.016Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Determining the first probability density function of linear random initial value problems by the random variable transformation (RVT) technique: a comprehensive study. In: Abstract and Applied Analysis 2014-ID248512, pp. 1–25 (2014). doi: 10.1155/2013/248512Casabán, M.C., Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Villanueva, R.J.: A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul. 32, 199–210 (2016). doi: 10.1016/j.cnsns.2015.08.009El-Wakil, S.A., Sallah, M., El-Hanbaly, A.M.: Random variable transformation for generalized stochastic radiative transfer in finite participating slab media. Phys. A 435 66–79 (2015). doi: 10.1016/j.physa.2015.04.033Dorini, F.A., Cunha, M.C.C.: On the linear advection equation subject to random fields velocity. Math. Comput. Simul. 82, 679–690 (2011). doi: 10.16/j.matcom.2011.10.008Dhople, S.V., Domínguez-García, D.: A parametric uncertainty analysis method for Markov reliability and reward models. IEEE Trans. Reliab. 61(3), 634–648 (2012). doi: 10.1109/TR.2012.2208299Williams, M.M.R.: Polynomial chaos functions and stochastic differential equations. Ann. Nucl. Energy 33(9), 774–785 (2006). doi: 10.1016/j.anucene.2006.04.005Chen-Charpentier, B.M., Stanescu, D.: Epidemic models with random coefficients. Math. Comput. Model. 52(7/8), 1004–1010 (2009). doi: 10.1016/j.mcm.2010.01.014Papoulis A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1991

Solving second-order linear differential equations with random analytic coefficients about regular-singular points

[EN] In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation.This work was partially funded by the Ministerio de Economia y Competitividad Grant MTM2017-89664-P. Ana Navarro Quiles acknowledges the funding received from Generalitat Valenciana through a postdoctoral contract (APOSTD/2019/128). Computations were carried out thanks to the collaboration of Raul San Julian Garces and Elena Lopez Navarro granted by the European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 2014-2020, Grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectivelyCortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2020). Solving second-order linear differential equations with random analytic coefficients about regular-singular points. Mathematics. 8(2):1-20. https://doi.org/10.3390/math8020230S12082Hussein, A., & Selim, M. M. (2012). Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Applied Mathematics and Computation, 218(13), 7193-7203. doi:10.1016/j.amc.2011.12.088Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Santos, L. T., Dorini, F. A., & Cunha, M. C. C. (2010). The probability density function to the random linear transport equation. Applied Mathematics and Computation, 216(5), 1524-1530. doi:10.1016/j.amc.2010.03.001Hussein, A., & Selim, M. M. (2019). A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique. Journal of Quantitative Spectroscopy and Radiative Transfer, 232, 54-65. doi:10.1016/j.jqsrt.2019.04.034Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation, 331, 33-45. doi:10.1016/j.amc.2018.02.051Cortés, J.-C., Jódar, L., Camacho, F., & Villafuerte, L. (2010). Random Airy type differential equations: Mean square exact and numerical solutions. Computers & Mathematics with Applications, 60(5), 1237-1244. doi:10.1016/j.camwa.2010.05.046Calbo, G., Cortés, J.-C., & Jódar, L. (2011). Random Hermite differential equations: Mean square power series solutions and statistical properties. Applied Mathematics and Computation, 218(7), 3654-3666. doi:10.1016/j.amc.2011.09.008Calbo, G., Cortés, J.-C., Jódar, L., & Villafuerte, L. (2011). Solving the random Legendre differential equation: Mean square power series solution and its statistical functions. Computers & Mathematics with Applications, 61(9), 2782-2792. doi:10.1016/j.camwa.2011.03.045Cortés, J.-C., Villafuerte, L., & Burgos, C. (2017). A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation. Mediterranean Journal of Mathematics, 14(1). doi:10.1007/s00009-017-0853-6Cortés, J.-C., Jódar, L., & Villafuerte, L. (2017). Mean square solution of Bessel differential equation with uncertainties. Journal of Computational and Applied Mathematics, 309, 383-395. doi:10.1016/j.cam.2016.01.034Khudair, A. R., Haddad, S. A. M., & Khalaf, S. L. (2016). Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method. Open Journal of Applied Sciences, 06(04), 287-297. doi:10.4236/ojapps.2016.64028Qi, Y. (2018). A Very Brief Introduction to Nonnegative Tensors from the Geometric Viewpoint. Mathematics, 6(11), 230. doi:10.3390/math6110230Ragusa, M. A., & Tachikawa, A. (2016). Boundary regularity of minimizers of p(x)-energy functionals. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 33(2), 451-476. doi:10.1016/j.anihpc.2014.11.003Ragusa, M. A., & Tachikawa, A. (2019). Regularity for minimizers for functionals of double phase with variable exponents. Advances in Nonlinear Analysis, 9(1), 710-728. doi:10.1515/anona-2020-0022Braumann, C. A., Cortés, J.-C., Jódar, L., & Villafuerte, L. (2018). On the random gamma function: Theory and computing. Journal of Computational and Applied Mathematics, 335, 142-155. doi:10.1016/j.cam.2017.11.04

A mean square chain rule and its application in solving the random Chebyshev differential equation

[EN] In this paper a new version of the chain rule for calculat- ing the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the cor- responding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.This work was completed with the support of our TEX-pert.Cortés, J.; Villafuerte, L.; Burgos-Simon, C. (2017). A mean square chain rule and its application in solving the random Chebyshev differential equation. Mediterranean Journal of Mathematics. 14(1):14-35. https://doi.org/10.1007/s00009-017-0853-6S1435141Calbo, G., Cortés, J.C., Jódar, L., Villafuerte, L.: Analytic stochastic process solutions of second-order random differential equations. Appl. Math. Lett. 23(12), 1421–1424 (2010). doi: 10.1016/j.aml.2010.07.011El-Tawil, M.A., El-Sohaly, M.: Mean square numerical methods for initial value random differential equations. Open J. Discret. Math. 1(1), 164–171 (2011). doi: 10.4236/ojdm.2011.12009Khodabin, M., Maleknejad, K., Rostami, K., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge Kutta methods. Math. Comp. Model. 59(9–10), 1910–1920 (2010). doi: 10.1016/j.mcm.2011.01.018Santos, L.T., Dorini, F.A., Cunha, M.C.C.: The probability density function to the random linear transport equation. Appl. Math. Comput. 216(5), 1524–1530 (2010). doi: 10.1016/j.amc.2010.03.001González Parra, G., Chen-Charpentier, B.M., Arenas, A.J.: Polynomial Chaos for random fractional order differential equations. Appl. Math. Comput. 226(1), 123–130 (2014). doi: 10.1016/j.amc.2013.10.51El-Beltagy, M.A., El-Tawil, M.A.: Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm. Appl. Math. Model. 37(12–13), 7174–7192 (2013). doi: 10.1016/j.apm.2013.01.038Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterran. J. Math. 12(3), 1123–1140 (2015). doi: 10.1007/s00009-014-0452-8Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comp. Math. Appl. 59(1), 115–125 (2010). doi: 10.1016/j.camwa.2009.08.061Øksendal, B.: Stochastic differential equations: an introduction with applications, 6th edn. Springer, Berlin (2007)Soong, T.T.: Random differential equations in science and engineering. Academic Press, New York (1973)Wong, B., Hajek, B.: Stochastic processes in engineering systems. Springer Verlag, New York (1985)Arnold, L.: Stochastic differential equations. Theory and applications. John Wiley, New York (1974)Cortés, J.C., Jódar, L., Camacho, J., Villafuerte, L.: Random Airy type differential equations: mean square exact and numerical solutions. Comput. Math. Appl. 60(5), 1237–1244 (2010). doi: 10.1016/j.camwa.2010.05.046Calbo, G., Cortés, J.C., Jódar, L.: Random Hermite differential equations: mean square power series solutions and statistical properties. Appl. Math. Comp. 218(7), 3654–3666 (2011). doi: 10.1016/j.amc.2011.09.008Calbo, G., Cortés, J.C., Jódar, L., Villafuerte, L.: Solving the random Legendre differential equation: Mean square power series solution and its statistical functions. Comp. Math. Appl. 61(9), 2782–2792 (2010). doi: 10.1016/j.camwa.2011.03.045Cortés, J.C., Jódar, L., Company, R., Villafuerte, L.: Laguerre random polynomials: definition, differential and statistical properties. Utilit. Math. 98, 283–293 (2015)Cortés, J.C., Jódar, L., Villafuerte, L.: Mean square solution of Bessel differential equation with uncertainties. J. Comp. Appl. Math. 309, 383–395 (2017). doi: 10.1016/j.cam.2016.01.034Golmankhaneh, A.K., Porghoveh, N.A., Baleanu, D.: Mean square solutions of second-order random differential equations by using homotopy analysis method. Romanian Reports Physics 65(2), 1237–1244 (2013)Khalaf, S.L.: Mean square solutions of second-order random differential equations by using homotopy perturbation method. Int. Math. Forum 6(48), 2361–2370 (2011)Khudair, A.R., Ameen, A.A., Khalaf, S.L.: Mean square solutions of second-order random differential equations by using Adomian decomposition method. Appl. Math. Sci. 5(49), 2521–2535 (2011)Agarwal, R.P., O’Regan, D.: Ordinary and partial differential equations. Springer, New York (2009

A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications

[EN] This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR-type epidemiological models based on discrete-time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.Spanish Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-P; Generalitat Valenciana, Grant/Award Number: APOSTD/2019/128; Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-PCortés, J.; El-Labany, S.; Navarro-Quiles, A.; Selim, MM.; Slama, H. (2020). A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications. Mathematical Methods in the Applied Sciences. 43(14):8204-8222. https://doi.org/10.1002/mma.6482S820482224314Hamra, G., MacLehose, R., & Richardson, D. (2013). Markov Chain Monte Carlo: an introduction for epidemiologists. International Journal of Epidemiology, 42(2), 627-634. doi:10.1093/ije/dyt043Becker, N. (1981). A General Chain Binomial Model for Infectious Diseases. Biometrics, 37(2), 251. doi:10.2307/2530415Allen, L. J. S. (2010). An Introduction to Stochastic Processes with Applications to Biology. doi:10.1201/b12537Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599-653. doi:10.1137/s0036144500371907Brauer, F., & Castillo-Chávez, C. (2001). Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics. doi:10.1007/978-1-4757-3516-1Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Some results about randomized binary Markov chains: theory, computing and applications. International Journal of Computer Mathematics, 97(1-2), 141-156. doi:10.1080/00207160.2018.1440290Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications. Journal of Computational and Applied Mathematics, 324, 225-240. doi:10.1016/j.cam.2017.04.040Casabán, M.-C., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 86-97. doi:10.1016/j.cnsns.2014.12.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Slama, H., Hussein, A., El-Bedwhey, N. A., & Selim, M. M. (2019). An approximate probabilistic solution of a random SIR-type epidemiological model using RVT technique. Applied Mathematics and Computation, 361, 144-156. doi:10.1016/j.amc.2019.05.019Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 195(2), 179-193. doi:10.1016/j.mbs.2005.02.004Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Van den Driessche, P. (2017). Reproduction numbers of infectious disease models. Infectious Disease Modelling, 2(3), 288-303. doi:10.1016/j.idm.2017.06.002Heffernan, J. ., Smith, R. ., & Wahl, L. . (2005). Perspectives on the basic reproductive ratio. Journal of The Royal Society Interface, 2(4), 281-293. doi:10.1098/rsif.2005.0042Khalil, K. M., Abdel-Aziz, M., Nazmy, T. T., & Salem, A.-B. M. (2012). An Agent-Based Modeling for Pandemic Influenza in Egypt. Intelligent Systems Reference Library, 205-218. doi:10.1007/978-3-642-25755-1_1