{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:58Z","timestamp":1787329318863,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"When immune cells detect foreign molecules, they secrete soluble factors that attract other immune cells to the site of the infection. In this paper, I study numerical solutions to a model of this behavior proposed by Kepler. In this model the soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the model cells is a Langevin process that is biased toward the gradient of the soluble factors. I have shown that the solution to this system exists for all time and remains positive, the supremum is a priori bounded, and the derivatives are bounded for finite time. I have also developed a first order split scheme for solving the reaction-diffusion stochastic system. This allows us to make use of known first order schemes for solving the diffusion, the reaction, and the stochastic differential equations separately.<\/jats:p>","DOI":"10.1137\/070701595","type":"journal-article","created":{"date-parts":[[2008,9,4]],"date-time":"2008-09-04T19:18:32Z","timestamp":1220555912000},"page":"3113-3135","source":"Crossref","is-referenced-by-count":5,"title":["Operator Splitting for an Immunology Model Using Reaction-Diffusion Equations with Stochastic Source Terms"],"prefix":"10.1137","volume":"46","author":[{"given":"Timothy A.","family":"Lucas","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,9,4]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n The Mathematical Theory of Finite Element Methods\n , 2nd ed., Springer-Verlag, New York, 2002.","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"R2","unstructured":"L. C. Evans,\n Partial Differential Equations\n , Grad. Stud. Math. 19, American Mathematical Society, Providence, RI, 1998."},{"key":"R3","unstructured":"G. B. Folland,\n Introduction to Differential Equations\n , 2nd ed., Princeton University Press, Princeton, N.J., 1995."},{"key":"R4","doi-asserted-by":"crossref","unstructured":"W. Hundsdorfer and J. Verwer,\n Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations\n , Springer-Verlag, Berlin, 2003.","DOI":"10.1007\/978-3-662-09017-6"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5193(71)90050-6"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1111\/j.1600-065X.2007.00500.x"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"T. B. Kepler,\n Microsimulation of Inducible Reorganization in Immunity\n , in Complex Systems Science in BioMedicine, T. S. Deisboeck and J. Y. 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