روش برون ‌يابي براي حل عددي يك مدل بيماري‌هاي عفوني بومي

Extrapolation Method for Numerical Solution of a Model for Endemic Infectious Diseases

در اين مقاله يك مدل بيماري عفوني را كه به شكل دستگاه معادلات انتگرال ولتراي نوع دوم غيرخطي است در نظر مي‌گيريم. اين مدل از نوع مدل SIR است. روش برون‌يابي به‌سوي حد ريچاردسون را براي حل اين سيستم طوري طراحي مي‌كنيم كه با يك فرايند تكرار، سيستم غيرخطي با درجۀ دقت خوب قابل حل باشد. الگوريتم حل چنين سيستم‌هايي به‌طور كامل تشريح مي‌شود. اين الگوريتم داراي نوعي ساختار تو در تو است كه باعث مي‌شود از اطلاعات پيشين در زمان‌هاي بعدي بتوان استفاده كرد و همين امر برنامه‌ نويسي اين الگوريتم را جالب كرده است. اين الگوريتم با هر زبان سطح بالا قابل برنامه‌نويسي است، كه ما اين فرآيند را با زبان برنامه‌ نويسي Mathematica انجام داده‌ايم. تحليل خطا هم از ديدگاه تئوري آناليز عددي و هم با استفاده از چند مثال نمونه به ‌طور شفاف نشان داده شده است، براي اين منظور طيفي از نمونه مسائلي را با استفاده از تبديلات لاپلاس طراحي كرديم كه جواب تحليلي داشته باشند.

Many infectious diseases are endemic in a population. In other words they present for several years. Suppose that the population size is constant and the population is uniform. In the SIR model the population is divided into three disjoint classes which change with time t and let , and be the fractions of the population that susceptible, infectious and removed, respectively. This model formulated as the following system of nonlinear Volterra integral equation. Material and methods We apply the Richardson extrapolation method for numerical solution of this model, so that the nonlinear system is solvable by an iterative process with a good accuracy. The algorithm of such systems completely described. This algorithm has a kind of nested structure, which cause we use the lag data in the future times, and it is the interesting section of programing of the algorithm. This algorithm is ready for programing with every program language, which we do this process by Mathematica programing software. Convergence and accuracy of the method is illustrated by either theoretical and numerical analysis, and some benchmark sample problems. For this aim by using Laplase transform, we sketch a spectrum of sample problems. These problems have analytical solution and appropriate for comparison with numerical solutions. Results and discussion We solve some test examples by using present technique to demonstrate the efficiency, high accuracy and the simplicity of the present method. The main advantage of the method is the applicability of method for a large interval of time, as the algorithm shows. Numerical results shows the accuracy of the method for a long time interval. Conclusion The following conclusions were drawn from this research.