روش برون‌یابی برای حل عددی یک مدل بیماری‌های عفونی بومی

نویسنده
بهمن بابایاررازلیقی
دانشگاه صنعتی قم، دانشکدۀ علوم پایه، گروه ریاضی
10.29252/mmr.5.1.29
چکیده
در این مقاله یک مدل بیماری عفونی را که به شکل دستگاه معادلات انتگرال ولترای نوع دوم غیرخطی است در نظر می‌گیریم. این مدل از نوع مدل SIR است. روش برون‌یابی به‌سوی حد ریچاردسون را برای حل این سیستم طوری طراحی می‌کنیم که با یک فرایند تکرار، سیستم غیرخطی با درجۀ دقت خوب قابل حل باشد. الگوریتم حل چنین سیستم‌هایی به‌طور کامل تشریح می‌شود. این الگوریتم دارای نوعی ساختار تو در تو است که باعث می‌شود از اطلاعات پیشین در زمان‌های بعدی بتوان استفاده کرد و همین امر برنامه‌نویسی این الگوریتم را جالب کرده است. این الگوریتم با هر زبان سطح بالا قابل برنامه‌نویسی است، که ما این فرآیند را با زبان برنامه‌نویسی Mathematica انجام داده‌ایم. تحلیل خطا هم از دیدگاه تئوری آنالیز عددی و هم با استفاده از چند مثال نمونه به‌طور شفاف نشان داده شده است، برای این منظور طیفی از نمونه مسائلی را با استفاده از تبدیلات لاپلاس طراحی کردیم که جواب تحلیلی داشته باشند.

کلیدواژه‌ها
روش برون‌یابی
بیماریهای عفونی
کلاس مستعد
کلاس عفونت‌زا
کلاس مصون
دستگاه انتگرالی ولترای غیر خطی

عنوان مقاله English

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

نویسنده English

Bahman Babayar-Razlighi
Department of Mathematics, Faculty of science, Qom University of Technology, Qom, Iran
چکیده English

Introduction

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.



Where , and are unknown functions and other constants and functions are known. The susceptibles are transferred at a rate equal to times the number of infectives, where is a constant. is the nonincreasing probablity function of remaining infectious units after becoming infectious, with and and is dominated by a decaying exponential, such as gamma distributed. Since the population size is constant, the birth rate must be equal to the death rate . The death rate is the same for susceptibles, infectives and removed individuals. The fraction of newborns are immunized so that the flow rate of immunized newborns into the removed class is . The initial susceptible and removed fractions be and and be the fraction of the population that was initially infectious and is still alive and infectious at time .

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.


The proposed algorithm is very suitable for mathematical programing.
Many cancer problems have such structure and the method is applicable for them.


This method has two characteristics, solve a nonlinear problem and use of previous solution in new interval. So the method is applicable for various kind of problems with little additional works../files/site1/files/51/%D8%A8%D8%A7%D8%A8%D8%A7%DB%8C%D8%A7%D8%B1.pdf

کلیدواژه‌ها English

Extrapolation method
Infectious diseases
Susceptible class
Infective class
Removed class
Nonlinear Volterra integral system
1. Hethcote H. W., Tudor D. W., "Integral equation models for endemic infectious diseases", J. Math. Biology, vol. 9 (1980) 37-47. 2. Miller I., Miller M., "John E. Freund's Mathematical Statistics with Applications", 8st ed., Pearson Education Limited, 2014. 3. Bailey N. J. T., "The mathematical theory of infectious diseases and its applications", Griffin, London (1975). 4. Niculescu S. I., Kim P. S., Gu K., Lee P. P. , Levy D., "Stability crossing boundaries of delay systems modeling immune dynamics in leukemia", Discrete and Continouos Dynamic systems, serie B, vol. 10, no 1 (2010) 129-156. 5. Babayar-Razlighi B., Soltanalizadeh B., "Numerical solution for system of singularnonlinear volterra integro-differential equations by Newton-Product method", APPL. MATH. COMPUT, vol. 219 (2013) 8375-8383. 6. Babayar-Razlighi B., Soltanalizadeh B., "Numerical solution of nonlinear singularVolterra integral system by the Newton-Product integration method", MATH, COMPUT, MODEL, vol. 58 (2013)1696-1703. 7. Babayar-Razlighi B., Ivaz, M. R. Mokhtarzadeh, "Convergence of product integration method applied for numerical solution of linear weakly singular Volterra systems",Bull. IranianMath, Soc., vol. 37 (2011) 135-148. 8. Babayar-Razlighi B., Ivaz K., Mokhtarzadeh M. R., Badamchizadeh A. N., "Newton-Product integration for a two phase Stefan problem with kinetics", Bull, Iranian Math, Soc., vol. 38, no. 4 (2012) 853-868. 9. Babayar-Razlighi B., Ivaz K., Mokhtarzadeh M. R., "Newton-Product integration for a Stefan problem with kinetics",J. Sci. Islam. Repub. Iran, vol. 22, no. 1 (2011) 51-61. 10. Babayar-Razlighi B., Khadem M., "Numerical solution of a free boundary problem from heat transfer by Legendre Wavelets", Proc.Iranian Conference on Mathematical Physics, Qom, Iran (Nov 2016) 92-95. 11. Babayar-Razlighi B., "Numerical solution of a heat conduction problem by the Legendre Wavelets", Proc. The 22th Seminar on Mathematical Analysis and It’s Applications, Bonab, Iran (Jan 2017) 319-322. 12. Brunner H., "Collocation Methods for VolterraIntegral and Related FunctionalEquations", 1st ed., Cambridge University Press (2004). 13. Ordokhani Y., Razzaghi M. "Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via a collocation method and rationalized Haar functions", Appl. Math. Lett., vol. 21, no. 1 (2008) 4-9. 14. Ordokhani Y., "Solution of nonlinear Volterra-Fredholm Hammerstein integralequations via rationalized Haar functions", Appl. Math. Comput., vol. 180, no. 2 (2006) 436-443. 15. Phillips G.M.M., Taylor P.J., "Theory and applications of numerical analysis", in Applied Mathematics, 2st ed., Elsevier Science & Technology Books, 1996. 16. Tao L., Yong H., "Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind", J. Math. Anal. Appl, vol. 324, (2006) 225-237.