ﺣﻞ ﻋﺪدي ﻣﺪل اﭘﯿﺪﻣﯿﮏ SIR ﺑﻪ ﮐﻤﮏ روش ﺗﻔﺎﺿﻞ ﻣﺘﻨﺎﻫﯽ ﻏﯿﺮاﺳﺘﺎﻧﺪارد

Numerical solution of the SIR epidemic model using the non-standard finite difference method

در اﯾﻦ ﻣﻘﺎﻟﻪ، ﯾﮏ ﺣﺎﻟﺖ ﺧﺎص از روش ﺗﻔﺎﺿﻞ ﻣﺘﻨﺎﻫﯽ ﮐﻪ روش ﺗﻔﺎﺿﻞ ﻣﺘﻨﺎﻫﯽ ﻏﯿﺮاﺳﺘﺎﻧﺪارد ﻧﺎﻣﯿﺪه ﻣﯽﺷﻮد ﺑﺮاي ﺣﻞ ﻋﺪدي ﯾﮏ ﻣﺪل رﯾﺎﺿﯽ از ﺑﯿﻤﺎريﻫﺎي اﭘﯿﺪﻣﯿﮏ ﻣﻮرد ﻣﻄﺎﻟﻌﻪ ﻗﺮار ﮔﺮﻓﺘﻪ اﺳﺖ. ﻃﺮح ﺗﻔﺎﺿﻞ ﻣﺘﻨﺎﻫﯽ ﻏﯿﺮ اﺳﺘﺎﻧﺪارد ﺳﺎﺧﺘﻪ ﺷﺪه داراي وﯾﮋﮔﯽﻫﺎي ﻣﻬﻢ ﻣﺪل ﭘﯿﻮﺳﺘﻪ از ﻗﺒﯿﻞ ﻣﺜﺒﺖ ﺑﻮدن، ﮐﺮاﻧﺪاري و ﭘﺎﯾﺪاري ﻣﯽﺑﺎﺷﺪ. ﭘﺎﯾﺪاري ﻧﻘﺎط ﺗﻌﺎدل ﺳﯿﺴﺘﻢ ﺑﺮرﺳﯽ ﺷﺪه اﺳﺖ. ﻃﺮح ﺗﻔﺎﺿﻞ ﻣﺘﻨﺎﻫﯽ ﻏﯿﺮ اﺳﺘﺎﻧﺪارد اراﺋﻪ ﺷﺪه ﻫﻤﮕﺮا ﺑﻪ ﻧﻘﺎط ﺗﻌﺎدل ﻣﺪل ﻣﯽﺑﺎﺷﺪ. در ﺣﻞ ﻣﺴﺎﺋﻞ ﻏﯿﺮﺧﻄﯽ، ﯾﮑﯽ از ﻣﺰاﯾﺎي ﻣﻬﻢ اﯾﻦ روش ﮔﺴﺴﺘﻪﺳﺎزي ﺟﻤﻼت ﻏﯿﺮﺧﻄﯽ ﺑﺎ اﺳﺘﻔﺎده از ﺗﻘﺮﯾﺐﻫﺎي ﻏﯿﺮ ﻣﺤﻠﯽ ﻣﯽﺑﺎﺷﺪ. در اﻏﻠﺐ ﻣﻮارد ﻓﺮﻣﻮلﻫﺎي ﺗﻔﺎﺿﻞ ﻣﺘﻨﺎﻫﯽ ﻏﯿﺮاﺳﺘﺎﻧﺪارد ﺣﺘﯽ وﻗﺘﯽ اﻧﺪازه ﻃﻮل ﮔﺎم ﺷﺒﮑﻪ ﺑﺰرگ در ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﻣﯽﺷﻮد ﭘﺎﯾﺪار ﻫﺴﺘﻨﺪ. ﻟﺬا در ﺳﯿﺴﺘﻢﻫﺎي دﯾﻨﺎﻣﯿﮑﯽ ﮐﻪ در ﺑﺎزهﻫﺎي زﻣﺎﻧﯽ ﺑﺰرگ ﻣﻮرد ﻣﻄﺎﻟﻌﻪ ﻗﺮار ﻣﯽﮔﯿﺮﻧﺪ اﺳﺘﻔﺎده از روش ﻏﯿﺮاﺳﺘﺎﻧﺪارد ﻣﻘﺮون ﺑﻪ ﺻﺮﻓﻪ ﺧﻮاﻫﺪ ﺑﻮد. ﻣﺜﺎلﻫﺎي ﻋﺪدي دﻗﺖ و ﮐﺎراﯾﯽ روش ﺗﻔﺎﺿﻞ ﻣﺘﻨﺎﻫﯽ ﻏﯿﺮ اﺳﺘﺎﻧﺪارد را ﺗﺎﺋﯿﺪ ﻣﯽﮐﻨﻨﺪ.

In this paper, a special case of the finite difference method which is called non-standard finite difference method is studied for the numerical solution of a mathematical model of epidemic diseases. The constructed non-standard finite difference schemes have the main properties of the continuous model such as positivity, boundedness, and stability. The stability of the equilibrium points of the system is investigated. The proposed non-standard finite difference schemes are convergent to the equilibrium points of the system. In solving nonlinear problems, one of the important advantages of this method is that nonlinear term discretized with nonlocal approximations. In most cases, non-standard finite difference schemes are stable even when large step sizes are considered. Therefore, using non-standard method will be cost-effective in dynamical systems that are studied over a large time interval. Numerical examples confirm the accuracy and efficiency of the non-standard finite difference method.