ISCA Journal of Biological Sciences _________________________________________________ ISSN 2278-3202 Vol. 1(2), 55-60, June (2012) ISCA J. Biological Sci. International Science Congress Association 55 Stability of the Six Equilibrium States between CN and G-CSF with Infectives Growth Rate Progression: A FFT StudyBalamuralitharan S.1 and Rajasekaran S. Department of Mathematics, Sri Ramanujar Engineering College, Chennai-48, INDIA Department of Mathematics, B.S. Abdur Rahman University, Chennai-48, INDIAAvailable online at: www.isca.in (Received 1st June 2012, revised 7th June 2012, accepted 8th June 2012)Abstract In this paper we have investigated a stability of the equilibrium states between CN (Cyclical Neutropenia) and G-CSF (granulocyte-colony stimulating factor) with infective growth rate progression. Mathematical models and FFT (fast fourier transform) simulations can be used as experimental tools for testing the control- measures and determining sensitivities of changes in parameter values. The paper deals with an investigation on a four parameters estimates consisting of cure rate (c), infective rate (I), absolute neutrophil count (A) and total carrying capacity (K) only. All six equilibrium points for this model are identified and their stability criteria are analyzed. Mathematical modelling of epidemics can lead to and motivate new results in mathematics. Keywords: Cure rate in infective, infective rate in neutropenia population, absolute neutrophil count, total carrying capacity. Introduction Neutropenia is defined as an absolute neutrophil count (ANC) of less than 1500/micro L. The ANC is equal to the product of the white blood cell count and the fraction of polymorph nuclear cells and band forms noted on the differential analysis1. The term neutropenia describes the situation where the number of neutrophilsin the blood is too low. Neutrophils are very important in defending the body against bacterial infections, and therefore, a patient with too few neutrophils is more susceptible to bacterial infections2. People with neutropenia get infections easily and often. Most of the infections occur in the lungs, mouth and throat, sinuses and skin. Painful mouth ulcers, gum infections, ear infections and periodontal disease are common. Most cases of neutropenia are acquired and are due to decreased granulocyte production or less often increased destruction. Neutropenia is a blood disorder that can affect anyone. White blood cells are just as important, but for a very different reason. One of their jobs is to protect you from infection. There are several kinds of white cells. Each has a special function. It is a condition in which an individual has an abnormally low number of a type of white blood cell called a neutrophil. Neutrophils are a specific kind of white blood cell that helps to prevent and fight off infections3. There function is to eat up the bad microorganisms that enter your body. Neutrophils make up fifty to seventy percent of the circulating white blood cells in the body. In cyclic neutropenia considerable evidence from human studies and from investigations of an analogous disease in collie dogs indicate that a regulatory defect at the hematopoietic stem cell level leads to oscillations in production of all types of blood cells. People of any age group can be affected by this disease4.In this Paper we examine the spread of CN among the G-CSF and establish Criteria for stability of the equilibrium states. Mathematical models can be used to compare and optimize control programs. The modelling process can contribute to the design of public health surveys, especially by suggesting data that should be collected. The process of formulating a mathematical model clarifies the assumptions and provides qualitative results such as thresholds5. And also we have solved a pair of differential equations which represents the growth rates of population and infective by employing the FFT. And solution graphs are presented for all possible cases. This model is characterized by a pair of First Order ordinary differential Equations reflecting the balances of growth rates of cure and infective in a neutropenia population. Basic notations: Q - total number of hematopoietic stem cells (10 cells/kg), X- number of tissue levels of G-CSF, g- growth rate in population, Ic - inhibition coefficient, c- cure rate in infective, I – infective rate in neutropenia population, k - total carrying capacity (g/I), A - initial number of absolute neutrophil counts, K, K – transfer rates, 0 T K , 0 B K - initial transfer rates, t-the time of dominates over the infective and cure rates. Material and MethodsEquation for Total number of hematopoietic stem cells (10cells/kg) (Q): Rate of change in total number of hematopoietic stem cells (Q) = growth rate in population (g) - reduction in population (Q) ()(1) rcdQgIQQ dt =--------- Equation for Number of tissue levels of G-CSF (X): Rate of change in number of tissue levels of G-CSF (X) = increasing of ISCA Journal of Biological Sciences ______________________________________________________________ ISSN 2278-3202Vol. 1(2), 55-60, June (2012) ISCA J. Biological Sci. International Science Congress Association 56 infective due to contract between susceptible and infective - decrease in the number of Infective due to death. ()(2) dXIQXXcX dt =-----------Equilibrium points: The system under investigation has three equilibrium points: 1. 0 Q = , 0 X = Fully washed out stage. 2. Qk = , 0 X = Healthy State. In this state healthy population can survive and infective population washed out. 3. Qk = , Ikc X I - Coexistence state. In this state both healthy and infective population can survive, it could happen only when I k c. When Ik = c this becomes same as 2. Stability of the Equilibrium States: This method affords stability information directly without solving the differential equations occupied in the system. This method is based on the main characteristic of constructing a scalar function. The stability behavior of solutions of linear and non-linear system is done by using the techniques of variation of constants formulae and integral inequalities. So this analysis is restricted to a small neighborhoods of operating point i.e., local stability. Further, the techniques used there in require explicit knowledge of solutions of corresponding linear systems7. Stability of the Equilibrium State 1: We consider the transfer rates K, K from study state (Q, X) i.e. , TB QQKXXK =+=+ Here K, KB is so small that the terms other than their first order can be neglected. Substituting in above equations (1), (2) and neglecting products and higher powers of K, K we get ,(3) TBrTBdKdKgKcK dtdt ==---- The characteristic equation for which is ()()0 gc ll -+= . The roots of which are gr , -c and these are of opposite signs. Hence the state is unstable Case 1.1:- When 0 T K 0 B K When 0 T K 0 B K , Total number of hematopoietic stem cells (Q) increases over the Number of tissue levels of G-CSF (X) in growth as well as its population strength. Stability of the Equilibrium State 2: The corresponding linearized perturbed equations are ,()(4) TBrTBdKdKgKIkcK dtdt =-=----The characteristic equation is ()(())0 gIkc ll +--= ----- (5) where roots are -g and Ik-c.Case2.1:- When k c I The roots of the above equation are one is positive and the other is negative. Hence the state is unstable. Case2.2:- when 0 T K 0 B K When 0 T K 0 B K , Population decreasing and the infective are increasing, after some time total population becomes infected and starts decay with population9. Case 2.3:- When 0 T K 0 B K Here no change in Number of tissue levels of G-CSF(X) and infective population declines. logrBgK  =    Case2.4:- When c k I The roots of the equation (5) are both negative. Hence the state is stable. By solving the equation (4) we get 00 () gt cIKt TTBBKKeKKe--==Case 2.5:- When 0 T K 0 B K When 0 T K 0 B K and gr = (c – Ik) here the total population and infective both are decreasing. Stability of the equilibrium state 3: ( Qk = , Ikc X I - This state exist only when Ik c, The corresponding differential equations are ,()()(6) TBrTTBdKdKgKIkcKK dtdt =-=-----The characteristic equation is ()(())0 gIkc ll ++-= . The roots of this equation are –and -(I k - c) both negative. Hence the state is stable. Case 3.1:- when 0 T K 0 B K ISCA Journal of Biological Sciences ______________________________________________________________ ISSN 2278-3202Vol. 1(2), 55-60, June (2012) ISCA J. Biological Sci. International Science Congress Association 57 When 0 T K 0 B K , and g= (Ik - c), here the total population and infective both are decreasing. A solution of differential equations: We get the differential equation representing the growth rate of infective.By solving the equation (3) we get, 00gt ct TTBB KKeKKe - ==where 0 T K , 0 B K are initial values of K and KB respectively. By solving the equation (4) we get, 00 () gt IKct TTBBKKeKKe== By solving the equation (6) we get. 000()()()rgtgt IkctIkct TTBBTIkcKKeKKeKeeIkcg----- -  ==+-  --Numerical solution of this equation is obtained by FFT method of transform for investigate the behaviour of the infective of this model10. Parameter estimation values of CN and G-CSF Parameter Name Value Used Unit Sources Q X G B 1.1 1500-7700 0.1 0 0.07 0.25 0.01 10 6 cells/kg I mg/kg mg/ml hours_1 hours_1I C. Foley S. Bernard M. Mackey (calculated) (calculated) (calculated) T. Hearn, (calculated) Results and DiscussionCase (1): when c I and A k: Computations have been carried for FFT of values for the system parameters as shown in the table 1 to estimate the strength of the infective (I). Table-1 t c I A k 0.1 0.3 0.2 1600 1550 0.2 0.4 0.2 1700 1650 0.3 0.5 0.2 1800 1750 0.4 0.6 0.2 1900 1850 0.5 0.7 0.2 2000 1950 Figure-1 Cure rate is greater than infective rate when Initial number of absolute neutrophil count greater than total carrying capacityCase (2): when c I and AN = k: Computations have been carried for FFT of values for the system parameters as shown in the table.2 to estimate the strength of the infectives (I) Table-2 t c I A N k 0.1 0.3 0.2 1600 1600 0.2 0.4 0.2 1700 1700 0.3 0.5 0.2 1800 1800 0.4 0.6 0.2 1900 1900 0.5 0.7 0.2 2000 2000 -6-4-202460.00.10.20.30.4 cure rate time-100100200 -6-4-20246 infective Angle(deg)-6-4-2024650010001500 time Absoulute Neutrophil Count -5050100150 -6-4-20246 total carrying capacity Angle(deg) ISCA Journal of Biological Sciences ______________________________________________________________ ISSN 2278-3202Vol. 1(2), 55-60, June (2012) ISCA J. Biological Sci. International Science Congress Association 58 Figure-2 Initial number of absolute neutrophil count is equal to total carrying capacity Case (3): when c I and A k: Computations have been carried for FFT of values for the system parameters as shown in the table 3 to estimate the strength of the infectives (I). Table-3 t c I A k 0.1 0.3 0.2 1550 1600 0.2 0.4 0.2 1600 1650 0.3 0.5 0.2 1650 1700 0.4 0.6 0.2 1700 1750 0.5 0.7 0.2 1750 1800 Figure-3 Initial number of absolute neutrophil count is less then total carrying capacity Case (4): when c I and A .64;ذ k: Computations have been carried for FFT of values for the system parameters as shown in the table 4 to estimate the strength of the infectives (I). Table-4 t c I A k 0.1 0.2 0.3 1200 1150 0.2 0.2 0.4 1250 1200 0.3 0.2 0.5 1300 1250 0.4 0.2 0.6 1350 1300 0.5 0.2 0.7 1400 1350 -6-4-2024650010001500 time Absolute Neutrophil Count -5050100150 -6-4-20246 total carrying capacity Angle(deg)-6-4-2024650010001500 time Absolute Neutrophil Count -5050100150 -6-4-20246 total carrying capacity Angle(deg) ISCA Journal of Biological Sciences ______________________________________________________________ ISSN 2278-3202Vol. 1(2), 55-60, June (2012) ISCA J. Biological Sci. International Science Congress Association 59 Figure-4 Cure rate is less than infective rate in susceptible when Initial number of absolute neutrophil count greater than total carrying capacity Case (5): when c I and A = k:Computations have been carried for FFT of values for the system parameters as shown in the table 5 to estimate the strength of the infective (I). Table-5 t c I A N k 0.1 0.2 0.3 1200 1200 0.2 0.2 0.4 1250 1250 0.3 0.2 0.5 1300 1300 0.4 0.2 0.6 1350 1350 0.5 0.2 0.7 1400 1400 Figure-5 Absolute Neutrophil Count is equal to total carrying capacity Case (6): when c I and A k: Computations have been carried for FFT of values for the system parameters as shown in the table 6 to estimate the strength of the infective (I). Table-6 t c I A N k 0.1 0.2 0.3 1200 1250 0.2 0.2 0.4 1250 1300 0.3 0.2 0.5 1300 1350 0.4 0.2 0.6 1350 1400 0.5 0.2 0.7 1400 1450 Figure-6 Absolute Neutrophil Count is less than total carrying capacity The objective of the paper is to examine the stability case in a mathematical model of CN between four factors in which a -6-4-202460.00.10.20.30.4 infective rate time-200-100100200300 -6-4-20246 cure rate Angle(deg)-6-4-2024620040060080010001200 time Absolute Neutrophil Count -5050100150 -6-4-20246 total carrying capacity Angle(deg)-6-4-2024620040060080010001200 time Absolute Neutrophil Count -5050100150 -6-4-20246 total carrying capacity Angle(deg)-6-4-2024620040060080010001200 time total carrying capacity -5050100150 -6-4-20246 Absolute Neutrophil Count Angle(deg) ISCA Journal of Biological Sciences ______________________________________________________________ ISSN 2278-3202Vol. 1(2), 55-60, June (2012) ISCA J. Biological Sci. International Science Congress Association 60 delayed G-CSF is comprised11. Six equilibrium points of the model are discovered. The cure rate is greater than the infective rate and the Initial number of absolute neutrophil count are greater than total carrying capacity (freg(X)= 2.5, Real(Y)= 0.125, Imag(Y)= 0.1, r(Y)= 0.16008, Phi(Y)= 38.65981, Power(Y)= 0.0032 and freg (X)= 1.25, Real(Y)= 920.86309, Imag(Y)= -1221.75144, r(Y)= 1529.92334, Phi(Y)= -52.9938, Power(Y)= 292583.17814 from figure1). The cure rate is greater than the infective rate and the Initial number of absolute neutrophil count is equal to total carrying capacity (freg(X)= 1.25, Real(Y)= 951.04076, Imag(Y)= -1221.75144, r(Y)= 1548.27489, Phi(Y)= -52.10195, Power(Y)= 299644.39014 from figure 2). The cure rate is greater than the infective rate and the Initial number of absolute neutrophil count is less than total carrying capacity (freg(X)= 1.25, Real(Y)= 958.36309, Imag(Y)= -1063.54076, r(Y)= 1431.635, Phi(Y)= -47.97779, Power(Y)= 256197.3472 from figure 3).The cure rate is less than the infective rate and the Initial number of absolute neutrophil count are greater than total carrying capacity (freg(X)= -1.25, Real(Y)= -0.30178, Imag(Y)= -0.01464, r(Y)= 0.30213, Phi(Y)= -177.22172, Power(Y)= 0.01141 and freg(X)= -1.25, Real(Y)= -822.11941, Imag(Y)= 716.94174, r(Y)= 1090.81886, Phi(Y)= 138.90945, Power(Y)= 148735.72208 from figure 4). The cure rate is less than the infective rate and the Initial number of absolute neutrophil count is equal to total carrying capacity (freg(X)= -1.25, Real(Y)= -852.29708, Imag(Y)= 716.94174, r(Y)= 1113.73954, Phi(Y)= 139.92987, Power(Y)= 155051.9705 from figure 5). The cure rate is less than the infective rate and the Initial number of absolute neutrophil count is less than total carrying capacity (freg(X)= -1.25, Real(Y)= -882.47475, Imag(Y)= 716.94174, r(Y)= 1136.99918, Phi(Y)= 140.90884, Power(Y)= 161595.89185 from figure 6). Conclusion In this paper we have investigated some results on the stability of equilibrium in a mathematical model of CN and G-CSF by FFT analysis.Some observations are made from the relationship between the cure rate of CN and G-CSF through the growth rates. Each and every one six stability of equilibrium points for this mathematical differential equation model are identified and their stability criteria are discussed. Under the limited and unlimited resources for first and second factors respectively, if the cure rate is greater than the infective rate for both the models, it is found that there are six equilibrium cases. 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