Analysis of Blood Transfused in a City Hospital. with the Principle of Markov-Dependence

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1 Applied Mathematical Sciences, Vol. 8, 2, no., HIKARI Ltd, Analysis of Blood Transfused in a City Hospital with the Principle of Markov-Dependence Usman Yusuf Abubakar*, Danladi Hakimi, Abdulrahman Ndanusa and Vincent Aforkeoghene Jevwia Department of Mathematics/Statistics Federal University of Technology Minna, Nigeria *Corresponding author Copyright 2 Usman Yusuf Abubakar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The management of blood bank is an integral part of health delivery system and it is centred on the donation and transfusion of blood. The four maor classification of blood is considered in the formulation of Markov model with a first- order dependence. The data of blood transfused in a city hospital was studied and the result revealed that the blood that is donated and received by the same blood group constitute the maor blood needed in the hospital with %,.%, 8.%, and 8.5% for blood groups AB,B,A and O respectively. There will be a little variation in the blood needed for transfusion in the future on the basis of the present when the initial values of the transition probabilities P2 (t), P (t), P (t), P2 (t), and P (t) (t = ) are.5,.22,.865,.65 and.88 respectively. The information about blood bank inventory is important for a successful health care delivery. If such information is available in every city hospital, it may be possible to move blood from one hospital to another during emergency. Mathematics Subect Classification: 6J28 Keywords: blood bank, transfusion, donation, Markov dependence, transition, probability, health care need and demand

2 862 Usman Yusuf Abubakar et al. Introduction Blood constitutes the red fluid that is pumped from the heart and circulates around the bodies of human and other vertebrates. It is divided into O, A, B, and AB groups. A blood bank is a facility in the hospital that procure, store and dispense blood from a donor to a recipient. The management of blood bank is an integral part of health delivery system. It is a recognition of this, that lead to the establishment of the National blood transfusion service and the inauguration of an advisory National technical committee in Nigeria. Many people are not aware that blood is perishable and cannot be stored indefinitely and many blood banks function as pipelines with little or no inventory. Therefore, there is a steady need for donors to cope with massive demands due to inuries resulting from Automobile crashes, communal clashes and bomb blast in recent time. The donation (supply) and the transfusion processes are quite complicated and require some explanations through the use of theoretical models. Consequently, Pegels et al () provided a model using the theory of absorbing Markov chains to determining the effects of the issuing policies on average inventory levels and determined blood shortage probabilities. In a related work, the author studied a blood bank collection scheduling and inventory control system reported in Pegels (6). Rockwell et al (62) gave inventory analysis as applied to hospital whole blood supply and demand. Recently, Jevwia (2), with the use of nonhomogenous Markov model showed that there is a variation in the need for blood for transfusion in the two seasons in the year at a general hospital. The prediction of blood needed for transfusion in the future using the model is not very interesting. It is therefore important to provide a Markov model in discrete states and in continuous time that could be used to make a prediction of blood needed in the hospital in the future on the basis of the present need. Theoretical Background of the Model According to Bhat (8), a continuous time stochastic process {x (t)} is an infinite family of random variables indexed by the continuous real variable t. That is, for any fixed t, x (t) is a random variable, and the collection of all of these (for all t) is a stochastic process. We think of t as time, so we may expect x (t), the random variable at time t to be dependent on x (t), where t < t but not upon x (t2), where t2 > t. We refer to the value of x (t) as the state of the process at time t. It is assume that x (t) are discrete state, continuous time stochastic process. Markov-Dependence Stochastic processes occurring in most real life situations are such that for a discrete set of parameters t, t tn T the random variable X (t), X (t2) X

3 Analysis of blood transfused in a city hospital 86 (tn) exhibit some sort of dependence. The simplest type of dependence is the firstorder dependence underlying the stochastic process. This is called Markovdependence, which may be defined as follows: We consider a finite (or countably infinite) set of points (t, t,.. tn, t), t < t < t2.< tn < t and t, tr T (r =, 2,..n) where T is the parameter space of the process {X(t)}. The dependence exhibited by the process {X (t), t T} is called Markov - dependence if the conditional distribution of X (t) for given values of X (t), X (t2)..x (tn) depends only on X (tn) which is the most recent known value of the process, that is, if P[X (t) < x x (tn) = xn, x (tn-) = xn-,., x (t) = x] = P[x (t) < x x (tn) = xn] (.) = F (xn, x: tn, t) The stochastic process exhibiting this property is called a Markov Process. In a Markov process, therefore, if the state is known for any specific value of the time parameter t, that information is sufficient to predict the next behaviour of the process beyond that point. As a consequence of the property given by (.), we have the following relation: F(x, x: t, t) =ys F (y, x,, t) df(x, y, t, ) (.) Where t < < t and S is the state space of the process x (t). Equations (.) is called the Chapman-Kolmogorov equation for the process. These are basic equations in the study of Markov processes. They enable us to build a convenient relationship for the transition probabilities between any points in T at which the process exhibits the property of Markov dependence Bhat (8). Formulation of Markov Model The supply of human blood to the city hospital blood bank is through donation from voluntary donors, some commercial donors and some from the relative (s) of the patient that needed blood. The demand and need for blood in the hospital arises from ailing, inury and planned surgery. The demand and supply of various blood groups in the hospital are therefore not simple processes and could be described as a random variable. Suppose that the pint of blood transfused in a month in the city hospital is described by a continuous random variable X (t) and the collection of the random variable for all t represents a stochastic process. It is supposed also that the random variable is governed by the first order Markov-

4 86 Usman Yusuf Abubakar et al. dependence stated in equation (.). A maor complication to the blood donation and transfusion is the existence of different blood types among humans and the matching. The eight types of blood that exist in human are; A+,A-,B+,B-,AB+,AB-,O+ and O-. These blood groups have complex substitutability pattern and could be regrouped into O, A, B, AB, Jevwia (2). Let the state space of the process be represented as follows: State: Blood group O State2: Blood group A State: Blood group B State: Blood group AB It is assumed that the states are mutually exclusive and exhaustive. On the basis of the observed data, the transition between the states is described by the transition diagram shown in figure. 2 Figure. Transition diagram for the Blood Transfusion Consequently, we have the following transition matrix p. p P P P P P P P P P

5 Analysis of blood transfused in a city hospital 865 The matrix P is homogeneous transition stochastic matrix and the transition probabilities pi are fixed and independent of time (stationary transition probabilities). The transition probabilities must satisfy Pi, i, =, 2,,. and Pi, i, 2,,. Jevwia(2) Following Howard(6), we let ai represent the transition rate of the Blood transfused from state i to state, i in a short time interval (t, t+ t) the Blood currently in state i, will be transfused or make a transition to state with probability ai t, i. If Xt is the state of the process at time t, then we have P( X t t X i a t ) it The probability of two or more state transitions is of order ( t) 2 or more and it is negligible if t is sufficiently small. Suppose that the transition rate do no change with time (ai s are constants) and a a i,,2,,... (2) i i We describe the process by a transition rate matrix A with component aἱ Suppose P i (t) is the probability that blood in the state ἱ at time t after the start of the process and let P (t + t) be the probability that the blood will be transfused to state a short time t later. Then P ( t t) p ( t) ai t pi ( t) ai t () i i,2, Equation () is obtained by multiplying the probabilities and adding over all ἱ that are not equal to because the blood could have be transfused to state from any other possible state ἱ. Putting (2) in () and rearranging the terms gives P ( t t) p Thus, we have ( t) t p ( t) a i i t

6 866 Usman Yusuf Abubakar et al. dp ( t) Pi ( t) ai i, 2 in matrix form, we have dp t P( t) A () Infact, equation () is an exact (not approximate) differential equations for Pἱ (t) in dp ( t) i p ik a k This a Chapman Kolmogorov differential equation in (.). It is a linear, first order differential equation with constant coefficients aἱ, s. The elements of A may be further related by extending the properties of P (t). In particular since for each ἱ then pi ( t) d pi ( t) t d() t d P i ( t ) t a (5) i

7 Analysis of blood transfused in a city hospital 86 That is each row of A must sum to zero, since every off diagonal is non-negative, hence equation (2). The elements of A are expected values of a negative exponential function Abubakar (2). The development of the equation that determine the Pἱ (t) functions, for this process can be simplified if the following assumptions are made: () The process satisfies the Markov property (2) The process is stationary () The probability of a transition from one state to a different state in a short time interval is proportional to t () The probability of two or more changes of state in a short interval t is zero P (t) is a row vector of the state probabilities at time t. to obtain the solution of (), the initial condition Pἱ (); ἱ=, 2,, must be specified. Taking the Laplace transform of () we have P( s) P() SI A (6) Thus P (t) is obtained as the inverse transform of P (S), Korve (2). Application General Hospital Minna is a typical city hospital in Niger state Nigeria. It is a government owned medical out- fit that provides health serves to the populace at subsidised price. Blood bank management is one of the important services rendered in the Hospital and it is highly patronised as showed in table. A summary statistics for years for the monthly transfused blood in the General Hospital Minna is presented in table. Table: A frequency table of blood transfused in the General Hospital Minna between 2-2. State Frequency From the data recorded in table, we obtained the transition count matrix

8 868 Usman Yusuf Abubakar et al. M Normalizing the matrix M using equation (2) gives the matrix 868 A The matrix A can be expressed as the expected value of the exponential distribution thus.52 Q IS Q s s s s Taking the inverse of the above matrix using maple software and also taking inverse Laplace transform of the entries from equation (6), we obtain the following equations. P2(t) =

9 Analysis of blood transfused in a city hospital 86 P (t)= P (t)= P2 (t) = P (t) = Results The values of the equations Pi(t) for i=,2, and =2,,, and t=,,2,... are presented in table and illustrated with graph in figure2 respectively. T P2(t) P (t) P (t) P2 (t) P (t)

10 862 Usman Yusuf Abubakar et al. Figure2. The graph of transition probabilities Discussion of Results The values of the equations Pi (t) for i=, 2, and =2,,. And t=,, 2,... are the transition probabilities from state i to state respectively as indicated in table2 and are illustrated with graphs in figure2. The initial values of the transition probabilities P2 (t), P (t), P (t), P2 (t), and P (t) (t = ) are.5,.22,.865,.65 and.88 respectively, corrected to 5 decimal places. These transition probabilities increased slightly (t = ) to.2,.,.825,.552, and.68 respectively. These set of probabilities indicates that the blood transfused from blood group O to blood groups A, B, and AB are about.,.2 and. percent respectively. The result shows that the reception of blood by blood group AB from blood group A and blood group B is about.% and.% respectively. The result also indicates that there will be little variation in the blood needed for transfusion in the future on the basis of the present. The blood group O that is needed to be donated to blood groups A, B, AB is.%,.% and 8.% respectively. The demand by blood group AB from blood group A and blood group B is 5.% and 6.% respectively. The blood that is donated and received by the same blood group constitute the maor blood needed in the hospital with %,.%, 8.%, and 8.5% for blood groups AB,B,A and O, when t = respectively. The information about the blood needed for transfusion and blood bank inventory are very important for a successful health care delivery. If such information is available in every city hospital, it may be possible to move blood from one hospital to another during emergency.

11 Analysis of blood transfused in a city hospital 862 References [] U. Y. Abubakar. The Exponential Distribution and The Application to Markov Models. Journal of Research In National Development, Vol. No. 2a December, (2), 2-5. [2] U. N. Bhat, Element of Applied Stochastic processes, John Wiley, New York (8). [] R. A. Howard, Dynamic programming and Markov processes. The MIT Press, Massachusetts (6). [] V. A Jevwia, Determining blood needed for transfusion in a general hospital with a Markov chain model. Unpublished MTech Thesis. Department of Mathematics/Statistics, Federal University of Technology Minna, Nigeria (2) [5] K. N, Korve. A Three State Continuous Time Markov model for the Asthma process. (Abacus), The ournal of the Mathematical Association of Nigeria. Volume 2, Number 2 (2), ( - 6). [6] C. C Pegels, E. J Andrew, An Evaluation of Blood- Inventory policies: A Markov Chain Application, Operations Research, Vol. 8 No. 6 (), [] C. C Pegels, A Blood Bank Collection Scheduling and Inventory Control System, AIIE Trans., (6), [8] T. H Rockwell, R.A Barnum, W.C Griffin., Inventory Analysis as Applied to Hospital Whole Blood Supply and Demand, J. Indust. Eng. 6, (62),. Received: November 2; Published: December 2, 2

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