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Journal of Service Science and Management, 2011, 4, 491-498 doi:10.4236/jssm.2011.44056 Published Online December 2011 (http://www.SciRP.org/journal/jssm) Copyright © 2011 SciRes. JSSM 491 Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems P. Tirupathi Rao1, M. Venkateswaran1, R. Vishnu Vardhan2 1Department of Statistics, S. V. University, Tirupati, India; 2Department of Statistics, Pondicherry University, Puducherry, India. E-mail: {drtrpadi, venky.swaran, rvvcrr}@gmail.com Received July 12th, 2011; revised September 3rd, 2011; accepted October 9th, 2011. ABSTRACT Evaluation of drug impact both on positive and negative sides is very important to monitor the drug administration during the treatment. The effectiveness of drug is a variable due to various chance and assignable causes. Evaluation of drug efficacy through conventional mathematical modeling is the existing practice. In this paper we proposed a Sto- chastic model for measuring the drug efficacy for Non-Clinical and short term drug administration practices. Sensiti- vity analysis is carried out to observe the model behavior. Development of computer software and desktop templates to this model will provide effective decision support systems for health caretakers. Keywords: Stochastic Model ing , Drug Efficacy, Self Drug Administration 1. Introduction Treatment with drugs is one of the customary methods to cure diseases. Drug administration on the disease control in under developed and developing countries is mostly on non-clinical environment, particularly for the short term and seasonal related disease treatments like cold, fever, influenza, cough etc. Major category/population of the patients in India are getting the treatment through non competent medical supervision. It leads to random pick- ing of drug at the choice of either individual own behalf or Medical Shop holders or Registered Medical Practi- tioners etc. This sort of situation provokes non-assurance of drug effectiveness on the control of the disease. Deci- sion making on drug picking has to be made on various parameters of drug administration like quantity of drug per unit time, number of times (frequency) of drug ad- ministration per unit time period and many other similar factors. The health care taking must be supported by competent screening procedures with proper health indi- cators on the efficacy of drug under usage. Drug dosage level less than the required and more than the sufficient are unwanted as the farmer leads to the body drug resis- tant and the later may cause hazards to general health of the patient. Therefore the limits on size of the drug dose are playing vital role in regulations of disease intensity. Random and erratic usage of drug without scientific approach will make drug administration more vulnerable. It always act as double edged weapon through either wanted or unwanted impact. Spontaneous selection of drug based on purely chance manner requires due attention of the researchers. Assessment on the levels of drug effect- tiveness on the targeted disease control is the need of the hour. Measuring the levels of both positive and negative effectiveness is possible through the construction of a relevant Mathematical model. Our study can measure the net effectiveness of drug by considering the linear com- bination of positive and negative influences of drug on the disease control. This study will assist as the health management tool for decision support system. It can as- sess the influence of drug either of proper usage or of abuse by obtaining various Statistical measures of lower and higher order. A suitable formulation of the bio-systems into mathe- matical formulation and transforming classic mathema- ticcal environment into statistical/empirical situations is pivotal. There is much literature evidence on modeling the drug efficiency in Deterministic environment. The work so far carried out is mostly concentrated on drug admini- stration for long term treatments. A few work is reported in the literature on measuring the drug effectiveness on the treatment of short term duration. Drug administration on clinical environment can be done in several different methods [1]. The model of drug resistance from gene amplifications could be studied th- rough the policies of optimal control [2]. Several asymp- totic properties of infinite dimensional model drug resis- Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems 492 tance evolutions are available [3]. The phamalogemmics can be studied through drug disposition, drug targets and its side effects [4]. The Legand efficiency indices are con- sidered as the guide spots in the discovery of drug effect- tiveness [5]. Drug efficiency indices were also constru- cted with structured based calculation [6]. The evaluation of drug efficacy for long term Treatment problems can be done through stochastic models. The environment of drug administration with ‘r’ alternatives and each alternative is of completely random is considered. The treatment is administered for N days for which the effectiveness of drug has varying chances. The selected drug for treating the disease is correct with certainty for N1 days, the choice of correct selection is equal to 1k for Nk days where k = 1, 2, , r. Further the total period of treatment is . The model has applied to study the 1 k i i NN drug effectiveness for long term treatments like cancer, T.B. and skin related diseases [7]. In this paper we have developed a stochastic model for evaluating the effect- tiveness of drug for the treatments of short durations. This study has focused the attention on calculating the drug efficacy when the choice of the drug is random and having positive and negative impacts. The net effective- ness of drug is a linear combination of both positive and negative impacts with some weight coefficients. 2. Stochastic Model and Statistical Measures In this study we develop a stochastic model for measure- ing the effectiveness of the drug by considering the fol- lowing assumptions based on the problems of the pa- tients; Problems of Drug selection and Problems of Drug administration. The patients may get ill health such as cold, sea- sonal fever, headaches and similar non-chronic and short term diseases at a random time. The patient is not having required knowledge on the drug administration parameters such as (i) The dosage level of drug; (ii) The frequency of drug administration per unit time (iii) time between two spells of drug administration; (iv) neither the pa- tient nor healthcare taker on his behalf is aware on the side effects of the drug. The patients may pick the drug either on his own choice or from the suggestion of a medical shop- keeper or from other similar people. He/She may initiate the usage of drugs with a group of medicines initially but some medicines may be missed during the administration period. There is every likely to skip some spells of taking medicines during the course period of drug admini- stration. The drug usage may be stopped abruptly at any point of time on various reasons like he/she may get relief from the problem, they may not enough stock of drug to use for the required course period. Patient will select the drug of option among avai- lable drugs. The usage of medicines may be either a single drug or a group of drugs. He/She administered the drug one or more pills for one time and more than one time in a day and more than one day in over all administration. This model can measure the drug effectiveness for in- termediate evaluation during treatment period. Due to various unexplained reasons and the varying levels of health conditions of the patient during the course period, the effectiveness of drug has to be considered as random variable and it is obtained as measure of efficiency (e). It is the ratio of the output achieved to the input used. Acheived Ouput eUsed Inputs Output can be obtained through various Statistical means and parametric observations with pre and post-drug ad- ministration datasets. The model is constructed to obtain stochastic proc- esses so as the behavior of a random variable Z can be analyzed. It is the Net effectiveness of drug, expressed as Z = a X + b Y; where a, b are the weight age coefficients corresponding to X, Y respectively. They are obtained through preparation of frequency tables and pattern iden- tifications in the data. X is a Bernoulli variable assuming values 1, 0 for the drug impact when it has positive effect (1) and drug non positive effect (0). “Positive effectiveness of drug” does implies that the drug has perform its task on healing the problem, what exactly it is meant for. “Non positive ef- fectiveness of drug” does implies that the task of drug assi- gnment may be un served as a neutral impact on overall effectiveness in healing the problem. Y is another Bernoulli variable assuming values 1, 0 for the drug impact has negative effect (1) and the drug has non-negative effect (0). “Negative effectiveness of the drug” does implies that usage of drug may cause some loss to the general health due to side effects in mis- matching of drug that is consumed and the health prob- lem that is under study. “Non-negative effectiveness of the drug” does mean that usage of drug may not harm the overall health of the patient or there is a neutral negative effectiveness. Let p1 be the probability of positive impact and p2 be the probability of non-positive impact. The probability distribution for the same is P (X = i) = p1; for I = 1; P (X = i) = p0; for I = 0; This probability distribution is ob- tained from the stochastic process of an indicator varia- ble {Xi; I = 0, 1}; further p1, p0 ≥ 0; p1 + p0 = 1. The rth Copyright © 2011 SciRes. JSSM Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems Copyright © 2011 SciRes. JSSM 493 order raw moment for the above probability distribution 1 22 2 10 10 11 .. .. .. app bqq ap bq . is ' 1r X p . Similarly, let q1 be the probability of Negative impact and q2 be the probability of non negative impact. The stochastic process of Y is {Yj; j = 0, 1} and the probabi- lity distribution is 6) The 3rd Central Moment is: 32 31 11 32 11 ..13. 2. ..13. 2. app p bqq q 1 P (Y = j) = q1; for j=1; P (Y = j) = q0; for j = 0; and q1 + q0 = 1. q1, q0 ≥ 0 the kth order raw moments of the 7) The 4th Central Moment is: distribution is . ' 1 kYq 42 41 111 42 1111 22 11 0111 ..14.6.3. .1 4.6.3. 6.... . app pp bqq qq ab pqpqpq 3 3 The random variables X and Y are assumed to be in- dependent as the occurrence of events say positive and negative effective nesses are no way influenced by one with the other so that bivariate stochastic process is {(Xi, Yj); i, j = 0,1} and its probability function is P(X = i, Y = j) = P(X = i). P(Y = j) for i, j = (0, 1) and the rth order raw 8) The coefficient of Skewness is: moment of joint probability distribution is ' 11 , r X Ypq . 2 323 111111 13 22 10 10 ..13.2.. 13.2. .. .. appp bqqq app bqq 2 Z as combined random variable can measure the net drug effectiveness by considering both positive and ne- gative impacts together. Usually ‘b’ has a negative coef- ficient so as Z value become Z = a X – b Y the study has to be focused on Z variable as a mixture of X and Y probability distribution. The Stochastic Process of Z is = {Zk, k = 0, 1}where Zi,j = a Xi – b Yj the probability dis- tribution is 9) The coefficient of Kurtosis is: (see Equation (9)) 10) The Moment generating function of Z is: 11 00 manb t zmn mn Mt pqe 11) Characteristic function of Z is: 11 11 ; ,0,10Otherwise. xy y 101 0 PZap pbqq xy xy 11 2 00 ;1 manb it zmn mn tpqefori 12) Probability generating function of Z 11 00 ma nb zmn mn PSpqS For all practical purposes we may consider b as (–1) (b). The Statistical measures are obtained using relation Z = aX + bY and b = (–1) b thus the results are: 3. Methodology 1) The rth order origin (raw) moment of Z is: The following methodology is considered for measuring the drug efficacy. The usual practices that are happened in clinical treatments are based on the readings of pre and post tests. In general the diagnosis procedures at clinical treatment are based on the screening tests. If we consider an example of screening tests of fever, the intensity of fever can be assessed with Temperature of the body, Pulse rate of the nerve, no of breathes of the patient, skin temperature, rectal temperature, etc. ' 11 0 1 rkr rk rk Z ap bq k . By assuming r = 1, 2, 3, 4, we can obtain the first four raw moments. With the help of these four, we can get Mean, S. D., Variance, C. V. Coefficients of Skewness and Kurtosis etc. 2) Average or Mean Effectiveness of Drug is: Fever may be caused due to so many reasons like in- fections, inflammation, indigestion, insect bite, etc and many unexplained also. Let us assume that a patient is getting treatment for a fever. He has no idea about the reason for getting fever. His objective is to get rid of fe- ver by consuming some pills. In this context the drug may give the effect on four fold namely: 11 . .ap bq. 3) Variability of Drug Effectiveness is: 22 10 10 .. ..app bqq . 4) Standard Deviation 1 22 2 10 10 app bqq. 5) Coefficient of Variation of Drug Effectiveness 42342322 1111111111 0111 22 22 10 10 .14.6.3..1 4.6.3.6..... .. .. a ppppbqqqqab pqpqpq app bqq (9) Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems 494 1) Positive effect: The drug shall decrease the tempera- ture through the means of suppressing the reasons of cau- sing fever. 2) Non-Positive effect: The drug may not decrease the temperature as it has no influence on suppression of the fever causing factors. 3) Negative effect: The drug may give adverse effects on general health of the patient causing unwanted side effects. 4) Non-Negative effect: The drug may not give unwan- ted side effects irrespective of its positive or non-positive effects. The concepts of Non-Positive and Non-negative ef- fects of drug, though they appears to be same, we have considered those two are significantly differed as the impact factor of non-positive effect of the drug is not equal to the impact factor of non-negative effect of drug. The coefficient of positive effectiveness (a), may ob- tained as influenced relation of many factors. It may other- wise defined as 1 r p i i pe ae r ; p i eis the Positive efficiency measure of ith factor; i = 1, 2, , r; As the usual parameters of a disease like fevers are Body tem- perature, Heartbeat/Pulse rate, number of breathes, skin temperature etc. Let us consider the first parameter namely temperature, 1 initial temperatureTemperature beforeusing drugafter using drug Quantity ofdrugused ptb ta eq . [Example: If tb = 103˚; ta = 102˚; q = 500 mg then 1 103 10210.20.002 500500 100 p e ]. Let Heart beat count is the 2nd parameter, then 2 initial heartbeatHeartbeat count b efore using drugafterusing drug quantity ofdrugused p e HB preHB post q . If number of breathes per unit time is the 3rd parameter then 3 HB preHB post quantityofdrug used p e . In general sense, p revious readingpostreading p arameter parameter useddrug quantity i th th p ii e reading ofpre testreading ofposttest q i p ii e Combining all the above, 1 r p i pi e ae r , where ‘r’ is the total number of factors on which the positive effe- cttiveness of drug is attained. Similarly the coefficient of Negative effectiveness (b) may also be influenced by non suitability or mismatching of drug to the disease under treatment. For example: usage of certain drug leads to increased loss of albumin through urine. 1 Albumni lossAlbumni loss b eforedrug useafter drug use quantityofdrug el Albpre testAlbposttest q , el1 is the total negative effectiveness of drug. If there are k types of factors that are making the drug negative ef- fectiveness. Then the overall negative effectiveness is obtained as 1 ; j rl l j e be r The coefficients of both po- sitive effectiveness and negative effectiveness are inf- luenced by all the relevant factors. This measure is statis- tically valid as the readings are considered on various parameters of the problem under study. The chances of positive effectiveness ‘p1’, Non-posi- tive effectiveness ‘p0’ are obtained as the proportions of their impact on overall usage. These values are usually obtained from the relative frequency distributions or from some historical data. These observations may also obtained from the pharmacology experimental studies. Let a drug is applied on ‘m’ living beings and of which ‘x’ are having the positive result and the rest (m-x) are not having the positive effectiveness. Then 1 x pm and 01 11 x mx ppmm . The chances of negative impact can also be studied on similar lines. Let a drug is applied on ‘n’ living beings of which ‘y’ are having the negative impact and the remaining n-y are not having negative results then the chances are Copyright © 2011 SciRes. JSSM Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems495 1 y q and n 01 11 ynx qq . nn 4. Numerical Illustration and Sensitivity In he insights of the drug efficiency, a data that Average drug effi- ci efficiency is an fficiency is an in- cr ariability of drug efficiency is an of having Positive Analysis order to get t is considered with inputs p1, q1, a and b. The outputs like average drug effectiveness, variability in the drug effect- tiveness, coefficient of variation, coefficient of Skewness, coefficient of kurtosis, etc are calculated with software MATHCAD. The numerical data is placed in Tables 1, mean response and the variability of the drug impact are analyzed. It is observed that Average drug efficiency is an increasing function of p1, and it is negative when p1 < q1, a < b where as Mean efficiency is an increasing func- tion of p1, and it is positive when p1 > q1, a < b when other parameters are constants. It is further observed that Average drug efficiency is an increasing function of p1, and it is negative when p1 < q1, a > b where as Mean effi- ciency is an increasing function of p1, and it is positive when p1 > q1, a > b when other parameters are constants. It is observed that Average drug efficiency is a decreas- ing function of q1, when p1 > q1, a < b where as Mean efficiency is a decreasing function of q1, and it is negative when p1 < q1, a < b when other parameters are constants. It is further observed that Average drug efficiency is a decreasing function of q1, and it is positive when p1 > q1, a>b where as Mean efficiency is a decreasing function of q1, and it is positive when p1 < q1, a > b when other pa- rameters are constants. It is observed from Figure 1 ency is an increasing function of p1, and it is negative when p1 < q1, a < b, where as Mean efficiency is an in- creasing function of p1, and it is positive when p1 > q1, a < b when other parameters are constants. It is further observed that Average drug efficiency is an increasing function of p1, and it is negative when p1 < q1, a > b where as Mean efficiency is an increasing function of p1, and it is positive when p1 > q1, a > b when other parame- ters are constants. It is observed that Average drug effi- ciency is a decreasing function of q1, when p1 > q1, a < b where as Mean efficiency is a decreasing function of q1, and it is negative when p1 < q1, a < b when other pa- rameters are constants. It is further observed that Avera- ge drug efficiency is a decreasing function of q1, and it is positive when p1 > q1, a > b where as Mean efficiency is a decreasing function of q1, and it is positive when p1 < q1, a > b when other parameters are constants. It is observed that the variability of drug increasing function of p1 when p1 < q1, a < b; it is de- creasing function of p1 when p1 > q1, a < b; Further it is observed that the variability of drug efficiency is a de- creasing function of p1 when p1 < q1, a > b; it is decreas- ing function of p1 when p1 > q1, a > b when other pa- rameters are constants. It is observed that the variability of drug efficiency is an increasing function of q1 when p1 > q1, a < b; it is decreasing function of q1 when p1 < q1, a < b; Further it is observed that the variability of drug efficiency is an increasing function of q1 when p1 < q1, a > b; it is a decreasing function of q1 when p1 < q1, a > b when other parameters are constants. It is observed that Average drug e easing function of a, it is negative when p1 < q1, a < b; whereas Mean efficiency is an increasing function of a, and it is positive when p1 < q1, a > b when other parame- ters are constants. It is further observed that Average drug efficiency is an increasing function of a, and it is negative when p1 > q1, a < b where as Mean efficiency is an increasing function of a, and it is positive when p1 > q1, a > b when other parameters are constants. It is observed that Average drug efficiency is a decreasing function of b, it is negative when p1 < q1, a < b; whereas Mean effi- ciency is a decreasing function of b, and it is positive when p1 < q1, a < b when other parameters are constants. It is further observed that Average drug efficiency is a decreasing function of b, and it is negative when p1 > q1, a < b where as Mean efficiency is a decreasing function of b, and it is positive when p1 > q1, a > b when other parameters are constants. It is observed that the v increasing function of a when p1 < q1, a < b; it is an increasing function of a when p1 < q1, a > b; Further it is observed that the variability of drug efficiency is an in- creasing function of a when p1 > q1, a < b; it is an in- creasing function of a when p1 > q1, a > b when other parameters are constants. It is observed that the variabili- ty of drug efficiency is an increasing function of b when p1 < q1, a < b; it is an increasing function of b when p1 < q1, a > b; Further it is observed that the variability of drug efficiency is an increasing function of b when p1 > q1, a < b; it is an increasing function of b when p1 > q1, a > b when other parameters are constants. 5. Summary and Conclusions Our study observed that the chance impact of the drug is giving an increasing impact on its average performance, decreasing impact on variability when p1 > p0; increasing impact of variability when p1 < p0. The coefficient of variation is a decreasing function of performance of positive impact when p1 > q1. Consisten- cy of drug performance may be increased by maintaining more positive impact than negative impact. The chance of having Negative impact of the drug is giving an de- creasing impact on its average performance, increasing impact in variability when p1 > p0. The coefficient of va- riation is an increasing function of performance of Copyright © 2011 SciRes. JSSM Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems Copyright © 2011 SciRes. JSSM 496 ewness and kurtosis for varying values of p1, q1, a, b. p1 β2 Table 1. Values of mean, variance, coefficients of sk q1 A B Mean Variance C.V. β1 0.1 0.5 0.7 0.8 –0.33 0.204 – 0. 1.3690722.01 0.2 –0.26 0.238 –1.878 0.08 2.126 0.3 –0.19 0.263 –2.699 0.046 2.07 0.4 –0.12 0.278 –4.391 0.013 2.007 0.6 0 0 0 0 – 9 0. 0 0. 0 0. 0 0. 0 – 7 0. 0 0. 0 4 0. 0 0. 0 3 0. 3. 0. – 0. 0.3 0.1 0.055 2.349 0.099 2.261 .5 0.7 0.80.02 0.278 26.344 0.013 2.007 0.7 0.09 0.263 5.697 0.046 2.07 0.8 0.16 0.238 3.052 0.08 2.126 0.9 0.23 0.204 1.964 0.072 2.01 0.1 .5 0.8 0.7–0.27 0.18 –1.572 0.233 2.598 0.2 –0.19 0.225 –2.496 0.212 2.458 0.3 –0.11 0.257 –4.608 0.109 2.206 0.4 –0.03 0.276 –17.515 0.029 2.039 0.6 .5 0.8 0.70.13 0.276 4.042 0.029 2.039 0.7 0.21 0.257 2.414 0.109 2.206 0.8 0.29 0.225 1.635 0.212 0.212 0.9 0.37 0.18 1.147 0.233 2.598 0.6 .2 0.4 0.50.14 0.078 2 3. 0.471 2.625 0.3 0.09 0.091 350.245 2.26 0.4 0.04 0.098 7.842 0.086 2.039 0.5 –0.01 0.101 31.765 .19E-03 1.967 6 0.7 .4 0.5–0.11 0.091 –2.741 0.073 2.26 0.75 –0.135 0.085 –2.163 0.121 2.427 0.8 –0.16 0.078 –1.75 0.165 2.625 0.85 –0.185 0.07 –1.433 0.188 2.832 6 0.2 .6 0.3 0.3 0.101 1.058 0.164 1.658 0.3 0.27 0.105 1.202 0.137 1.726 0.4 0.24 0.108 1.369 0.108 1.753 0.5 0.21 0.109 1.571 0.083 1.761 6 0.7 .6 0.30.15 0.105 2.163 0.056 1.726 0.75 0.135 0.103 2.38 0.056 1.699 0.8 0.12 0.101 2.646 0.059 1.658 0.85 0.105 0.098 2.98 0.068 1.597 6 0.7 .1 0.5–0.29 0.055 0.808 0.66 1.864 0.2 –0.23 0.062 –1.083 0.427 2.071 0.3 –0.17 0.074 –1.601 0.208 2.223 0.4 –0.11 0.091 –2.741 .30E-022.26 6 0.7 0.6 .50.01 0.139 37.269 6.50E-06 2.114 0.7 0.07 0.17 5.892 7.23E-03 2.006 0.8 0.13 0.206 3.492 0.023 1.901 0.9 0.19 0.247 2.615 0.04 1.807 7 0.6 0.1 .5–0.23 0.062 –1.083 0.146 1.287 0.2 –0.16 0.068 –1.635 0.089 1.571 0.3 –0.09 0.079 –3.121 0.028 1.869 0.4 –0.02 0.094 –15.297 .75E-042.087 7 0.6 0.6 .50.12 0.136 3.069 0.059 2.256 0.7 0.19 0.163 2.124 0.12 2.257 0.8 0.26 0.194 1.696 0.186 2.234 0.9 0.33 0.23 1.454 0.25 2.199 6 0.7 0.4 .5–0.11 0.091 –2.741 0.073 2.26 0.6 –0.18 0.114 –1.876 0.153 2.247 0.7 –0.25 0.141 –1.504 0.235 2.208 0.8 –0.32 0.173 –1.299 .09E-012.16 6 0.7 0.4 0.15 0.135 0.043 1.538 0.097 1.532 0.2 0.1 0 0.047 2.163 0.056 1.726 0.25 .0650.052 3.492 0.023 1.901 0.3 0.03 0.057 7.979 44E-03 2.042 7 0.6 0.4 0.5 –0.02 0.094 15.297 4.75E-04 2.042 0.6 –0.08 0.12 –4.33 0.014 1.953 0.7 –0.14 0.151 –2.777 0.036 1.83 0.8 –0.2 0.187 –2.163 0.056 1.726 7 0.6 0.4 0.15 0.19 0.039 1.039 0.458 2.046 0.2 0.16 0.043 1.299 0.309 2.16 0.25 0.13 0.049 1.696 0.186 2.234 Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems497 Figure 1. Graphical presentations of mean and variance effects of drug. Copyright © 2011 SciRes. JSSM Stochastic Modeling on Drug Efficacy in Self Drug Administration Health Problems Copyright © 2011 SciRes. JSSM 498 Negative impact when p1 > q1. The chance of having Positive impact of the drug is giving an increasing impact on its average performance, increasing impact on varia- bility when a > b. The coefficient of variation is a de- creasing function of performance of positive impact of drug when a > b. Consistency of drugs performance may be increased by maintaining more positive impact than negative impact. This model will help the individual pa- tients in quantification of the problem severity with drug abuse/misuse. Development of software to this model will assist in the health monitoring of self health care takers fort their decision support systems. The scope for future work may be done with multinomial cases instead of Bernoulli cases. This work may be extended to more suitable contexts. 6. Acknowledgements All the authors of the paper are very much greateful to the editorial board, reviewers of the Paper, and Editors of the Journal of Service Science and Management (JSSM) for their valuable suggestion in improving the quality of the paper. REFERENCES [1] S. H. Daukes, “A Consideration of the Various Methods of Drug Administration,” Post Graduate Medical Journal, Vol. 5, No. 49, 1929, pp. 1-7. doi:10.1136/pgmj.5.49.1 [2] J. Smieja, et al., “Optimal Control for the Model of Drug Resistance Resulting from Gen Amplification,” Prepera- tion of the 14th IFAC World Congress, Beijing, China, Vol. 50, 1999, pp. 71-75. [3] A. Swierniak et al., “Asymptotic Properties of Infinite Di- mensional Model Drug Resistance Evolution,” Proceed- ings of the European Control Conference (ECC), Brus- sels, 1997. [4] W. E. Evans and H. L. McLeod, “Pharmacogenomics— drug disposition, drug targets, and side effects,” New England Journal of Medicine, Vol. 348, No. 6, 2003, pp. 538-549. doi:10.1056/NEJMra020526 [5] C. Abad-Zapatero and J. T. Metz, “Ligand Efficiency In- dices as Guideposts for Drug Discovery,” Drug Discovery Today, Vol. 10, No. 7, 2005, pp. 464-469. doi:10.1016/S1359-6446(05)03386-6 [6] C. Hetenyi, et al., “Structure-Based Calculation of Drug Efficiency Indices,” Journal of Structural Bioinformatics, Vol. 23, No. 20, 2007, pp. 2678-2685. doi:10.1093/bioinformatics/btm431 [7] P. Tirupathi Rao and R. Bharathi, “Stochastic Modeling in the Evaluation of Drug Efficiency,” International Jour- nal of Statistics and System, Vol. 5, No. 3, 2010, pp. 427-438. |