Optimization Programming Problems For Control And Management Of Bacterial Disease With Two Stage Growth/Spread Among Plants

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1 Internatinal Jurnal f Engineering Science Inventin ISSN (Online): 9 67, ISSN (Print): Vlume 5 Issue 8 ugust 06 PP.0-07 Optimizatin Prgramming Prblems Fr Cntrl nd Management Of Bacterial Disease With Tw Stage Grwth/Spread mng Plants E.S.V. Narayana Ra and P. Tirupathi Ra Dept. f Statistics, Pndicherry University, Puducherry bstract: This study has prpsed fur nn linear prgramming prblems fr ptimal cntrl f bacterial spread ver the plants at different ages n different species. The bjective functins and cnstraints f the said NLPPs are frmulated thrugh the develped mathematical mdels f bacterial disease grwth and spread using bivariate stchastic prcesses []. This study will help the agricultural care takers fr develping the relevant decisin supprt systems t get the indicatrs n spread f bacterial diseases and its cntrl. Keywrds: Nn Linear Prgramming Prblems, Bacterial Disease Cntrl, Optimizatin, Bivariate Stchastic Prcesses, Decisin Supprt Systems. I. Intrductin Studies n bacterial diseases in plants have significant imprtance fr farmers fr making its cntrl and regulating with ptimal management appraches. Bacterial spread n grwing plants may lead t huge damage f crps. ssessing the intensity f disease thrugh mdeling is the cre area f research which has attracted the attentin f many researchers. Prper management f disease cntrl is pssible nly when its spread is prperly assessed. Designing f treatment prtcls are directly linked with suitable study methds f disease intensity. Due t many explained and unexplained reasns the influencing factrs f bacteria spread amng plants are stchastic. Thus prbabilistic tls need t be used t study the dynamics f bacterial transmissin. In rder t develp the cntrlling devices n the disease spread, ptimizatin prgramming mdels were frmulated with the bjectives f minimizing the average sizes f bacteria in different stages f plants and maximizing the vlatility in bacterial size by using the variance measures. The rates f arrival r nset f fresh bacteria t different staged(aged) plants, rates f grwth f bacteria within each state, rates f death f bacteria in each stage, migratin rates f bacteria frm different stages, transitin rates f bacteria frm ne stage t ther stages, etc are the study parameters f these mdels. Explring f these parameters will help the crp caretakers fr measuring the dynamics f bacterial intensities s as suitable interventin r treatment prtcls can be designed. Using stchastic simulatin mdel, Xu and Ridut (000) demnstrated the imprtance f initial epidemic cnditins, especially the spatial pattern f initially infected plants and the relatinships f spati-tempral statistics with underlying bilgical and physical factrs. The stchastic ptimisatin and prgramming mdels fr drug administratin in cancer chemtherapy was develped by Tirupathi Ra et al., (00, 0, 0, 0). Madhavi et al. (0) have prpsed a stchastic prgramming prblem fr ptimal drug administratin fr cancer chemtherapy. Their studies have fcused n explring the decisin parameters like drug dsage levels, drug administrative perid, drug vacatin perid, number f drug cycles and frequency f drug administratin within a cycle etc., during chemtherapy. With the mtivatin f the abve said wrks, the researchers have prpsed this study in the cntext f grwth and spread f plant diseases. II. Optimizatin Mdeling The current study prpsed sme nn-linear prgramming prblems fr ptimal cntrl f bacterial spread ver the plants at different ages f different species. The wrks n develpment f stchastic mdels fr grwth and spread f bacteria [] are cnsidered here fr frmulating the bjective functins as well as cnstraints. The bjectives f the said prblems are frmulated in tw dimensins such as minimizing the average size f bacteria and maximizing the variance f bacterial size hsted n the plants. The purpse f this study is t derive the decisin parameters such as regulating rates f bacterial grwth, transitins frm ne stage t ther and bacterial lss. The subjective cnstraints are frmulated with a view f aviding the risky levels f bacterial grwth. The study has cnsidered the live data btained frm the labratries f plant pathlgy situated in ndhra Pradesh. Page

2 Optimizatin Prgramming Prblems Fr Cntrl nd Management Of Bacterial Disease With.. Ntatins and Terminlgy: n : number f units f bacteria in stage-i (nursery stage plants) at time t ; m : number f units f bacteria in stage-ii (transplantatin stage plants) at time t ; Here ne unit dentes the number f bacteria in a square area (mm ); and : the rates f grwth f bacteria due t immigratin frm external means per unit time t stage-i and stage-ii plants respectively; and : the rates f internal grwth (birth) f bacteria per unit time in stage-i and stage-ii plants respectively; : the rate f transitin f bacteria per unit time frm stage-i t stage-ii plants; and : the rates f emigratin f bacteria per unit time frm stage-i and stage-ii plants t the ther areas respectively; and : the rates f lss (death) f bacteria per unit time in stage-i and stage-ii plants respectively. ( ) ; B ( ) ; J N 0, M 0 : Initial sizes f bacteria existing at nursery and plantatin stages f plants C 0, D 0, E 0 : Integral Cnstants while slving the differential equatins C : Upper limit f average number f bacterial units in stage-i plants at a pint f time t ; C : Upper limit f average number f bacterial units in stage-ii plants at a pint f time t ; C : Upper limit f variance f bacterial units in stage-i plants at a pint f time t ; C : Upper limit f variance f bacterial units in stage-ii plants at a pint f time t ; III. Stchastic Prgramming Prblems In this sectin, fur prgramming prblems are frmulated. They are. Optimal Prgramming fr Minimizing the Expected Bacteria size in Stage-I plants. Optimal Prgramming fr Minimizing the Expected Bacteria size in Stage-II plants. Optimal Prgramming fr Maximizing the Variance size f Bacteria in Stage-I plants. Optimal Prgramming fr Maximizing the Variance size f Bacteria in Stage-II plants.. Optimal Prgramming fr Minimizing the Expected Bacteria size in Stage-I plants This prgramming prblem is frmulated with the bjective f minimizing the average number f bacterial units in stage-i plants with the cnstraints f average number f bacterial units and als with the variance f bacterial units in different stages f plants. In rder t achieve the abve bjective, the grwth f bacterial units size shall nt beynd the warning limits. Hence, the average number f bacterial units in stage-i plants at a pint f time t shuld nt be mre than sme threshld limits say C and C. Similarly with the variances f bacterial units in stage-i and stage-ii plants at a pint f time t shuld nt be mre than sme threshld limits say C and C. Then the prgramming prblem is. M in Z N e.. Subject t the cnstraints, C.. N. e ( B ) M. e C.. C. e C.. Page

3 Optimizatin Prgramming Prblems Fr Cntrl nd Management Of Bacterial Disease With J J M. e ( ) ( B ) ( B )( B ) ( B ) ( B )( B ) ( B )( B )( B ) ( B ) t M. e ( ) C. e D. e E. e C ( )( B ) ( B )( )( B ) ( B ) ( B )..5 and the decisin parameters are α 0, α 0, β 0, β 0, τ 0, ε 0,ε 0, δ 0, and δ 0..6 Fllwing the same prcedure mentined in the sectin. the ther three prgramming prblems can be frmulated... Optimal Prgramming fr Minimizing the Expected Bacteria size in Stage-II Plants This prgramming prblem is frmulated with the bjective f minimizing the average number f bacterial units in stage-ii plants with the cnstraints f average number f bacterial units in different stages f plants and variance f number f bacterial units in different stages plants. M in Z M. e ( B) Subject t the cnstraints, C.. M. e C ( B ).... C. e C.. J J M. e ( ) ( B ) ( B )( B ) ( B ) ( B )( B ) ( B )( B )( B ) ( B ) t M. e ( ) C. e D. e E. e C ( )( B ) ( B )( )( B ) ( B ) ( B ) and the decisin parameters are α 0, α 0, β 0, β 0, τ 0, ε 0,ε 0, δ 0, and δ Optimal Prgramming fr Maximizing the Variance size Bacteria in Stage-I plants This prgramming prblem is frmulated with the bjective f maximizing the variance number f bacterial units in stage-i plants with the cnstraints f average number f bacterial units in different stages and variance f number f bacterial units in different stages.... M a x Z C e Subject t the cnstraints, C.. M. e C ( B )..5.. C. e C.. J J M. e ( ) ( B ) ( B )( B ) ( B ) ( B )( B ) ( B )( B )( B ) ( B ) t M. e ( ) C. e D. e E. e C ( )( B ) ( B )( )( B ) ( B ) ( B )..5 Page

4 Optimizatin Prgramming Prblems Fr Cntrl nd Management Of Bacterial Disease With and the decisin parameters are α 0, α 0, β 0, β 0, τ 0, ε 0,ε 0, δ 0, and δ Optimal Prgramming fr Maximizing variance f size f Bacteria in stage-ii plants This prgramming prblem is frmulated with the bjective f maximizing the variance f number f bacterial units in stage-ii plants with the cnstraints f average number f bacterial units in different stages and variance f number f bacterial units in different stages. M a x Z J J M. e ( ) ( B ) ( B )( B ) ( B ) ( B )( B ) ( B )( B )( B ) M. e ( ) C. e D. e ( )( B ) ( B )( )( B ) ( B ) ( B) Subject t the cnstraints, ( B ) t C.. M. e C ( B ) E. e.... C. e C.. J J M. e ( ) ( B ) ( B )( B ) ( B ) ( B )( B ) ( B )( B )( B ) ( B ) t M. e ( ) C. e D. e E. e C ( )( B ) ( B )( )( B ) ( B ) ( B ) and the decisin parameters are α 0, α 0, β 0, β 0, τ 0, ε 0,ε 0, δ 0, and δ 0..6 IV. Numerical Illustratin In rder t understand the behaviur f ptimizatin prblems, data set frm agricultural labratries situated in ndhra Pradesh state were cnsidered. Decisin parameters like grwth, lss and transitin rates f bacterial units amng tw stages f plants are btained at different values f initial number f bacterial units at stage-i (N), initial number f bacterial units at stage-ii (M), upper limit f variance f bacterial units at stage-i (C ), upper limit f variance f bacterial units at stage-ii (C ) and time. ll the btained values f ; ; ; ; ; ; ; ; ; Z; Z; Z; Z using LINGO.0 are presented in tables frm. t.. Table-.: Values f ; ; ; ; ; ; ; ; ; Z fr changing values N 0 ; C ; t and fixed values f M 0 ; C : N 0 M 0 C C t Z E E E E E E E E E E E E E E E Table-.: Values f ; ; ; ; ; ; ; ; ; Z fr changing values M 0 ; C ; t and fixed values f N 0 ; C : N 0 M 0 C C t Z Page

5 Optimizatin Prgramming Prblems Fr Cntrl nd Management Of Bacterial Disease With Table-.: Values f ; ; ; ; ; ; ; ; ; Z fr changing values N 0 ; C ; t and fixed values f M 0 ; C : N 0 M 0 C C t Z E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table-.: Values f ; ; ; ; ; ; ; ; ; Z fr changing values M 0 ; C ; t and fixed values f N 0 ; C : N 0 M 0 C C t Z E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E V. Discussin nd nalysis 5.. Observatin f Optimal size f Bacterial units (Z ) in Stage-I Plants with varying values f N, C, t : Frm table., it is bserved that the ptimal (minimum) size f expected bacterial units (Z ) is a decreasing functin f initial number f bacterial units (N 0 ) in stage-i plants when ther parameters are cnstants. h birth rate ( ) and death rate ( ) f bacteria in stage-i plants are increasing functins f (N 0 ) when ther parameters are cnstants. Hence it may cnclude that even thugh the initial number f bacterial units in stage-i plants is increasing, the expected size f these bacteria remains same as the birth and death rates are ne and the same. It is als bserved that the ptimal (minimum) size f expected bacterial units (Z ) is a increasing functin f upper limit f variance f bacterial units (C ) in stage-i plants when ther parameters are cnstants. ll the rates f bacteria in stage-i plants are nt influenced by upper limit f variance f bacterial units in stage-i (C ). Hence it may cnclude that the increased variance f bacterial units in stage-i plants is nt influencing the expected size f bacterial units. Further, it is bserved that the ptimal (minimum) size f expected bacterial units (Z ) is a decreasing functin f time perid t when ther parameters are cnstants. Further, arrival rate ( ), birth rate ( ) and death rate ( ) f bacteria in stage-i plants are increasing functins f time where as arrival rate ( ) and birth rate ( ) in stage- II plants are increasing and decreasing functins respectively. Hence it may cnclude that even thugh the time 5 Page

6 Optimizatin Prgramming Prblems Fr Cntrl nd Management Of Bacterial Disease With elapsed, the expected size f these bacteria in stage-i plants remain same as the birth and death rates are ne and the same. 5.. Observatin f Optimal size f Bacterial units (Z ) in Stage-I plants with varying values f M, C, t : Frm table., it is bserved that the ptimal (minimum) size f expected bacterial units (Z ) is an increasing functin f initial number f bacterial units (M 0 ) in stage-ii plants when ther parameters are cnstants. h birth rate ( ) and death rate ( ) f bacteria in stage-i plants are nt influenced by (M 0 ). Hence it may cnclude that even thugh the initial number f bacterial units in stage-ii plants is increasing, the expected size f these bacteria remains same as the birth and death rates are cnstant. It is als bserved that the ptimal (minimum) size f expected bacterial units (Z ) is an increasing functin f upper limit f variance number f bacterial units (C ) in stage-ii plants when ther parameters are cnstants. ll the rates f bacteria in stage-ii plants are nt influenced by upper limit f variance f bacterial units in stage-ii plants (C ). Hence it may cnclude that the increased variance number f bacterial units in stage-ii plants is nt influencing the expected size f bacterial units in stage-ii plants. Further, it is bserved that the ptimal (minimum) size f expected bacterial units (Z ) in stage II plants is a decreasing functin f time perid t when ther parameters are cnstants. Further, arrival rate ( ), birth rate ( ) and death rate ( ) f bacteria in stage-i plants are nt influenced by time and same is happening with arrival rate ( ) in stage-ii plants. Hence it may cnclude that even thugh the time elapsed, the expected size f bacteria in stage-ii plants remains same as the birth and death rates are cnstant. 5.. Observatin f Optimal Variance f Bacterial units (Z ) in Stage-I Plants with varying values f N, C, t : Frm table., it is bserved that the ptimal (maximum) size f variance f bacterial units (Z ) is a decreasing functin f initial number f bacterial units (N 0 ) in stage-i plants when ther parameters are cnstants. h arrival rate ( ) and birth rate ( ) f bacteria in stage-ii plants are nt influenced by (N 0 ) when ther parameters are cnstants. Hence it may cnclude that even thugh the initial number f bacterial units in stage-i plants is increasing, the vlatility f these bacteria remains cnstant as the birth and death rates are nullified in stage-i plants. It is als bserved that the ptimal (maximum) size f variance f bacterial units (Z ) in stage-i plants is an increasing functin f upper limit f variance f bacterial units (C ) in stage-i when ther parameters are cnstants. h arrival rate ( ) and birth rate ( ) f bacteria in stage-ii plants are nt influenced by (C ) when ther parameters are cnstants. Hence it may cnclude that even thugh the upper limit f variance f bacterial units (C ) in stage-i is increasing, the vlatility f these bacteria remains cnstant due t nullified arrival, birth and death rates in stage-i plants. Further, it is bserved that the ptimal (maximum) size f variance f bacterial units (Z ) in stage-i plants is an increasing functin f time when ther parameters are cnstants. The arrival rate ( ) is cnstant and birth rate ( ) f bacteria in stage-ii plants is decreasing functin f time when ther parameters are cnstants. Hence it may cnclude that even thugh the time is elapsed, the vlatility f these bacteria remains cnstant due t nullified arrival, birth and death rates in stage-i plants. 5.. Observatin f Optimal Variance f Bacterial units (Z ) in Stage-I plants with varying values f M, C, t : Frm table., it is bserved that the ptimal (maximum) size f variance f bacterial units (Z ) in stage-ii plants is a decreasing functin f initial number f bacterial units (M 0 ) in stage-ii plants when ther parameters are cnstants. ll the rates f bacteria in stage-i plants and stage-ii plants are nt influenced by (M 0 ). Hence it may cnclude that fr the increasing initial number f bacterial units in stage-ii plants, the vlatility f these bacteria is decreasing as the arrival, birth, transitin and death rates f bacteria are cnstant. It is als bserved that the ptimal (maximum) size f variance f bacterial units (Z ) in stage-ii plants is a decreasing functin f upper limit f variance f bacterial units (C ) in stage-ii plants when ther parameters are cnstants. ll the rates f bacteria in stage-i plants and stage-ii plants are nt influenced by (C ). Hence it may cnclude that fr the increasing upper limit f variance f bacterial units in stage-ii plants, the vlatility f these bacteria is decreasing as the arrival, birth, transitin and death rates are cnstants. Further, it is bserved that the ptimal (minimum) size f variance f bacterial units (Z ) in stage-ii plants is a decreasing functin f time when ther parameters are cnstants. The arrival rate, birth rate and transitin rates f bacteria in stage-i plants are decreasing functins f time. Death rate f bacteria in stage-i plants and all the rates in stage-ii plants are nt influenced by time. Hence it may cnclude that fr the increasing time, the vlatility f these bacteria is decreasing as the arrival, birth and transitin rates in stage-i plants are increasing. 6 Page

7 Optimizatin Prgramming Prblems Fr Cntrl nd Management Of Bacterial Disease With References []. Madhavi, K., Thirupathi Ra, P. and P.R.S. Reddy (0). Optimal Drug dministratin fr Cancer Chemtherapy thrugh Stchastic Prgramming. merican Jurnal f pplied Mathematics and Mathematical Sciences, Vl. (), pp []. Narayana Ra, E.S.V. and Tirupathi Ra, P. (06). Bivariate Stchastic Mdeling fr Spread f Bacterial Diseases amng Plants. IOSR-Jurnal f Mathematics, Vl. (), pp. -9. []. Segarra, J., Jeger, M. and Van den Bsch, F. (00). Epidemic Patterns and Dynamics f Plant Disease. Phytpathlgy, Vl. 9, pp []. Tirupathi Ra, P., Srinivasa Ra, K. and Padmalatha, K (00). Stchastic mdel fr ptimal drug administratin in cancer chemtherapy. Internatinal Jurnal f Engineering Science and Technlgy, Vl. (5), pp [5]. Tirupathi Ra, P., Flra Evangel, D. and Madhavi, K. (0). Stchastic Prgramming n Optimal Drug dministratin fr Tw- Stage Cancer Treatment Prblem. Internatinal Jurnal f Green Cmputing, Vl. (), pp. -0. [6]. Tirupathi Ra, P., Naveen Kumar, B.N. and P.R.S. Reddy (0). Stchastic Prgramming n Optimal Drug dministratin fr Three-Stage Cancer Chemtherapy Treatment. merican Jurnal f Operatinal Research, Vl. (), pp [7]. Tirupathi Ra, P., Jayabarathiraj, J. and Vishnu Vardhan, R (0). Stchastic Optimisatin Mdels fr Cancer Chemtherapy. British Jurnal f pplied Science & Technlgy, Vl. (8), pp [8]. Xu, X-M and Ridut, M.S. (000). Stchastic Simulatin f the Spread f Race Specific and Nn-specific erial Fungal Pathgens in Cultivar Mixtures. Plant Pathlgy,Vl.9, pp Page