Mathematical Model on Branch Canker Disease in Sylhet, Bangladesh

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 13, Issue 1 Ver. IV (Jan. - Feb. 2017), PP Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh Rahman, S.M.S. 1 Islam, M.A. 2 Shahrear, P. 3* Islam, M.S M.S. (Thesis), Department of Mathematics, Shahjalal University of Science an Technology, Sylhet-3114, Banglaesh 2. Professor, Department of Mathematics, Shahjalal University of Science an Technology, Sylhet-3114, Banglaesh 3. Associate Prof., Department of Mathematics, Shahjalal University of Science an Technology, Sylhet, Banglaesh 4. Senior Scientific Officer, Plant Pathology Division, Banglaesh Tea Research Institute, Srimangal, Banglaesh Abstract: Members of the Camellia genus of plants may be affecte by many of the iseases, pathogens an pests that mainly affect the tea plant (Camellia sinensis). One of the most common iseases which are foun in tea garen is known as Branch Canker (BC) [1]. Recent outbreaks of these iseases is not only hampering the prouction of tea, but also stupenously hampering our national economy. In this paper, a simple SEIR moel has been use to analyze the ynamics of Branch Canker in the tea garen. The stability of the system is analyze for the existence of the isease free equilibrium. We establishe that there exists a isease free equilibrium point that is locally asymptotically stable when the reprouction number R o < 1 an unstable when R o > 1. Theoretically, we analyze the BC moel. Finally, we numerically teste the theoretical results in MATLAB. Keywors: Mathematical Moel, Tea isease, Stability Analysis, Reprouction Number, Epiemiology, Equilibrium, ODE. I. Introuction Like other living being, plants are boun to be victim of iseases, which may also come as an epiemic form creating a heavy toll of lives of plants. We are accustome to having ifferent kins of rinks in our aily life. Out of all such kins of all rinks, tea is one of the most popular rinks of the worl, which are prouce more than 50 countries aroun the worl. Tea is basically prouce from the tea plant. But sometimes these plants are affecte by lots of iseases. Generally these iseases are cause by pathogens, a group of micro organs, which weaken the plants, kill the metabolism an block the transportation of foo of host plants. The iseases graually make the plants weak an estroy immune system leaing plants to eath ultimately. Actually, the flush shoots of the tea plant are suitable to use as rink. 81.4% of total tea is prouce in Asia followe by Africa an Latin America having the prouction of 15.5% an 2% respectively [2]. In Banglaesh, there are 169 tea estates [3] comprising aroun thousan hectares prouce million kg [3] of mae tea which is 0.11% of entire GDP creating employment for almost 0.30 million people. Comparing to the other tea growing countries, the tea prouction per hectare in Banglaesh is relatively much low (1,239 kg/ha) [3] if the cultivation is consiere. Among other factors, isease is the key one. Tea plants are force to grow uner monoculture conitions where iseases may attack in a sporaic or epiemic form. While it is in epiemic form, it irectly hits our national economy negatively. Although majority of the iseases of tea plants are of fungal origin, more than 400 pathogens [4] are responsible for other isease like 18 algal isease an few epiphytes [5] which attack the foliage, stem an root of tea plants. In Banglaesh, common tea iseases an their positions are state in following table: Name of Diseases Causal organisms Infecte parts Status Branch canker Macrophoma theicola Stem & Branch Major Re rust Cephaleuros parasiticus Stem & Branch Major Charcoal stump rot Ustulina zonata Stump & Root Major Ustulina eusta Stump & Root Major Violet root rot Sphaerostilbe repens Root Minor The total crop loss in tea sector in Banglaesh causes ue to some iseases, specifically, Re rust isease causes 18-20%, Gall 20-25%, Horsehair Blight 11-13% [6], Branch canker an Black rot cause 25-35% loss. Re rust is a leaing DOI: / Page

2 Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh steam iseases in tea garen, which is basically cause by alga, Cephaleuros parasiticus karast [7][8] an Black rot is the most vicious leaf isease of tea cause by Corticium theae Bernar [9][10]. Root iseases are the common features of tea garen. Charcol stump rot (Ustulina estua) an Violet root rot (Spherostible repens) are two of the most common root iseases in Banglaesh. Charcol stump rot not only attacks tea but also many other plants like Albizzia ooratissima, A. Procera, Derris robusta, Tephrosia cania etc [11]. Violet root attack many plants of all ages starting from two years an is unreporte on the hills. Figure: 1 Branch Canker Due to Macrophoma theicola petch fungas, Branch Canker (Figure 1), a stem isease of tea plants is commonly seen which tolls a heavy loss in tea prouction in Banglaesh [1][5][12]. Aroun 15-25% crop loss is recore ue to the attack of this isease. Microclimate, the climate of a very small or restricte area that iffers from the surrouning area, is significantly affecte by the architectures of tree plantation since there are Tillah, Flat an Kunchi may be foun in the same area. Such topographical varieties irectly influence on access of solar rays, moisture, heat an air flow, thus affect plant growth an iseases evelopment [13]. In fine, in this paper, we have trie to evelop the Branch Canker moel (can be name as BC moel) to erive the isease free equilibrium an the enemic an the enemic equilibrium.we have one a qualitative analysis of the moel. Stability conitions for the isease-free equilibrium an the enemic equilibrium are erive as well as numerically teste the theoretical results in MATLAB. II. Formulation of the Branch Canker (BC) Moel We nee to formulate five classes of plants such as, susceptible class, asymptomatic infecte class, symptomatic infecte class, quarantine class an recovere class by using SEIR moel [14-18] with a view to stuying the compartmental mathematical Branch Canker moel. Each group or compartment is shown in Figure 2. S(t) represents the number of iniviuals which is not yet infecte with the isease at time t or those which is susceptible to the isease. E(t) represents the number of asymptomatic infectious iniviuals (plants infecte with isease not showing the symptoms) at time t. I(t) is the number of symptomatic infectious iniviual (plants infecte with isease showing the symptoms) at time t. Q(t) is the number of quarantine iniviuals (plants which have been infecte with isease but o not to sprea the isease among the other susceptible) at time t. R(t) is the number of recovere iniviuals (plants infecte though recovere are also unable to be infecte further or to be contagious) at time t. Therefore, the compartment SEIR moel for BC is shown in figure 2. Symptomatic as well as asymptomatic infectious iniviuals are able to sprea the isease to other plants, which are susceptible. The ynamics of the moel for each compartment is then governe by the following set of ifferential equations [19]: DOI: / Page

3 Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh S t = - ( b + s )SI + R - S Figure 2: Branch canker isease (BC) moel E SI ( ) E E t I SSI ( ) I I t (1) Q t = r ( E + I ) - µq - Q R Q E I R R t In the following tables, the variables an parameters use in the moel are efine: Table 1: Moel Variables Variables S(t) E(t) I(t) Q(t) R(t) Description Susceptible class Asymptomatic infectious class Symptomatic infectious class Quarantine class Recovere class Table 2: Moel Parameters Parameters l r m Description Birth an Death rate Transmission co-efficient (asymptomatic) Transmission co-efficient (symptomatic) Recovery or removal rate contact rate from asymptomatic infecte class an symptomatic infecte to quarantine class Contact rate from Quarantine class to Recovere iniviuals class Contact rate from Recovere class to Susceptible class III. Equilibrium Analysis Equilibrium conition states us that the left han sie (L.H.S) of the system equations of B.C. will be zeros, i.e., S t = 0 E I ; = 0; t t = 0; Q t = 0 ; R t = 0 DOI: / Page

4 Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh For the B.C. moel given by the systems of Equation (1) the following properties hol when plants with constant size are assume. The equilibrium point without infection calle isease free equilibrium is given by: S=1, E=0, I=0, Q=0, R=0 Thus we can take the isease free equilibrium as (1,0,0,0,0). The equilibrium point with infection calle enemic equilibrium is given by: R * S S ( s) I SI * E E SI s * I I ( E I) * Q Q Q ( E I) R R * (2) IV. Moel Analysis We nee to examine the behavior of the moel plant near the equilibrium solutions (the isease free an the enemic) to analyze the stability of each of the two equilibriums of the BC. We will investigate the local stability of the steay state of the isease moel by the eigenvalues of the associate Jacobian matrices of the system equations [20,21]. Therefore, for the system of (1); the Jacobian matrix is: s I 0 s S 0 I S 0 0 J= si 0 ss (3) At the isease free equilibrium i.e. at I=0 the Jacobian matrix woul be as follows (consiering the total fraction of Susceptible S=1), the Jacobian matrix becomes é J 0 = ë ( ) 0 ( ) 0 0 ( ) r + l r + l + 0 r r -m 0 0 l l m - ù û Now from equation (3) we have characteristic polynomial, ( a + ) ( a + ) ( a + m) ( a + r + l + ) ( a - + r + l + ) = 0 DOI: / Page

5 Hence the eigenvalues are as follows: a 1 = -, a 2 = -, a 3 = -m, a 4 = - r + l + ( ), a 5 = - r + l + ( ) Theorem 1: The isease free equilibrium (DFE) is stable if R 0 < 1. Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh The isease free equilibrium for the moel only exists if all the eigenvalues are non-positive. This hols of, - r + l + ( ) < 0 ( ) Þ < r + l + Þ r + l + < 1 \ R 0 < 1 i.e. the basic reprouctive number, R 0 = r + l + At enemic of equilibrium, the Jacobian matrix becomes: s s * * I S * * I 0 S * * * J si 0 ss é = ë - SI ( + ) - 0 -( b r + l + a + ) SI r + l + SI r + l + -( r + l + ) 0 R ( + )I + R ( + )I + R ( + )I r r -m 0 0 l l m - The eigenvalues of J * coul all be negative, positive, zero or any combinations of the three alternatives on the basis of various combinations of moel parameters. Thus the enemic equilibrium can be stable, unstable or sale with respect to the ifferent values of the various moel parameters poole at a given time [22-24]. ù û V. Numerical Simulation an results for BC Moel We are going to iscuss the numerical simulation an finings from BC moel in this section. We have solve the system for the moel by using MATLAB. The baseline variables an use parameters estimate for Banglaesh are shown in the following tables. Although ata we use are actually base on approximations. We foun that our approximations are actually correct enough to present the moel appropriately. When there is no treatment, we consier the treatment parameters as zero i.e. 0 an 0. Table 3: Values of Branch Canker Disease moel parameters DOI: / Page

6 Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh Parameters Values m l r When there is no treatment, i.e. 0 an 0, we have plotte the following graphs in MATLAB: (a) Situation of the class S(t) (b) Situation of the class E(t) (c) Situation of the class I(t) () Situation of the class Q(t) DOI: / Page

7 Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh (e) Situation of the class R(t) Figure 3: Situation of ifferent plant classes with respect to time when there is no treatment VI. Result an iscussion from the figure 3 Figure 3 epicts us that when there are no treatments i.e. λ=0, µ=0, the curve for the recovery class R(t) becomes continuously ecreasing. It means the plant in the recovery class ecreases with respect to time. So for the recovery from the isease, the treatment is a must. When there are treatments, we consier the treatment parameters as an Table 4: Values of Branch Canker Disease moel parameters Parameters Values m l r When there are treatments i.e. when an , we have plotte the following graphs in MATLAB: (a) Situation of the class S(t) (b) Situation of the class E(t) (c) Situation of the class I(t) () Situation of the class Q(t) DOI: / Page

8 Mathematical Moel on Branch Canker Disease in Sylhet, Banglaesh (e) Situation of the class R(t) Figure 4: Situation of ifferent plant classes with respect to time when there exist treatments VII. Result an iscussion from the figure 3 From the figure 4, we observe that when there exist treatments, the curve for the recovery class R(t) start to increase. An it is going on till a certain perio of time. Then when the treatments are one, it is starts to ecrease again. Also the reuction of the plant in the E(t), I(t) an Q(t) classes becomes quicker when there exist treatments i.e. λ= an µ= So, we can conclue asserting that treatments can recover the isease easily. VIII. Conclusion In this paper we have presente the Branch Canker moel. This analysis of such moel is base on SEIR moel. To the best of our knowlege, for the first time we use SEIR moel to investigate the ynamics of tea isease in Sylhet, Banglaesh. We have introuce a compartment that efforts on recovering isease from tea garen. The resulting moel has been solve analytically. The equilibria an their stabilities have been also presente. Numerical simulations of the system have been presente to show the variations of the plant in ifferent situations. Data that have been use for the simulations are base on the tea iseases in Sylhet, Banglaesh. We foun that our approximations are actually correct enough to represent the BC moel appropriately. References [1]. Ali, M. an Huq, M., 1993, Branch Canker (Macrophoma) In Tea. pp [2]. Foo an Agriculture Organization, [3]. PDU (Project Development Unit), Statistics on Banglaesh Tea Inustry Banglaesh Tea Boar, Srimangal, Moulvibazar. pp [4]. Chen, Z.M. an Chen. X.F., The iagnosis of tea iseases an their control in Chinese. Shanghai Sci. Tech. publishers, Shanghai, China. pp [5]. Sana, D. L., 1989, TEA SCIENCE. pp: [6]. Ali, M. an Huq, M., 2009, Horse Hair Blight of tea an its management. pp [7]. Huq, M., Ali M. an Islam, M.S., Re rust isease of tea an its management. Memo. No.1. BTRI. pp [8]. Sarmah, K.C., Re rust- Cephaleuros parasitiaus Karst. Disease of tea an associate crops in North-East Inia. Memo. No. 26. In. Tea Asso. pp [9]. Ali, M.A., Black rot isease of tea. Pamphlet no. 14. Banglaesh Tea Res. Inst. Srimangal. pp [10]. Ali, M., 1993, Black Rot isease of Tea. pp [11]. Ali, M., 1993, Charcoal Stump Rot isease of tea, pp [12]. Banglaesh Tea Research Institute (BTRI), Srimangal-3210, Moulvibazar. [13]. Islam, M.S. an Ali., M., Incience of Major Tea Diseases in Banglaesh. Banglaesh J. Agril. Res. 35 (4): [14]. Daley, D.J. an Gani, J., Epiemic Moeling: An Introuction. NY: Cambrige University Press. [15]. Hethcote, H. W., The mathematics of infectious iseases. Society for inustrial an Applie Mathematics, 42, pp [16]. Weiss, H. an Tech, G. A Mathematical Introuction to population Dynamics uner construction, pp [17]. BjØrnsta, O., May 16, SEIR moels, pp [18]. Brauer, F. an Castillo-Chảvez, C Mathematical Moels in Population Biology an Epiemiology, Springer-Verlag NY, Inc, pp [19]. Shepley L. Ross, 2005, Differential Equations, pp [20]. Anton, H. an Rorres C., 2006, Elementary Linear Algebra, pp [21]. Lipschutz S., an Lipson M.L., Linear Algebra, pp [22]. Roussel, M.R., September 13, Stability Analysis for ODEs, pp [23]. Lungu, E. M., Kgosimore, M. an Nyabaza, F., February 2007, LECTURE NOTES: MATHEMATICAL EPIDEMIOLOGY, pp [24]. Roy N., 2012, Mathematical Moeling of Han-Foot-Mouth isease: Quarantine as a Control Measure, pp. 39. DOI: / Page