International Journal of Mathematics and Mathematical Sciences

Volume 2017, Article ID 4865015, 13 pages

https://doi.org/10.1155/2017/4865015

## Banana* Xanthomonas* Wilt Infection: The Role of Debudding and Roguing as Control Options within a Mixed Cultivar Plantation

^{1}Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda^{2}School of Agriculture, Policy and Development, University of Reading, Agriculture Building, Whiteknights, Reading RG6 6AR, UK^{3}Bioversity International, Plot 106, Katalima Road, P.O. Box 24384, Naguru, Kampala, Uganda

Correspondence should be addressed to Juliet Nakakawa; gu.ca.kam.snc@awakakan

Received 14 September 2017; Accepted 16 November 2017; Published 13 December 2017

Academic Editor: Irena Lasiecka

Copyright © 2017 Juliet Nakakawa 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

An optimal control framework is designed in which the use of clean planting materials, debudding, disinfection of tools, and roguing are considered as control measures of Banana* Xanthomonas* Wilt (BXW) within a plantation of multiple cultivars. A model for a special case of two cultivars (AAA- and ABB-genome cultivars) was analyzed. By Pontryagin’s Maximum Principle, we characterized and discussed possible control strategies that substantially reduce the infection levels of BXW within a plantation of ABB- and AAA-genome cultivars. A combination of both prevention and containment controls yielded the greatest decline in the infection levels in both cultivars. Additionally, for effective BXW management, it is important to assess the endemic level of the plantation before application of controls, and once implemented, this should be maintained even when the disease is undetectable to eliminate possible resurgence.

#### 1. Introduction

Banana is a major crop grown in the East and Central African region. It provides food security and income to over 20 million people in the region [1]. Since 2001, Banana* Xanthomonas* Wilt (BXW) has been reported as a major threat to the banana production and livelihoods of millions of people have been affected [2]. The pathogen caused by* Xanthomonas campestris *pv.* musacearum* (Xcm) was first reported in Ethiopia on* Ensete*, a related crop to banana in 1964 [3]. The disease is transmitted by insect vectors, birds, bats, contaminated farming tools, and infected suckers used for setting up new plantings. It attacks all the commonly grown cultivars although some are more susceptible than others. Disease symptoms include yellowing and wilting of leaves, male bud wilting, premature ripening and rotting of fruit, yellow ooze observed on the cross section cut of the pseudostem, and eventually death of the entire plant.

Since the reporting of BXW disease, scientists have disseminated information on identification based on symptoms, mode of spread, and how to implement cultural control practices that have been used to manage similar diseases such as Moko and Bugtok [4, 5]. Cultural control practices for management of BXW have been categorized as preventative or containment. Among the preventative practices, we have debudding which is the removal of the male bud by twisting with a forked stick as soon as the last cluster is formed [6]. This approach avoids contact between the tool and the potentially infected tissues. This is done within 3 weeks since flowering to prevent insect vector from transmitting the disease.

Use of clean planting materials is another preventative approach being advocated for although there are no proper screening facilities for the seed system. Farmers basically obtain suckers for new plantings either from their plantations or from neighboring plantations. Such suckers could be latently infected but since the available detecting tool (lateral flow device) is not yet accessible to the farmers, there is a risk of transferring latently infected plants to new plantations. Nevertheless, to reduce this risk, farmers are constantly cautioned on the use of these suckers and the need to monitor them as they develop. The third preventative approach is the disinfection of tools between plants using sodium hypochlorite solution (usually known by the trade name of the commonest brand in Uganda: Jik) or flames of fire [5, 7].

Two approaches have been implemented as containment practices. These are single disease stem removal (SDSR) dealing with removal of the infected stem from the mat at stool level and roguing involving removal of the entire mat from which the infected stem arose. The latter is more effective as it eliminates the risk of vertical transmission from mother stem to attached suckers. However, it is expensive to implement as it requires more labor [7]. Kubiriba et al. [8] evaluated the SDSR approach and noted that it was effective especially if the manifesting symptoms are via the inflorescence part of the plant.

Jogo et al. [9] studied the extent to which cultural control practices have been adopted by farmers and assessed the socioeconomic factors that influence adoption of these practices among smallholder farmers in Uganda. Their results indicate that among the key determinants were labor and perceived effectiveness of the technologies being implemented. This indicates knowledge gaps in identifying the combination of control options which is cheap enough to implement but causes a large decline in number of infected stems.

The best combination of cultural control practices also depends on the main production systems within the region. The two main groups of cultivars include the East African highland banana (EAHB) AAA-genome grown in western Uganda (1700 masl) and the “kayinja” (beer banana) ABB-genome cultivar grown in the central Uganda (1300 masl) [9]. The intensity of management among these cultivars is different with the former being managed intensively than the later [9]. Also ABB-genome cultivars are more susceptible to insect infection due to the greater production of sweet nectar that attracts the insects as compared to AAA-genome cultivars. These features influence BXW management and control adoption by farmers in these respective regions.

Even though application of all control approaches has been advocated, few farmers are in position to implement them. Therefore, it is necessary to examine the approaches and identify the most cost-effective combination for management of BXW disease.

The use of optimal control approaches/models allows the determination of the most cost-effective intervention. Optimal control modeling has been extensively used to understand biomedical problems and suggest optimal control combinations that will minimize the costs [10–14]. Our model focuses on determining the optimal control combination within smallholder plantations with mixed cultivars given that in banana cropping system different cultivars are managed differently and the risk of infection is also different.

Thus, we propose a Healthy-Infected model for the spread of BXW via inflorescence infection, vertical transmission, and tool-mediated spread. We formulate an optimal control model with debudding, use of clean planting material (reducing the proportion of infected suckers), roguing/SSDR, and disinfection of tools used for cultivating between and within cultivars as our control options. The optimal control model is analyzed using the Pontryagin’s Maximum Principle [15–17] and numerical simulations assessing different combinations of control options. The paper is organized as follows: in Section 2, the formulation of the optimal control model for a plantation with multiple cultivars () is presented. Mathematical analysis and numerical simulations of the optimal control model considering a special case of two cultivars are given in Section 3 and lastly conclusions drawn are given in Section 4.

#### 2. Model Formulation

Consider a population of banana stems with different cultivars each subdivided into healthy and infected classes for . We assume negligible latent and incubation period for simplicity of the model, also because omission of this compartment does not entirely affect the dynamics of the disease. It is assumed that in a unified plantation several blocks of each cultivar are maintained. The effort going into maintenance of each block will depend on preference of farmers in a specific region. The healthy compartment contains all banana stems that are disease-free but at a risk of being infected either by contaminated tools or by insect vectors during inflorescence formation. The populations per cultivar are maintained at equilibrium by a logistic function at a constant rate and a constant carrying capacity . The equilibrium is a result of various activities within the plantation including replanting, desuckering, sucker emergence, and harvesting. The infected class is increased by the following scenarios.(1)Let denote the proportion of healthy suckers from infected mats; then a proportion indicates the suckers recruited into the infected compartment arising from vertical transmission.(2)Disease transmission by contaminated tools is modeled using mass action incidence term with as the coefficient of effective contact between healthy and infected stems by tools. Assuming random use of tools, spread is independent of plantation density but rather dependent on the proportion of infected stems within the plantation. Therefore, the incidence term due to contaminated tools is given as for the interaction between cultivars and .(3)Transmission by vectors via the inflorescence infection is considered to be frequency dependent, that is, with as the coefficient of effective contact by vectors between cultivars and . This incidence term is reasonable because spread by vectors depends on the number of flowering stems and vector activity resulting in infection at a particular time.

Once infected, infected stems showing symptoms are removed at a constant rate for cultivar . This rate also depends on the available resources and farmers’ desire to control that particular cultivar. In some regions within the central Uganda, farmers tend to concentrate on the cooking banana ignoring other cultivars like kayinja which tends to increase the risk of vector spread.

We now incorporate the main cultural control practices that are being implemented in the management of BXW disease into the model formulation.(I) as the control is associated with ensuring that only clean planting materials are used for replanting after mat removal or infilling of space. This involves the ability to detect and remove infected suckers from mats such that they are not used for replanting. Since there is no proper screening for the seed system, farmers rely on suckers from their plantations or neighboring plantations for replanting. It is necessary to determine whether a sucker/planting material is clean. The lateral flow device developed could be used to detect even latently infected material but currently it is not available for farmer and even then it would be expensive. shows the failure to detect and remove infected suckers so that they are reluctantly used for replanting.(II) is the control associated with disinfection of tools used during cultivation, pruning, or harvesting between cultivars and . shows the effort placed in controlling within cultivar spread. This involves use of sodium hypochlorite solution or heating briefly in a fire to destroy bacteria before cutting into another stem. Besides the expense of sodium hypochlorite solution, use of fire flames may not be feasible as it requires moving to the fire and is likely to damage the cutting edge by softening or embrittling the blade depending on how often it is used. indicates failure to disinfect tools when cutting into cultivars and .(III) is the control associated with debudding (removal of male bud by forked stick). This requires continuous surveillance and monitoring such that the male buds are removed immediately after the last cluster is formed to eliminate vector spread. represent failure to implement this control which results in vector spread.(IV)Roguing or single stem removal of infected stem. Let indicate control associated with removal of infected stems with as the roguing rate of cultivar . This involves detecting BXW infected stems and removing them effectively without further spread by tools or mat removal.

Combining all the cases above, the following system can be used to describe the dynamics of BXW when control measures are being implemented.with nonnegative initial conditions given as , , and .

The term represents the proportion of healthy suckers that are recruited into healthy population arising from infected suckers given application of control. With , only healthy plants are maintained in the plantation. Thus, indicated full effort placed in the implementing of controls while indicated failure to implement the controls. The objective function to be minimized is given aswith being the maximum time for which the control practices are implemented.

The constants are the weight constants on the infected plants of cultivar while , , , and are the corresponding costs for controls , , , and , respectively, in terms of labor or cash. The costs associated with use of clean planting materials () and debudding are directly proportional to the rates at which these controls are implemented. Thus, it is reasonable to consider the objective function with linear controls , strategy. Later these will be adjusted to include quadratic control terms to make the problem mathematically more tractable. The quadratic cost functions are used as the simplest form of describing nonlinear costs involved in implementation of the controls disinfection of tools and roguing [10].

The target is to determine an optimal control solution for with their corresponding state solution which minimizes the objective function subject to the system on the defined control space: Thus,

Applying all the control options would be the most rapid method in stopping the spread of BXW but this is often too costly [9]. Therefore, here we determine combinations of control measures that would effectively lead to disease eradication at minimum total cost of implementation.

#### 3. Optimal Control Model: Special Case of Two Cultivars

To analyze the optimal control model, we consider a special case whereby, on a particular plantation, only two cultivars, that is, AAA-genome cultivars and ABB-genome cultivar , are considered. This is justified by the common practice in the affected areas of growing these two types, one for food and trade and one for beer.

First, we formulate our optimal control problem considering linear cost function for use of clean planting material and debudding controls with an intention of determining , , , , for for that minimizesubject to

From system (6), it is noted that, in the absence of disease, no control practice is implemented (). The population growth per cultivar is determined by Moreover, by solving for , we obtain as .

Thus, given , , for , the state solutions , are positively invariant for all . That is, the set is positively invariant. Furthermore, the following holds:

##### 3.1. Existence of the Optimal Control Solution

Consider linear controls and and quadratic controls and such that the costs associated with controls , , , and are , ,, and , respectively. We show that the optimal control solution exists and then we characterize the system using Pontryagin’s Maximum Principle [17, 20] to obtain the optimal control solution.

Theorem 1. *There exists an optimal control solution for objective function (5) subject to model system (6).*

*Proof. *Let denote the state variables and denote the control variables. Let the integrand for the objective function be defined as To prove existence of the optimal control solution, we apply Theorem 4.1 in [21] by checking the following assumptions. (A1)The set of control and corresponding variables is nonempty.(A2)The control set is closed and convex.(A3)The right-hand side of the state system is bounded above by a sum of bounded control and the state and can be written as a linear function with coefficients dependent on time and state.(A4)The integrand is a convex function on .(A5)There exists constant and such that the integrand satisfies Since the equations of system (6) have bounded coefficients and the solution exists in finite time intervals, Theorem 9.2.1 in [22] guarantees the existence of the solution to the model system which gives assumption (A1). By definition of the set , (A2) is also satisfied. From system (6), the right-hand side of the equations is continuous and linear in the controls indicating that it can be written as , where and . By the boundedness of the solution, for . Thus (A3) is also satisfied.

For (A4), the integrand is convex; that is, it can be easily shown given . Let and be two vectors in the integrand; then Lastly there exist constants and such that Thus, assumption (A5) is satisfied.

###### 3.1.1. Characterization of the Optimal Control Solution

Pontryagin’s Maximum Principle is applied to obtain the necessary conditions that an optimal control pair must satisfy. The principle converts the optimal control problem given by objective function (5) subject to system (6) into a problem of minimizing the Hamiltonian with respect to the control set . is the nonempty adjoint vector function corresponding to the state variables . The necessary conditions to be satisfied arewith the transversality condition .

We define the Hamiltonian equation The differential equations governing the adjoint variables are obtained by differentiating the Hamiltonian with respect to the state variables. The adjoint system is obtained by determining written aswhere with the transversality condition . For the optimal controls, we solve for by differentiating with respect to the respective controls, and considering the bounds, we obtain the optimal control solution as follows.

For the linear controls , , bounded such that , the optimal control solution is given as

Let and referred to as switching functions for the controls and , respectively. In the region where and are not equal to zero, we say that the controls and are bang-bang controls. In this case, applications of these controls are switched between the lower and the upper bound through the period of implementation. To address the case where and are zeros for some time interval , we note that all the derivatives with respect to the corresponding controls vanish in this time interval. The functions and are differentiated with respect to time and substitution is consequently done until the controls and , respectively, reappear. The optimal controls are then referred to as singular arcs. The singular arcs and are then determined by equating the time derivatives to zero and solving. If the th derivative of the switching functions , is nonzero and the Generalized Legendre Clebsch Condition given by is satisfied, then the singular controls have order [23]. Thus, for the system (Section 3), we have the optimal control solution given as For the quadratic controls, given the bound , where , the optimal controls are given as

These can be summarized aswhere , , , are solutions to the adjoint system (18) with the transversality condition. The optimal controls for use of clean planting materials and debudding tend to suggest existence of singular arcs. In this case, the controls are implemented whenever the resources are available which better suits application than the bang-bang case where they are just switched between the lower and the upper limit. According to Ledzewicz and Schätler [23], singular controls (if they exist) tend to be either the best (minimizing) or the worst (maximizing) strategies and in either case they are essential in determining the structure of optimal controls. However due to the complexity of the optimal control model, this can not be verified. Thus we consider quadratic terms for all the controls in the objective function for the following reasons: (1) Minimizing linear control terms in the objective function is similar to minimizing quadratic terms. (2) With varying seasons, the cost associated with implementation of these control may vary unproportionally to the rate of implementation.

###### 3.1.2. Quadratic Control Terms for Debudding and Use of Clean Planting Materials

Considering quadratic terms for the objective function to be minimized and by applying Pontryagin’s Maximum Principle, we have the following Hamiltonian function: The existence of controls in this case can be proved using similar techniques as in Theorem 1. The adjoint variables are the same as system (18). Thus the optimal control solution can then be given asThe optimal control characterizations are inversely proportional to the associated weighted costs of implementation indicating that the effectiveness of the control strategy greatly depends on the necessary costs.

Next we solve numerically the optimality system consisting of state system (6) with corresponding initial conditions, adjoint system (18) with the transversality condition, and optimal control characterization assuming the costs terms associated with the controls are quadratic (26).

##### 3.2. Numerical Simulation

In this section, numerical results for the optimal control model given in Section 3 are studied. The backward-forward sweep method with the fourth-order Runge-Kutta algorithm is applied to solve the optimality system. The algorithm has been implemented by various authors to obtain optimal control solutions of related systems [11, 14, 17]. For easy reference we summarize it below.(i)Make an initial guess of the control; is always sufficient.(ii)Using the initial conditions , , , , and the values of solve for forward in time according to the differential equations in the optimality system given.(iii)Using the transversality condition and the values of and solve the adjoint system backward in time according to the corresponding differential equations in the optimality system.(iv)Update the control set by entering the new values of and into the characterization of the optimal control.(v)Verify for convergence the following: if the values obtained are sufficiently close to the corresponding ones in the previous iterations, then the output of the current values is the optimal control solution; otherwise return to step (ii).

This algorithm was implemented in Matlab by modifying the optimal control code presented by [17].

The numerical values used for the numerical simulation as reflected in Table 1 are obtained from literature and some are selected from given ranges based on the banana growing system in Uganda, for which the feasible solution is guaranteed.