MODELING THE DYNAMICS AND TRANSMISSION OF CASSAVA MOSAIC DISEASE IN TANZANIA

Cassava mosaic disease (CMD) is caused by cassava mosaic virus (CMV) and is transmitted by the whitefly vector called Bemisia tabaci. In this paper, the deterministic model for transmission dynamics of CMD is formulated by considering the whitefly vector, cassava resistant and susceptible breeds, and infected cassava. The basic reproduction number R0 and sensitivity index for each parameter with respect to basic reproduction number R0 are computed to determine which parameters are sensitive to the dynamics of cassava mosaic disease. Analysis shows that the death rate of whitefly vectors, the infection rate for susceptible vectors, the number of vectors that can be supported and the rate of loss of infected cassava due to disease are the most sensitive parameters to the dynamics of cassava mosaic disease. Numerical simulation indicates that, cassava new infections increase as the number of vectors that can be supported increase and acquire cassava mosaic disease. It shows that if control measures are not considered, then the susceptible breed and cassava resistant breed will be wiped out after five and ten months respectively. To control the disease, farmers are encouraged to apply control strategies such as spraying of insecticide, using of vector-resistant varieties, phytosanitation which involve the removal of infected cassava plants from the farm, crop hygiene and the use of free stem cutting method.


Introduction
Cassava (Manihot esculenta) is one of the crops which was firstly introduced in West Africa from Brazil at the end of 16 th Century by Portuguese and spread to other African countries [3,19].Cassava is grown in tropical and subtropical areas which experience low rainfall as the crop survives in drought climate [10], and this makes cassava a major staple food in the world.
According to FAO, about 700 million people depend on cassava as their main food in Africa [20].Production of cassava in Africa is becoming low due to a number of causes, notably pests and diseases [6].Cassava Brown Streak Disease (CBSD) and Cassava Mosaic Disease (CMD) are the most important biotic constraints which have led to decrease in yields [13,1].Cassava mosaic virus (CMV) contaminates the cassava leaves and is transmitted by the whitefly vector called Bemisia tabaci [3].There are other 500 different plants including weeds and crops which are host to whitefly vector [15,17].Different causes for transmission of cassava mosaic disease have been reported, this includes the use of infected cassava stem, the use of infected plant materials by the farmers [12] as well as the use of CBSD resistant breed which later becomes vulnerable to cassava mosaic disease [22,25].
The infected cassava plant is characterized by leaf mosaic patterns and it can persist during the premature stage of cassava leaf development.The cassava leaves which are infected by the disease are warped, reduced in size and distorted with yellow color separating the ordinary green color which is the health part of the leaves.They then deteriorate and the new leaves bend [7].Tanzania is among the countries that face this problem and the disease has been spreading at a fast rate leading to food shortages [24].According to Tanzania Commissions for Science and Technology (COSTECH) production of cassava in Tanzania is only 8t/ha which is lower compared to 20t/ha that can be produced, the main causes of lower production are pests and diseases.
Studies have been conducted to analyze the transmission dynamics of cassava mosaic disease and the impact of different control strategies.Holt et al: [8] studied the model with susceptible and infected cassava, and susceptible and infectious vectors.The study show that using infected cutting tools and elimination of infectious cassava have a little effect on the occurrence of the disease.Hebert M.P [4], use the Markov chain models to find the probability of eliminating the disease by using the stochastic process models.The model was applied to CMV, the numerical and analytical results show that the vector aggregation is growing in intricacy as a well as the possibility of a disease to be recognized in host plant.Lawrence et al: [14] use the system of differential equation to find the equilibrium value of the whitefly vector and the cassava plants.The result was analyzed using the finite difference method to assess the spatiotemporal spread of the disease.Results obtained were compared to the field data and the implication of controlling the CMV through the practical were explored.The study concluded that using of ACMD resistant strains of cassava and windbreaks will have positive results on cassava yields.This paper studies the dynamics of cassava mosaic disease by considering cassava resistant breed which only catch cassava mosaic disease through unhealthy cutting and susceptible breed which catch mosaic disease through unhealthy cutting and contact with whitefly vectors before implementing controls.

Model Development
The model is formulated by modifying the model which was developed by Holt et al. [8] to include breed which catches cassava mosaic disease through unhealthy cutting and susceptible breed that catches mosaic disease through unhealthy cutting and through contact with whitefly vector.The model consists of two groups of population.The first group includes the cassava population (N C ) which is divided into resistant (S r ) and Susceptible (S C ) breeds, and infected cassava (I C ). Second group includes the whitefly vector population (N V ) which consists of susceptible vector (S V ) and infectious vector (I V ).
Cassava resistant breed is replanted at a rate r 1 and is infected by cassava mosaic disease through unhealthy cutting at a rate β 1 and they are harvested at a rate ρ 1 .The term k 1 , represents the maximum plants for cassava resistant breed which can be planted.Cassava susceptible breed is replanted at a rate r 2 , and is infected by cassava mosaic disease following contact with infected whitefly vector and unhealthy cutting at a rate β 2 while it is harvested at a rate ρ 2 .
The maximum plants of cassava susceptible breed that can be planted is k 2 .Infected cassava flourish following infection of cassava resistant breed through unhealthy cutting at a rate β 1 , and infection of cassava susceptible breed through unhealthy cutting and contact with infected whitefly vector at a rate β 2 and they decrease due to the effect of cassava mosaic disease at a rate a and harvested at a rate ρ 3 .Susceptible vector is recruited by birth at a rate b and catch infection following contact with infected cassava at a rate β 2 .Also, k 3 is the maximum number of vectors that can be supported.Infected vector is recruited when susceptible vector catch infection following contact with infected cassava at a rate β 3 and γ is the death rate of whitefly vector.

Assumptions of the Model
The model assumes that, all whitefly vectors are born susceptible to cassava mosaic disease.
The replanted cassava for both breeds are susceptible to CMD.The whitefly vector cannot transmit cassava mosaic disease to cassava resistant breed except through unhealthy cutting.
Cassava susceptible breed gets cassava mosaic disease through contact with infected whitefly and through unhealthy cutting.Susceptible vectors can be infected when they come into contact with the infected cassava.The interaction between cassava and vector population is shown in Figure 1.Variables and parameters are described in Table 1 and 2  The maximum number of resistant breed that can be planted.
The maximum number of susceptible breed that can be planted Maximum number of vectors that can be supported

Model equations for the two groups
The total population of cassava is given as S r + S C + I C = N C and the total population of vector is given as

Basic Properties of the Model
Invariant Region: Metzer matrix is used to show the feasible region, in which the variables are positive ∀t ≥ 0. To deduce the feasible region; the model system (1a)-(1e) can be written as: (2) where x = (S r , S C , I C , S V , I V ) T and a constant term F = (0, 0, 0, 0, 0) T such that: , for; In equation ( 3), A is a Metzler matrix ∀x ∈ R 5 and due to the fact that F ≥ 0, the model system

Positivity of the solutions:
Let the initial condition be S r (0), S C (0), I C (0), S V (0), I V (0), the solutions S r , S c , I c , S v , I v of the model system (1a) -(1e) are positive ∀t > 0. We show that, the solution of the model system (1a) -(1e) are positive by starting with equation (1a) that: Separate the variables and integrate both sides of the equation, This give the values of S r as: At initial condition time, t = 0, equation ( 7) above becomes Thus, S r (0) ≥ 0, ∀t > 0.
Apply the same procedure to the remaining equations (1b), (1c), (1d) and (1e): We get Here we conclude that, the requirement to study the dynamics of CMD is satisfied considering that, all the solutions of the model (1a) -(1e) are positive and bounded in the region:

Cassava Mosaic Free Equilibrium
The steady state when there is no cassava mosaic disease is called cassava mosaic free equilibrium.We compute cassava mosaic free equilibrium when I c = I v = 0.At this state, the total cassava plants is the sum of susceptible and resistant breeds.However, the population of the vector at this state consists of susceptible whitefly vector.Cassava mosaic free equilibrium is given by: (15) The basic reproduction number is denoted by R 0 .It refers to an expected number of secondary infections from an infected whitefly when introduced into a susceptible population of cassava plants [5].If R 0 > 1, the infectious whitefly can transmit the cassava mosaic disease to more than one cassava plants, and if R 0 < 1, an infectious whitefly transmits the cassava mosaic disease to less than one cassava plants, hence the disease is dying out.The basic reproductive number will be determined by next generation matrix [11] as follows: Assume that, f i (x) is the rate of cassava and whitefly new infections and where V + i (x) are the terms that are transferred into the compartment and V − i (x) are the terms that are transferred out of the compartment such that: [5].
, where i, j = 1, 2, ..., m and x 0 indicates the cassava mosaic free equilibrium.From the model system (1a)-(1e), f i and V i are defined by: ( 17) Matrices F and V are obtained by differentiating equation ( 17) and ( 18) respectively, with respect to I c and I v so that: .
The next generation matrix is given by: (21) The basic reproduction number R 0 for cassava plants and vector is a dominant eigenvalue of the next generation matrix FV −1 [18].The basic reproduction number R 0 is therefore given by: From equation (22), basic reproduction number R 0 is determined by all parameters from the model.The basic reproduction number R 0 increases in proportion to and r 2 , and decreases as γ, ρ 3 , ρ 2 , a and ρ 1 increase.

. Sensitivity analysis
Sensitivity index of a parameter tells how a parameter is sensitive to the disease.In this section, sensitivity index of each parameter with respect to basic reproduction number R 0 is derived to determine how each parameter influences the disease.If f is a parameter in reproduction number R 0 then, sensitivity index of f with respect to R 0 is given by:

Parameters Adoption
Parameter values from the literature and assumed ones are used.Table 3 summarizes the parameter values, range and their sources.Using forward normalized sensitivity index for each parameter with respect to basic reproduction number R 0 , sensitivity index for β 2 is derived as follows: Full computation gives: We apply the same method to obtain sensitivity indices for other parameters.decrease when a, ρ 1 , ρ 2 , ρ 3 and γ increase.The most sensitive parameter is the death rate of whitefly vectors γ, the increase of this parameter decrease the basic reproduction number R 0 .

. Global Stability of Cassava Mosaic Free Equilibrium
The global stability of cassava mosaic free equilibrium is established by approach used by Castillo-Chavez [2].When this approach is used, system (1a) -( 1e) is written as follows: (27) where X 1 presents the noninfectious classes and X 2 infectious classes.X (F 0 ) present mosaic free equilibrium.Mosaic free equilibrium is said to be globally asymptotically stable if eigenvalues of matrix H are negative and matrix G is a Metzler matrix [9].We thus define X 1 , X 2 and X F 0 by: (29) .
Matrices H 1 and H are defined by: (32) , where Matrix H has negative eigenvalues and matrix G is Metlzer matrix since elements in the main diagonal are negative and the off diagonal elements are positive provided the rate of planting cassava is greater than the rate at which they are harvested and the recruitment rate of whitefly vectors is greater than their death rate.Therefore, when the basic reproduction number R 0 , is less than one (R 0 < 1) and greater than one (R 0 > 1), then the disease free equilibrium is globally asymptotically stable and unstable respectively.

. Global Stability of Cassava Mosaic Free Equilibrium
Due to non-linear nature of the model, it is not possible to obtain cassava mosaic equilibrium explicitly.To prove existence of cassava mosaic equilibrium, we state and prove the following theorem: Proof: Approach in Tumwine et al. [23] and Massawe et al: [16] is adopted in proving existence of cassava mosaic equilibrium.We use the sum of cassava plants and whitefly vectors when their rate of change is zero.When we consider total cassava plants at cassava mosaic equilibrium, we have: This lead to : Since S * r < k 1 , S * C < k 2 and all the parameters are positive.Then: showing that: S * r > 0, S * C > 0 and I * C > 0. Using the same approach for whitefly vector we have S * V > 0 and I * V > 0. This shows that cassava mosaic equilibrium exists.

Global Stability of Cassava Mosaic Equilibrium
The global stability of cassava mosaic equilibrium is investigated by logarithmic Lyapunov function which is given by: (37) where G 1 , is a positive constant which is to be chosen carefully, P i is a variable in a compartment i and P * present a compartment variable at equilibrium point.Using system (37) the Lyapunov function is defined by; (38) Differentiate the Lyapunov function (38) above with respect to time, we get (39) From equations (39), we have: At cassava mosaic equilibrium, equation (40) becomes: (41) this simplifies to: (42) where: (43) The function F(Ω) is negative or zero in Ω, therefore dL dt ≤ 0 in Ω and it is zero for Ω = Ω * .Since dL dt = 0 when Ω = Ω * and dL dt ≤ 0 in Ω then the largest invariant set in Ω when dL dt = 0 is a singleton Ω * which is cassava mosaic equilibrium point.By LaSalles invariant principle, the casssava mosaic equilibrium Ω * is globally asymptotically stable when R 0 > 1.

Numerical Simulation of the Basic Model
In this section, we simulate model system (1) to determine the long term impact of cassava mosaic disease.We simulate the dynamics of cassava mosaic disease by considering sensitive parameters.
The dynamics of cassava mosaic disease is demonstrated in Figure 2. All susceptible vectors contract the disease before five months, this is reflected by susceptible cassava which also decrease due to the disease.Cassava resistant breed takes longer to get cassava mosaic disease as demonstrated in Figure 2. Figure 3 illustrates cassava and vector populations.From Figure 6 the graphs demonstrate the variation of vector carrying capacity k 3 to the susceptible class of cassava and susceptible class of vector.The graphs show as the carrying capacity of whitefly vectors increase the number of susceptible cassava breed decrease, the number of susceptible vector increase.

Conclusion
In this paper, the deterministic model for transmission dynamics of CMD which includes population of cassava and whitefly vector is presented and analyzed.The sensitivity analysis was performed to identify sensitive parameters.Analysis shows that the number of vectors that can be supported, the rates at which vectors acquire disease and the carrying capacity of susceptible cassava breed, play important role in the transmission dynamics of cassava mosaic disease.New infections will increase as the carrying capacity of susceptible cassava and the rate of infection of vectors increases.To improve cassava productivity, campaigns to eradicate cassava mosaic disease should focus on strategies which reduce vectors' population.These strategies include spraying insecticide, use of vector-resistant varieties, phytosanitation which involve the removal of infected cassava plants from the place that will be used for the new plantings, crop hygiene and the use of free stem cutting method.

FIGURE 1 .
FIGURE 1. Compartmental Model for the transmission dynamics of Cassava Mosaic Disease

Figure 4
Figure 4  demonstrates the variation of the rate of loss of infected cassava to the infected classes.It shows the behavior of infected cassava and infected vectors when the parameter a vary, the increase of a lead to the decrease of infected cassava and the decrease of infected 033, R0 = 42.9395 a = 0.3, R0 = 14.8009 a = 0.6, R0 = 10.4918(B) Infected vector.

FIGURE 4 .
FIGURE 4. Variation of loss of infected cassava rate in infected class.

Figure 5 ,
Figure5, shows the variation of vector mortality rate γ to the infectious vector and infected cassava class, if the rate of vector mortality increase the number of infected vector and infected cassava decreases.

FIGURE 5 .
FIGURE 5. Variation of vector mortality rate in infectious class.

FIGURE 6 .
FIGURE 6. Variation of vector carrying capacity to the susceptible class.

Figure 7
Figure7shows the impact of varying the carrying capacity of susceptible breed of cassava to the infected cassava class and infected vector .It shows that as the carrying capacity of susceptible breed increases the number of infected cassava and infected vector increases.

FIGURE 7 .
FIGURE 7. Variation of cassava susceptible breed carrying capacity to the infected class.

TABLE 1 .
respectively.Variables' Descriptions 2The rate at which cassava susceptible breed is infectedρ 3The rate at which infected cassava is harvested aThe rate of loss of infected cassava due to disease b Recruitment rate for whitefly.β 3 Vector infection rate γ The death rate of whitefly vectors k l

TABLE 4 .
Table 4 summarizes sensitivity indices for all parameters with respect to basic reproduction number R 0 .Sensitivity Indices.
From the Table4parameters β 2 , β 3 , β 1 , k 1 , k 2 , k 3 , b,r 1 , r 2 have positive indices, this means that the basic reproduction number R 0 increase in proportion to these parameters.Parameters a, ρ 1 , ρ 2 , ρ 3 and γ have negative indices.This means that the basic reproduction number R 0