CO-DYNAMICS OF CASSAVA VIRUS DISEASE WITH FARMING AWARENESS AND CONTROL MEASURES: A CASE OF LAKE ZONE IN TANZANIA

: The cassava virus diseases co-dynamics model is formulated and analysed using ordinary differential equations theories. The modelling involves cassava plant and whitefly vector population, with aspects of Farmer's awareness level. The evaluation of farmers' awareness inverted numerically and found that increasing awareness levels among farmers decrease disease transmission and spread. The data from ten district councils in the Tanzania lake zone fit and estimate the model parameters. In addition, time-dependent controls were incorporated to reduce the burden on cassava farmers. The findings revealed that with limited resources, uprooting and burning the infected cassava plants and awareness campaign programs significantly reduce the transmission. Therefore, to overcome the burden caused by cassava virus diseases, the farmers are recommended to uproot and burn the infected cassava plants from the farm. Also, it should be updated on controlling disease through educational campaign programs to enhance farmers' awareness.

agricultural details [22]- [24]. Accessible pesticide information campaigns help farmers know the severe risks pesticides have on human health and the environment and minimise adverse effects [25]. Adopting awareness programs to educate farmers results in better comprehensive development for the cultivars and farmers. Farmers primarily receive information on the use and risks of pesticides orally. Farmers who are self-aware use vastly enhanced agronomic techniques, protecting public health and lowering environmental hazards [26]. Therefore, awareness is essential in crop pest management. Television, radio, and mobile telephone are beneficial media providing agricultural practices and crop protection information. This article presents a mathematical model for CMD and CBSD (CVD co-infection), incorporating farming awareness levels. The main focus is to compare a fundamental advantage favouring the combined strategy by involving the following CVD co-infection control management. This is the immediate removal of the infected plants by uprooting and burning, killing the whitefly vectors through spraying pesticides and other means, and increasing farming awareness through the education campaign among cassava farmers through media to minimise the spread of CVD. The three-dependent optimal control problem is formulated and solved using the Pontryagin Minimum Principle to determine the optimal level of control strategies for cost-effectiveness.

MODEL DESCRIPTION AND FORMULATION
The deterministic mathematical model of Codynamics of Cassava mosaic disease (CMD) and Cassava brown streak disease (CBSD) was formulated and analysed following the theory of  The compartmental diagram in Figure 1 summarises the transmission dynamics of CVD coinfection. The susceptible cassava plants contacted an infected whitefly vector and moved to the exposed class before becoming infected. Cassava plants can be infected with CMV, CBSV or both; the susceptible whitefly vector acquires viruses by inoculating the infected cassava plants and becomes infected. Due to the finite size of the cassava field, we assume logistic growth for the cassava population, with a net growth rate r and carrying capacity 1 K . The Farmer can replant both susceptible and exposed cassava cuttings stem without knowing. Therefore, specifies the frequency to which exposed cassava is selected relative to susceptible cassava replanting material.
Hence susceptible cassava plant population can be described by the following logistic population growth term 1 1 and by replanting exposed cuttings materials is 1 1 where E  is the number of cassava material replanted with the viruses. The cassava plant moves to the exposed class after acquiring viruses through contact with the infective whitefly vector at a rate  . After developing the symptoms of one disease, the cassava plant infected with the single disease will become dually infected with the other Cassava Virus disease. The circumstance of a cassava plant being infected by both viruses simultaneously will be referred to as CVD co-infection.
The CMV infectious class can move to the CBSV infectious class to form the CVD class at a rate , and the CBSV infectious class can move to the CMV infectious class to create the CVD class at the rate since each disease is joining the CVD class in a different time. All cassava classes are harvested at a rate during the harvesting cassava tubers period.

Equations of the model
Now using the model assumptions given above, the co-dynamics of CVD are described by the following systems of nonlinear differential equations:

Invariant Region of the Solution
The CVD dynamic model system (1) has three subpopulations where all parameters and variables are nonnegative 0 t  .

Proof
We use the box invariant method to establish the feasible region of the CVD co-infection model solution [32]- [35]. In the co-infection dynamical system (1) In a reduced Metzler matrix ( )

Theorem 1
Let the initial values of the state variables of the model system be Then, the solution set

Proof
We consider the first equation of model (1) are positive for all time 0 t  . As a result, the CVD co-infection transmission model stated in the system (1) is epidemiologically significant and mathematically well-posed.

CVD co-infection free equilibrium point
The basic reproduction number 0  of the model (1), obtained using the next-generation method

Theorem 2
The CVD co-infection free equilibrium point is locally asymptotically stable if Where: Here, we employ the determinant and trace method to examine the local stability of the CVD coinfection-free equilibrium point. However, first, we must prove that the determinant and trace of the matrix (8) are positive and negative, respectively.
The determinant of the matrix (8) is then calculated using the software Mathematica; we get As we can see , it is the product of positive parameters and the Det J C to-be 0  we should have 0 1 .
Also, it is clear that the trace of the matrix (20) is negative, which is Hence, the CVD co-infection-free equilibrium point is locally asymptotically stable if 0 1 .

Global stability of CVD co-infection free equilibrium point (CVDCFE).
To investigate the issue of the global stability of CVD co-infection free equilibrium point, we use the Metzler matrix method, as stated by [36]. So, we pose the following theorem:

Proof:
We separate the model system (1) To prove the global stability of the CVDCFEP, we are required to show that matrix 1 has real negative eigenvalues and 3 C is a Metzler matrix containing all nonnegative off-diagonal entries.
Based on the model system (1), we will have We will now have the matrices below employing the general system non-transmitting and transmitting compartments. Now when we calculate eigenvalues for the matrix 1 The results indicate that the eigenvalues are negative and real, proving that the system asymptotically stable at 0 C X . Furthermore, considering the matrix, 3 C we find it is a Metzler stable matrix since all of its off-diagonal elements are positive. Therefore, CVDCFEP for system (1) is globally asymptotically stable.

OPTIMAL CONTROL
This section aims to introduce the control mechanisms that possibly reduce the burden on farmers These control strategies need to be adjusted to minimise the number of infectious cassava plants, whitefly vectors and the cost of implementing the control strategies. It is ought to consider the optimal control problem with objective functional of the form objective function J ; consequently, it is required to find the optimal control such that where ( ) the set of admissible controls. Note that in all cases, setting the control to zero indicates that no effort is being made to control the disease, and setting it to one indicates that the most effort possible is being made. Now we apply the following theorem to show the existence of optimal control for systems (18) and (19).

CHARACTERISATION OF THE OPTIMAL CONTROLS
The representation of the optimal controls depends on Pontryagin's Maximum Principle [38]. To achieve this, one needs to transform the optimal control problem into the problem of minimising In more compact, the Lagrangian is defined by Then, the expanded form of the Lagrangian is given by ( )

Proof.
We are taking the partial derivative of the Lagrangian H with respect to state variables. That is,  *  *  *  1  3  3  3  5  5  8  3  4  *   *  *  *  *  *  *  *  3  3  3  3  3  6  7  3  2   1  1  1  1   1  1  1 The optimality system consists of the state system (18) coupled with the adjoint system with initial and transversal conditions and the characterisation of optimal control. It can be noticed that which implies that the second partial derivative of the Hamiltonian H with respect to controls 1 u , 2 u and 3 u are positive. Therefore, the optimal problem is minimum at controls 1 u , 2 u , and 3 u .

Parameter estimation of CVD co-infection model
This section presents the numerical analysis using the data obtained from ten district councils in two regions (Mwanza and Geita) in Tanzania     and Jittama et al., [42], while others were assumed based on the environmental implications. The numerical results for MLE for the CVD co-infection model parameters are summarised in Table 3.

Sensitivity Analysis of Model Parameters
Sensitivity analysis was obtained on the fundamental parameters to check and identify parameters influencing the basic reproductive number. We used the approach specified by Blower, S. M. and Dowlatabadi, H. [43] to conduct sensitivity analysis, as described in [44]. The sensitivity indices are given as where p is the model parameter.
The sensitivity indices of the model are presented in figure 3 below

Impact of Farmer's Awareness toward cassava virus disease
The impact of the farmers' awareness is evaluated through model (1)

Control Scenarios
The following four combination optimal control strategies were considered and evaluated for their impact on the eradication of CMD, CBSD and CVD-coinfection: The parameter used in the simulation is shown in Table 3  The results show that uprooting and burning infected cassava plants in the system, spraying pesticides and farming awareness will reduce the spread of the disease. We can observe this tendency from 9(a)-(c), which displays that the CMD, CBSD and CVD co-infected cassava decreases by intensifying the removal of the infected plants and the infected whitefly vector decreases by strengthening pesticides application due to the increase of farmers awareness.
Moreover, Figure 9(d) shows that the infected whitefly vector decreases when applied control. The combination of strategies 12 , uuand 3 u gives the best results to optimise the objective function ( ).

COST-EFFECTIVENESS ANALYSIS
The Incremental Cost-Effectiveness Ratio (ICER), as suggested by Kinene et al. [11], is a more formal approach to evaluating the cost-effectiveness of the strategies. The ICER compares two competing alternative strategies using limited resources and limited resources. When comparing two intervention strategies in ICER, one strategy should be compared incrementally with the nextless-effective alternative. It is termed additional costs per additional health outcome. In other words, ICER can be defined as the ratio of the cost difference between two strategies to the averted difference between the total number of infections. That is,   Comparing strategies C and B, strategy B's ICER is less than strategy C's ICER. Therefore, strategy C is more costly and less effective than strategy B. Thus, strategy C is excluded from alternatives, and ICER is recalculated for the remaining strategies. With Strategy C dropped, Table 5    Finally, the findings show that strategy B is less expensive than strategy A. Strategy B (uprooting and burning infected plants and a farmer's awareness campaign) is the best of all possible strategies due to its cost-effectiveness and health benefits. Then the later mathematical model was modified to incorporate three time-dependent control to obtain an optimal control problem. The existence of optimal control is proved and later analysed. Finally, the optimality system is numerically solved, and its results are presented. The findings suggest that a strategy that includes a combination of two controls (uprooting and burning and a farmer's awareness campaign) is effective and plays a significant role in minimising the epidemic.
Similarly, it was found that any strategy under consideration, combined with awareness campaigns, appeared to be more effective than that ignored awareness. Finally, the cost-effectiveness of the