MODELLING THE EFFECT OF FUSARIUM OXYSPORUM IN TRANSMISSION DYNAMICS OF FUSARIUM WILT DISEASE OF CASHEW PLANTS IN CONTEXT OF SOUTH-EASTERN TANZANIA

Cashew nuts are the most important cash crop in Tanzania's south-eastern region, where cashew nut farming is the primary source of income for the majority of people. Aside from their importance, cashew plants are vulnerable to Fusarium wilt, a devastating disease. In particular study a system of equations for the model is formulated and analyzed qualitatively using the stability theorem of ordinary differential equation. The stability analysis indicates that the system is stable under the specified conditions. Furthermore, the analysis shows that an increase in transmission is determined by the rate of contact between susceptible plants and infected plants via root contact, whereas numerical results show that decomposed infected plants contribute to the growth of Fusarium oxysporum, increasing the disease


INTRODUCTION
Fusarium wilt is the most destructive plant disease affecting various plants worldwide [1]- [3]. This disease was first recorded in Australia 1876 affecting bananas [4]. Fusarium wilt has been the most damaging agricultural disease affecting numerous plants across the globe, including cotton, tomatoes and conifers [5]- [9]. On the other hand Fusarium wilt also affects cashew plants and was first confirmed to be a destructive disease in various parts of Tanzania in 2012, including Magawa village in the Coast region, Nanganga, Lindumbe, and Mnongodi villages in Mtwara region [2].
The cashew plant is grown all over the world and produces cashew apples and cashew nuts.
The continents that produce Cashew nuts are Asia, Africa, South America and Australia, where 40% of global production comes from Africa. Tanzania is the third largest producer in Africa and accounts for about 20% of Africa's total production [10]. Most of the production in Tanzania (about 80%) comes from the south-eastern part of the country, Lindi and Mtwara regions [11].
Cashew nut is the main source of income for many families in the south-eastern regions of Tanzania, but cashew nut also accounts for up to 10% of the overall amount of foreign exchange earnings in Tanzania [12]. Regardless of high quantity contribution to the nation and continent at large, the production of Cashew nut in the south-eastern region of Tanzania is below production standard of Cashew nut [11].
The standard production of cashew nut is 800 kg per hectare while the actual production decreases from 800 kg to 500 kg per hectare. Several factors including drought, decreasing soil fertility, insect pests and diseases have been attributed to the failure to achieve the ultimate production goal [13], [14]. The disease factor, among others, is the key reason that influences the production of cashew nuts. Pestalotia leaf stain, Fusarium wilt, cashew leaf and nut blight, anthracnose, dieback, damping off, and cashew powdery mildew disease are the most significant cashew plant diseases [11]. Cashew plants are affected by Fusarium wilt disease, which contributes to harvest losses of up to 100% if not managed and thus reduces yield [2].
The Fusarium wilt disease is induced by Fusarium Oxysporum fungus [5], [6], [15], [16]. The infection is initiated by chlamydospores through roots or wound penetration [7]. At any stage of the plant growth the symptoms of Fusarium wilt may appear. The symptoms of Fusarium wilt disease is the yellowing and wilting of the leaves, xylem vessels develop into slightly red color and are noticeable as lines or dots within the stem [1], [16]- [18]. The entire leaves wilt and ultimately the tree dies within two to three weeks from the onset of symptoms [2], [19]. The focus was on understanding the evolution of different states associated with disease transmission.
Mathematical modelling play a vital role in analyzing the dynamics of disease transmission in various population including plants. The study by [20], [21] analyzed disease in plants and suggested the possible control measures. On other hand, [22], [23] investigated the epidemiological model of fungus in plants and analyzed the disease severity and provide enough information to make appropriate decisions. This study therefore intends to evaluate the impact of Fusarium oxysporum in transmission of Fusarium wilt disease in cashew plants in South-eastern Tanzania.

MATHEMATICAL MODEL FORMULATION
We formulate mathematical model for Fusarium wilt disease by considering two populations, Cashew plants and Fusarium Oxysporum fungus population. Four compartment variables comprise the cashew plants as susceptible plants () S , exposed plants () E , infected plants ( ) I and recovered plants () R . Owing to the disease dynamics, the plant may shift from one compartment to another. The cashew plant, which is free of the disease but can be infected when it comes into contact with fungus, is susceptible plant. The root of susceptible plant exposed through its contact with the infected soil or infected Cashew plant via roots or wounds [3], [24], [25], [26], [27]. The total population of Cashew plants is denoted by N S E I R = + + + . The population of Fusarium oxysporum contains two subclasses namely, Macroconidia spores ( ) and Chlamydospores ( ). Macroconidia spores are the spores that appear on the surface of dead infected plants and then transform into Chlamydospores before they return to the soil [1], [17].
The model formulated under the assumption that, the growth rate of Cashew plants population  [3].
The parameter d represents the saturation constant rate of Chlamydospores fungus into the soil.
Progression rate of exposed cashew plants to infected represent by  . The infected cashew plant may die naturally at the rate of  or may die from the disease for a period of two to three weeks at the rate of  . The treated infected Cashew plants may recover after 120 days at the rate of  and become susceptible at the rate of  [19].
The Macroconidia spores follows logistic function with growth rate a and carrying capacity 2 k .
Fungus (Microconidia) attack parenchymatous tissue of the dead plants and sporulate profusely on the surface of dead plants as a Macroconidia spores at the rate of  [17], [28], [29]. The parameter  is defined as where  and  are transformation rates of Microconidia to Macroconidia from exposed and infected plants respectively. After the decomposition of dead cashew plants, Macroconidia spores transformed into Chlamydospores fungus at the rate of  [30]. Chlamydospores leaves the compartment through decay at the rate of  [1].
Basing on the assumptions made, the following flow chart represents the summary and model

Model equations
From the Figure 1, we formulate ordinary differential equations as: Subject to the following initial conditions: FATU CHILINGA, ALFRED K. HUGO, ALFRED M. WANYAMA

Positivity and Boundedness of the Model
The set off points ( , , , , , ) x x x x x x x = we investigate the system of equation (1).We formulate the following lemma using the approach of Lungu et al, [31]: Lemma1. The system (1) can be transformed into following differential inequalities This is direct contradiction to our lemma (1). For the state variable in our model, we always take Hence in the region ℜ + 6 the model equation is epidemiologically significant and may be utilized to analyze Fusarium Wilt in Cashew plants. 7

Stead state
The model (1) has the disease free equilibrium DFE as The basic reproduction number 0  , is defined as the average number of secondary infections produced by infectious individual in its life time [32]. The computation of reproduction number was done by next generation method as proposed by Van den Driessche and Watmough [33].
The model basic reproduction number obtained by choosing the largest eigenvalue in absolute terms that is the spectra radius of the matrix 1 FV − (next generation matrix) is given by The basic reproduction number generated via two causative sources that is contaminated environment and infected cashew plants. The the average length the time used for the chlamydospores to be saturated in the soil 1 d and the average length of time spends for transformation process from Macroconidia spores to Chlamydospores

3.2.Short -Term Behavior of the Disease-Free Equilibrium Point
All the equations in the model system (1) are considered and examined at the disease-free equilibrium point; where the short term behavior is described by using the Jacobian matrix of the model system, necessary and sufficient condition for local stability of the system is evaluated through real parts of the eigenvalues. The system at DFE presented as  ) 0 The eigenvalues of matrix (7)  Since all roots are negative, then the disease free has a permanent behavior.

3.3.Long -Term Behaviors of the Disease-Free Equilibrium Point
Long-term behavior of the disease-free equilibrium point of the Fusarium wilt model is described by considering Metzler matrix method the approach of [34]. We rewrite the model system (1)  A has a real and negative eigenvalue and also 3 A is a Metzler matrix. Metzler matrix is a square matrix with all elements off the diagonal are non-negative. Consider equation (15) we have ( ) A is a Metzler stable matrix since all of its elements in off the main diagonal are non-negative. Hence, the disease-free equilibrium point for Fusarium wilt disease has a long-term behavior and leads to the theorem 2.

Long-Term Behavior of Endemic Equilibrium Point
The Long -term behavior of the endemic equilibrium point EE of the model is explored by considering following continuous and differentiable Lyapunov function below

Effect of Chlamydospores on Disease Transmission
The number of plants exposed to the disease appears to increase exponentially as the contact rate between susceptible plants and chlamydospores increases from 600 to 1100 in 9 months. It then fell to 286 after 2.5 years as a result of disease progression, as shown in Figure 2.

Effect of Disease Transmission through Plant Roots contact
As the rate of contact between susceptible cashew plants and infected cashew plants increases, the plants that exposed to diseases increase as shown in figure 2. The rate of contact between plant roots has a significant impact on disease transmission. This implies that as the rate of interaction between plant roots rises, disease transmission tends to increase.

Effect of Fusarium Oxysporum in Disease Dynamics
As the number of exposed plants gets bigger, the infected plants raises as well, due to the progression rate of exposed to infected plants, while Macroconidia decrease as Macroconidia transform into chlamydospores, as shown in Figure 6. Then, as time passes, the exposed plants become fewer, while the infected become more numerous due to an increase in progression rate, and Macroconidia become fewer as the transformation rate to chlamydospores increases.   Figure 7 shows that as the number of macroconidia increases, the number of exposed and infected plants increases. This is because the rate at which exposed plants become infected, as well as the decomposition of dead infected plants, increases the amount of fungus in the soil. Figure 8 shows that the number of exposed plants decreases, while the number of infected plants and chlamydospores increases, as more exposed plants become infected, and more decomposed dead infected plants increase the fungus in the soil. Figure 9. The comparison between infected plants with exposed plant. 17 TRANSMISSION OF FUSARIUM WILT DISEASE OF CASHEW PLANTS Exposed plants increases at first due to the incubation period, then decrease after the fungus mature, while infected plants continue to increase over three years as shown in figure 9.  Figure 11. Effect of Fusarium oxysporum fungus on exposed plants As the number of chlamydospores increases in the soil, the number of Macroconidia decreases due to the rate of transformation to chlamydospores, whereas exposed plants increase at first before decreasing as a result of progression to infected plants, as shown in Figure 11. The number of chlamydospores increases as the number of Macroconidia decreases. This is due to the rate at which Macroconidia transform into chlamydospores. Figure 13, effect of Chlamydospores to susceptible plants 19 TRANSMISSION OF FUSARIUM WILT DISEASE OF CASHEW PLANTS Figure 14, impact of Chlamydospores to exposed plants Figure 15, relationship between Chlamydospores and infected plant As the number of chlamydospores in the soil increases, the number of susceptible plants decreases due to plant exposure to the disease, as shown in Figure 13. Figure 14 shows that for the first 8 months, as the chlamydospores increase, so do the exposed plants; however, exposed plants decrease after two years, possibly because exposed plants shift to infected plants after the incubation period. Figure 15 depicts the direct proportion between the two populations; as chlamydospores increase, so do infected plants.

CONCLUSION
A specific study investigated and discussed the role of the Fusarium oxysporum fungus in the transmission of Fusarium wilt disease in cashew plants. To gain some quantitative insight into the dynamics of the Fusarium wilt disease, the two populations were used to build a mathematical model that included the cashew plant and the Fusarium oxysporum fungus. The study's findings illustrated that the contact rate of chlamydospores and infected plants with susceptible plants via root contact has a high impact on disease transmission, resulting in significant economic losses, particularly for communities or individuals who rely on cashew farming. The growth of the Fusarium oxysporum fungus in the soil also has an impact on the epidemic. Fusarium oxysporum growth rate is aided by an increase in the decomposition of infected plants in the field.