A theoretical model for the prevention of Banana Moko (Musa AAB Simmonds)

Introduction

The banana is a fruit of great economic importance and food sovereignty, because it is found in the shopping basket of people across different social strata and because of its nutritional content. However, its production is threatened by re-emerging diseases such as Moko, caused by the bacterium Ralstonia solanacearum race 2 philotype II (Fegan & Prior, 2006), which causes wilting and deterioration of the plant. Symptoms are usually visible when there is a great spread of bacteria and adjacent plants may have already been infected (Viljoen et al., 2016). Moko is a peculiar manifestation of bacterial wilt; it is a quarantine pest that, once inside the host, moves through the vascular bundles. Being a vascular disease, this bacterium not only affects the vegetative part but also the daughter hill, promoting the spread of the disease, which is accelerated by the high optimal minimum, optimal and maximum temperatures of 10°C, 35°C and 41°C, respectively, infecting triploid plantains, heliconia (Heliconia spp.) and other ornamental Musacea plants (Jiang et al., 2017).

The development of mathematical models has contributed through the use of a wide range of techniques to the study of epidemics and diseases, helping to answer biological questions and raising new questions related to the epidemiology and ecology of pathogens and the diseases they cause; in most cases, mathematical models lead directly to applications in the control of the disease (Jeger et al., 2018). Some prediction models that calculate the propagation threshold R₀ have evaluated the control of some diseases in plantain and bananas, such as banana wilt by Xanthomonas (BXW) using mathematical models, describing a deterministic SI-type epidemic model for control of BXW focusing on the integrated management of the disease through cultural control as in Nannyonga et al. (2015), who considered the optimal control strategies associated with the prevention of transmission by the use of contaminated tools. The researchers assumed a model with three modes of transmission: vertical (from the mother plant to its child), horizontal (indirect) from the vector to plant, and through contaminated agricultural tools (Nannyonga et al., 2015).

Likewise, Nakakawa et al. (2016) presented a mathematical model for BXW propagated by an insect vector. The mathematical model they formulated takes into account inflorescence infection and vertical transmission from the mother corm to the daughter hills, but not tool-based transmission by humans (Nakakawa et al., 2016). In this context, a dynamic system is formulated based on ordinary two-dimensional differential equations that interprets the dynamics of incidence of banana Moko disease, including prevention and treatment.

The model

A population model with nonlinear ordinary differential equations is presented, which interprets the dynamics of the banana Moko, including a constant rate of disease prevention in the population of susceptible plants over time. A variable population of plants and a logistic growth of replanting are assumed, taking into account the maximum capacity of plants in the study region. The variables and parameters of the model are: x(t), the average number of susceptible banana plants; y(t), the average number of diseased banana plants; and P(t) = x(t) + y(t), total number of banana plants at one time t, shown in Figure 1.

Figure 1. Banana Moko's disease diagram with prevention.

The model parameters are: γ, constant overseeding rate; k, load capacity (maximum capacity) of banana plants in the study region; and β, probability of transmission of infection. Preventive controls are: g, fraction of infected banana plants removed; and f, fraction of susceptible banana plants that receive prevention of contagion of the bacteria. The dynamic system that interprets the infectious process including prevention and elimination, is formed by the following two nonlinear differential equations:

With initial conditions x(0) = x₀, y(0) = y₀, P(0) = x(0) + y(0), γ, k > 0, 0 < f, g, β < 1, P ≤ k, x(t) ≡ x, and y(t) ≡ y.

The region of eco-epidemiological sense is defined where the trajectories of the plant infection dynamics make sense,

Stability and sensitivity analysis

We start by finding the equilibrium populations, the constant solutions of the system, where the population variation of susceptible plants and variation of infected plants become zero, that is, $\frac{dx}{dt}=0;\frac{dy}{dt}=0$

We solve this non-linear algebraic system for x and y, determining the equilibrium point with agronomic sense, free of infected plants E₁ = (k, 0). A breakeven point with disease invasion without susceptible plants E₂ = (0, k), and a balance point with susceptible plants and infected plants $E_3=(\widehat{x},\widehat{y}),$ with

Considering,

We write x and y like this

In coexistence of populations $\widehat{x}>0$ and $\widehat{y}>0$, which is true when ξ₀ > 1 and ρ < 1.

Since P = x + y, the total plant population in equilibrium is, $\widehat{P}=\widehat{x}+\widehat{y}.$ That is,

Therefore, $\widehat{P}=k(\rho -1).$

ξ₀, indicates the average number of infected plants that an infected plant produces during the infectious period (before being killed) in the population of susceptible plants and is considered the threshold of infected plants. We can consider this threshold as a function that depends on f and g,

To determine the stability of each equilibrium point (E), we apply the Hartman-Grobman theorem (Perko, 2011), linearizing the system of non-linear Equation (1) – Equation (2), obtaining the linearization matrix (Jacobian matrix) of the form:

With the following partial derivative elements,

These elements of the matrix J (E) are the coefficients of the linear system

Where, U = (u, v)^t (transposed vector).

We analyze the balance points with an agronomic sense E₁ = (k, 0) y E₃ = $(\widehat{x},\widehat{y}).$ For E₁, we obtain the Jacobian matrix,

Because it is a triangular matrix, the eigenvalues (λ_i, i = 1,2)

where λ₂ < 0 since the threshold ξ₀ < 1.

We conclude that the free equilibrium point of Moko disease is locally and asymptomatically stable.

For case $E_3=(\widehat{x},\widehat{y}),$ in matrix (10) we obtain the trace and the determinant of J(E₃), respectively,

We conclude that the equilibrium point with susceptible plants and infected plants is locally and asymptomatically stable if the threshold inequalities (7) and the inequalities are met,

These analytical results are shown in the phase planes of Figure 2, made with Maple 18 software (free trial available; SageMath is an openly available alternative), for different scenarios varying initial conditions.

Figure 2. Local stability of the susceptible plant population and infected plant population corresponding to ξ₀ = 7 y ξ₀ = 0.79.

Results and conclusions

Local sensitivity is a measure of the relative change in a variable when its parameters change (Chitnis et al., 2008; Hamby, 1994; Rodrigues et al., 2013). That is,

Where,

y p: β, g, f, are previously defined parameters.

The indices of local sensitivity of the epidemic threshold with respect to each parameter are $I_{{\xi }_0}^{\beta }$ = 1, $I_{{\xi }_0}^f$ = -0,43 y $I_{{\xi }_0}^g$ = -1. These values indicate that the parameter that most influences the threshold value is β proportionally and g inversely proportional.

It is concluded that mathematical simulation models are a useful tool for research in banana Moko disease. With them it was determined that the elimination of banana plants infected with the disease plays an essential role in the good agronomic management of the crop.

As a research perspective, consider the problem of a simulation model including a piecewise function for the rate of elimination of infected plants, in the form

Where T is the time it takes for agricultural institutions to confirm the presence of plantain Moko and suggest the elimination of infected plants.

The analysis of the following optimal control problem is also proposed as a research perspective, applying the principle of the Pontryaguin maximum (Kopp, 1962).

The functional objective of direct and indirect costs is proposed:

Linked to the system of differential equations:

With initial conditions p_s(0) = and p_s0, p_i(0) = p_i0, P(0) = p_s0 + p_i0,      γ, k > 0, 0 ≤ u₁, u₂, β ≤ 1, P ≤ k, n_i > 0, i =1,2,3, P_s≡x and P_i≡y.

It is about finding optimal control $({\overline{u}}_1(t),{\overline{u}}_2(t))$ such that:

Where,

is the space of admissible controls and L² is the space of integrable functions.

Data availability

All data underlying the results are available as part of the article and no additional source data are required.