Previously, mathematical models were used only for health problems in humans with epidemic models SI, SIS, SIR, SIRS among others; currently, they are a useful tool in various areas. With these models the representation of biological systems can be achieved and thus knowledge of the behavior and the dynamics of certain diseases are facilitated, helping to predict and make future decisions that facilitate control ( Jeger et al., 2018).

A population model with nonlinear ordinary differential equations is proposed, which interprets the dynamics of the Moko of plantain, with the following assumptions: •

A fraction h of reseeding with the disease is included.

The variables and parameters of the model are: x t the average number of susceptible banana plants, and y t the average number of infected banana plants and P t = x t + y t total number of banana plants at time t , respectively.

The model parameters are γ: constant replanting rate; k : carrying capacity (maximum population) of banana plants in the study region; and β: probability of infection transmission. The preventive controls are g : fraction of infected banana plants eliminated 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 ( Figure 2), is formed by the following two non-linear differential equations: dx t dt = 1 − h γ 1 − x t + y t k x t − β y t x t + y t 1 − f x t (1) dy t dt = h γ 1 − x t + y t k x t + β y t x t + y t 1 − f x t − gy t (2)

With initial conditions x 0 = x 0 , y 0 = y 0 , P 0 = x 0 + y 0 , γ , k > 0 , 0 < f , g , β < 1 , P ≤ k .

The region of eco-epidemiological sense is defined where the trajectories of the plant infection dynamics make sense, Ω = x y ∈ R + 2 : 0 ≤ x ≤ k 0 ≤ y ≤ k . (3)

And the variation of the population of diseased plants eliminated in time, is given by the following equation, dω t dt = gy t (4)

Is defined r t as the load (incidence) of banana plants by infected hills in replanting and dr dt the variation of this population with respect to time, dr t dt = h γ 1 − x t + y t k x t (5)

We define P r t the economic loss due to overseeding with infected hills, such that P r t = P r r t (6) where P r is the cost per hill planted and treated.

Deriving equation (6) with respect to t and substituting the derivative dr dt , we obtain the differential equation for the economic loss due to replanting of infected hills d P r t dt = P r h γ 1 − x t + y t k x t (7)

Similarly, we defined the economic loss by infected banana plants eliminated P ω t , P ω t = p ω ω t (8)

Where P ω is the cost of removing each plant with Moko.

Deriving with respect to t and substituting the derivative dω dt , we obtain Pω t dt = p ω gy t (9)

The total loss by replanting and by elimination of infected plants in time is P e t = P ω t + P r t , and its variation in time is given by the differential equation, P e t dt = p r h γ 1 − x t + y t k x t + p ω gy t (10)

The model simulations were made with Maple software version 18 (Maple, RRID:SCR_014449) using the values of the populations and initial losses, as the values of the parameters indicated in Table 1. An open-source alternative software that can perform equivalent functions is RStudio version 1.4 (RRID:SCR_000432).

Evaluating different scenarios in the conditions of the strategies of prevention and elimination of infected plants, we can observe (as shown in Figure 3a) that implementing 10% of prevention strategies, eliminating 80% of the infected plants and doing a reseeding with 10 % of infected hills, the population of susceptible plants is sustained over time. This would be a good agronomic management for the sustainability of the crop, of the harvest and would avoid greater economic losses, associating only with losses of approximately 100 or 200,000 thousand pesos with a trend to stabilize at that value. However, the losses due to the elimination of infected plants would be greater when implementing this strategy in a greater proportion, highlighting that it is a methodology that must be implemented in large proportions to avoid the spread of the disease. On the contrary, in the reseeding of 30% infected hills with the same prevention conditions ( Figure 3b), the economic losses associated with the elimination of infected plants tend to an exponential increase in costs week after week and replanting in week 50, with an approximate value of 2,000,000 pesos and a tendency to increase in the following weeks.

By evaluating different scenarios, we can demonstrate the importance of early elimination of infected plants, showing when the prevention strategies are implemented by 80% and only 10% of the infected plants are eliminated, the susceptible population tends over time to decrease. In terms of economic losses in these scenarios, in week 50 it can be observed that they are greater by approximately 1,000,000 pesos for reseeding by 30% ( Figure 4b) compared to reseeding of 10%, which causes losses of approximately 200,000 pesos ( Figure 4a). We can also see that the losses associated with infected plants and their elimination, in both cases will have an exponential increase week after week.

Finally, in the best of cases it is observed that by using both strategies in a medium proportion f 60 % − g 70 % , susceptible plants tend to be sustained over time and infected plants tend to be controlled. In this scenario it is observed that economic losses are generally lower; for example, in the reseeding of infected plants by 10% ( Figure 5a), after week 50, approximately 100,000 pesos would be lost, while for the reseeding by 30% ( Figure 5b) approximately 1,000,000 pesos would be lost. Additionally, the losses associated with infected plants and their elimination in week 50 will no longer have an exponential increase but tend to stabilize at 1,200,000 pesos and 5,000,000 pesos respectively.

In Figures 6a, we show the behaviour of the infected plant population over time with reseeding by 10% and in figure and 6b with reseeding by 30% in different scenarios, demonstrating the importance of early elimination of infected plants to prevent the spread of the illness. In Figure 6a the red line corresponds to the low elimination of infected plants (f 80% -g 10%) showing an increase in time of the exponential type of the population, while in the black line in both scenarios it is shown that the population of infected plants tends to be controlled, in the 10% replanting it tends to be controlled, while in the 30% replanting the population tends to stabilize at approximately 150 infected plants over time. Finally, the blue line shows that when both strategies are applied in a medium proportion, the population of infected plants tends to be controlled and disappear in time either at 10 and 30% overseeding, associating this with good agronomic management and sustainability of the crop, avoiding greater economic losses and reduction of production costs.

In conclusion, cultural control strategies in banana Moko disease, such as disinfestation of tools, footwear, and weed pruning are important agronomic practices for disease control. However the identification of planting infected hills plays an essential role in preventing the spread of the disease and stopping the increase of foci infection, as well as in the early elimination of infected plants, thus leading to a decrease in production costs.

The analysis shows that to reduce production costs, both prevention strategies can be implemented to a lesser extent, thereby reducing labour, etc., but still keeping the disease in a controlled state.