Parameter Estimation in a PDE Model for the Spatial Spread of Cocoa Black Pod Disease

Abstract

In this paper, we develop an epidemiological model with both environmental (primary infection from the environmental spores reservoir) and direct transmission (secondary infection from an infected host to a susceptible pod). This model simulates the spatiotemporal evolution of cocoa black pod disease caused by Phytophthora megakarya. Since reliable parameter estimation is a central issue for modeling realistic biological systems, we used a mechanistic–statistical approach to estimate model parameters from real observations of a specific cocoa plot. In addition, to refine numerical simulations of the pathosystem, data describing the shade intensity all over the plot were exploited and led to increased model predictions accuracy and also highlighted a higher number of infected pods located in areas of the plot with higher shading intensity. Recommendations in terms of promoting cocoa farming in systems with low shading intensity may be evident if these results are confirmed. Our results also highlight the importance of the environmental spore reservoir in black pod disease dynamics.

Appendices

Appendix A

In this appendix, we provide numerical simulations for susceptible pod compartment S (Fig. 15) and spores produced by infected pods $P_1$ (Fig. 16) for the set of parameters $\hat{\theta }$.

Fig. 15

Spatial dynamics of the susceptible pods compartment S with $P_2^0$ defined via a covariate function (namely the shading function) and with $\theta =\hat{\theta }$

Appendix B

In this Appendix, we provide estimation results ($\hat{\theta '}$) for the “benchmark” estimation where we define $P_2^0$ as a piecewise constant function. We also present the data histograms for the spatiotemporal raw residual analysis for the two estimation strategies.

• Estimation results with $P_2^0$ defined as a piecewise constant function.

  The “benchmark” estimation of model parameters through the optimization function led to convergence after around 3000 iterations. The optimal value computed for the log-likelihood function is $-7.2923e+04$, and the corresponding set of parameters $\hat{\theta '}$ (with the corresponding asymptotic standard errors) are given in Table 5.

• Raw residual analysis for the two estimation strategies.

Fig. 16

Spatial dynamics of the spore produced by infected pods $P_1$ with $P_2^0$ defined via a covariate function (namely the shading function) and with $\theta =\hat{\theta }$

Fig. 17

Histograms of residuals, with $P_2^0$ defined as a piecewise constant function, for weeks $\in \{2, 10 , 20, 29, 35, 4\}1$; the horizontal axis gives the residual values $|I_\mathrm{obs}(t_i,x_i) - I_{{\widehat{\theta }}}(t_i,x_i)|$

Fig. 18

Histograms of residuals, with $P_2^0$ defined via a covariate function (namely the shading intensity data), for weeks $\in \{2, 10 , 20, 29, 35, 41\}$; The horizontal axis gives the residual values $|I_\mathrm{obs}(t_i,x_i) - I_{{\widehat{\theta }}}(t_i,x_i)|$

Table 5 Epidemiological parameter estimates with $P_2^0$ defined as a piecewise constant function