Mathematical Model for Rice Blast Disease Caused by Spore Dispersion Affected from Climate Factors

Abstract

Rice blast disease, caused by the fungus Pyricularia oryzae, is one of the diseases that reduce rice yields in Thailand. The fungus’s dispersal influences this disease outbreak. This study aims to develop a mathematical model for spore dispersion, namely a non-local diffusion equation in terms of the dispersal kernel, by adding a factor that causes spore dispersion due to rain splash and adjusting the infection term based on weather conditions such as air temperature and relative humidity. The model assessed the existence and uniqueness of solutions by Banach’s fixed point theorem and used a finite difference method to solve them. The numerical simulation confirmed the existence and uniqueness part of the analysis. Because the climate data was used effectively for developing the disease, the rice blast disease pandemic was widespread in the study areas.

1. Introduction

Rice is an important crop in Thailand as it plays a vital role in socio-economic development. In August 2020, the International Rice Research Institute (IRRI) of Thailand reported that each person consumes an average of 115 kilograms of rice per year [1], which means rice is a staple food for Thai people. In addition, there are 13 million; around 60% of Thai people are farmers that plant rice. Rice disease is one of the factors that affect rice production. In Thailand, the rice-growing season is classified into dry season rice and wet season rice. For dry season rice, rice is planted between December to June. Moreover, from May to July is the wet season rice period. Rain usually falls in a manner suitable for rice planting, and rice will be harvested by December. Rainfall was identified as a common factor of foliar disease outbreaks [2]. In 2020, Jarroudi et al. stated that many crop foliar diseases are caused by fungal pathogens [3]. Therefore, rainfall is one component that leads to the dispersion of fungal pathogens.

The dispersal of fungal pathogens or spores causes widespread plant diseases. Spore dispersal is a two-step process. The first step is spore release. After that, spores disperse away from the parent by many factors. The spores can spread from a distance of a few inches to hundreds of kilometers depending on many factors such as wind, water, insect, humans, and animals [3,4]. Spores produced on infected hosts are transmitted to new susceptible host plants due to wind currents or other reasons. The adhesion of the spore to the plant surface is the first step in the infection process. The adhered spores spend a short or long time on the plant surface, depending on the surface wetness duration and the wind. However, the infection process depends on three important factors: the susceptible host, virulent pathogens, and favorable environmental conditions (temperature, humidity, rainfall, and wetness). Temperatures between 18–28 °C, particularly in the range of 23–26 °C [5], and relative humidity (more than 9 h) >90% during crop season were within the optimum range required for disease development [6]. At the end of the pathogen production process, pathogens can be spread from the factors mentioned above. It means that new susceptible host plants can become infected, going on until they are completely removed. Therefore, spore dispersal is an important factor in widespread plant diseases.

Many researchers want to study spore dispersal. They have constructed different mathematical models to describe dispersion behavior and spore density change. In 1991, Yang et al. [7] developed and evaluated a model to simulate the splash dispersal of plant diseases from a point source over a homogenous ground cover using the diffusion equation. Several factors limit the splash dispersal of plant pathogens. First, most dispersion occurs within plant canopies, which generally results in a non-homogeneous ground surface. Second, splash dispersion frequently happens in tandem with wind transport processes. His model contained the rain splash component, but the wind component was not included in his model. Burie et al. [8] used the SEIR model with a set of PDEs to describe spore dispersal in 2007. The model did not consider the wind and rain-splash borne dispersal. In 2020, Jarroudi et al. proposed a mathematical model to describe wind-borne spore dispersal. Spore dispersal was modeled using the non-local diffusion equation, which includes the non-local term and the dispersal kernel. The dispersal kernel contained the wind component. The outcomes of said work indicated that spore dispersal by wind, depending on the wind velocity, causes spore liberation. Other elements that influence the aerial dissemination of spores across plant fields include the wind direction angle, the weight and shape of spores, and plant architecture. In addition, it was found that the main routes of spore dispersal are wind-borne and rain-splash borne [3]. Moreover, the rain splash component was not included in their model.

Diffusion is a concept found in physics (particle diffusion), chemistry, biology, sociology, economics, and finance. The basic principle of diffusion is the same in all of fields: an object undergoing diffusion grows outward from points or locations where there is a higher concentration of the object. The non-local diffusion equation is described in Equation (1). Let $J:{\mathbb{R}}^N\to \mathbb{R}$ be a nonnegative, radial, continuous function with [9,10]:

$${\int }_{{\mathbb{R}}^N}J\left(r\right)dr=1.$$

Then, we have non-local evolution equations of the form:

$$\frac{\partial u(x,t)}{\partial t}=(J\ast u-u)(x,t)={\int }_{{\mathbb{R}}^N}J(x-y)u(y,t)dy-u(x,t),$$

(1)

where $u(x,t)$ is a density at time t and position x and $J(x-y)$ is the probability of a pathogen migrating from location y to the position x. Then, ${\int }_{{\mathbb{R}}^N}J(y-x)u(y,t)dy=(J\ast u)(x,t)$ is the rate at which individuals are arriving at position x from all other places, and $-u(x,t)=-{\int }_{{\mathbb{R}}^N}J(y-x)u(x,t)dy$ is the rate at which they are leaving location x to travel to all other sites.

This research aims to develop a non-local diffusion equation by including a component that causes spore dissemination by rain splash and changing the infection term based on meteorological factors.

2. Methodology

This research investigates the effects of wind and rain on the dispersal of the spores in rice blast disease. First, we added a component to the non-local diffusion equation that causes spore spread by rain splash (Equation (2)). Second, we changed the proportion of infected plants (Equation (3)) from a constant to a function that varies on environmental parameters, including air temperature and relative humidity. Third, we verified the existence and uniqueness of the developed model’s solutions using Banach’s fixed point theorem. In addition, we estimated the solution by using the explicit and trapezoidal rules. Finally, we used the weather data from the Prachinburi Rice Research Center (PRRC) to simulate the proposed model.

2.1. Model Development

Jarroudi et al. provided a mathematical model of spore dispersal carried by the wind [3]. Assume that both time t and position x are continuous variables with bounded domains in ${\mathbb{R}}^2$, as defined in Equation (2):

$$\begin{array}{c}\hfill \begin{array}{c}\frac{\partial H(x,t)}{\partial t}=-\alpha u(x,t)H(x,t),x\in \Omega ,t>0,\\ \frac{\partial u(x,t)}{\partial t}={\int }_{\Omega }J(x-y)[u(y,t)-u(x,t)]dy+f\left(u\right)(x,t),x\in \Omega ,t>0,\end{array}\end{array}$$

(2)

where $\Omega \subset \mathbb{R}$, $H(x,t)$ and $u(x,t)$ are the densities of healthy host individuals and spore, respectively, at time t and position x. $\alpha$ is the proportion of the plant that becomes infected. The first term of the right-hand side of (2) accounts for the individuals arriving or leaving position x from other locations. $J(x-y)$ is the probability of a pathogen migrating from position y to position x, and $f\left(u\right)(x,t)$ is the production of the spores by infectious hosts at time t and position x. This section is divided into two parts: the first covering the development of a plant disease epidemic and the second covering spore dispersion.

The probability of infecting a susceptible individual can be calculated from the product of the fraction of spore deposition on the crop ($\mu$) and the probability that a spore located on a susceptible individual initiates the infection ($\phi$) shown in Equation (3):

$$\alpha =\mu ·\phi .$$

(3)

The probability that an individual pathogen migrates from position y to position x or the spore dispersal density denoted by J is called the dispersal kernel. The wind velocity components affect the dispersal kernel, which corresponds to the density probability function:

$${\displaystyle J\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\mu }^2}{2\pi v^2}\exp \left(-\frac{\mu }{v}\left|x\right|\right)}\hfill & \mathrm{if}\mu >\gamma ,\hfill \\ {\displaystyle \frac{{\mu }^2}{8\pi v^2}{K}_0\left(\frac{\mu }{v\sqrt{2}}\left|x\right|\right)}\hfill & \mathrm{if}\mu <\gamma ,\hfill \end{array}\right.}$$

(4)

where $\mu$ is the fraction of spore deposition on the crop, v is the wind velocity, $\gamma$ is the probability for wind changing directions, and $K_0$ is the zero-order modified Bessel function of the second kind.

At time t and position x, the spores production can be described as follows:

$$f\left(u\right)(x,t)=\kappa \alpha {\int }_0^t\left(uH\right)(x,t-s)\beta \left(s\right)ds,$$

(5)

where $\kappa$ is the average number of pathogens generated per infected host reaching the end of the sporulation stage, $\alpha$ is the proportion of the plants that becomes infected, and $\beta \left(s\right)$ is the sporulation curve, which can be expressed as Equation (6):

$$\beta \left(s\right)=\left\{\begin{array}{cc}0& \mathrm{if}s<\tau ,\\ {\displaystyle \frac{a^b{(s-\tau )}^{b-1}{e}^{-a(s-\tau )}}{\Gamma \left(b\right)}}& \mathrm{if}s\ge \tau ,\end{array}\right.$$

(6)

where $\tau$ is the latent period, s is the age of infection, and $a,b$ are the shape parameters of the sporulation curve. The age of infection begins when the spore penetrates the host tissue through sporulation and enters the latent stage.

Weather conditions influence pathogen life-cycle characteristics. This study considers two meteorological variables that are important to the condition at the consideration: the air temperature and relative humidity. The weather varies throughout the season; therefore, it is dependent on the time. From the system (2), we adjust the proportion of plants that will become infected (Equation (3)) from a constant value to a function that depends on environmental factors ($\alpha )$.

The probability that a spore on a susceptible individual initiates infection at time t ($\phi \left(t\right)$) is determined by three factors: (i) the maximum infection efficiency, ${\phi }_{max}$; (ii) the infection efficiency based on the temperature at the time t, ${\phi }_T\left(t\right)$; and (iii) the infection efficiency depended on dew period at time t, ${\phi }_D\left(t\right)$. An equation can thus be used to get $\phi \left(t\right)$:

$$\alpha \left(t\right)=\mu \phi \left(t\right)=\mu [{\phi }_{max}·{\phi }_T\left(t\right)·{\phi }_D\left(t\right)].$$

(7)

The infection efficiency depending on temperature at time t [11] can be described in Equation (8):

$${\phi }_T\left(t\right)={\left(\frac{T\left(t\right)-T_{IEmin}}{T_{IEmax}-T_{IEmin}}\right)}^{x_{IE}}{\left(\frac{T_{IEmax}-T\left(t\right)}{T_{IEmax}-T_{IEmin}}\right)}^{y_{IE}},$$

(8)

where $T\left(t\right)$ is the temperature at time t, $T_{IEmax}$ is the maximum temperature, $T_{IEmin}$ is the minimum temperature, and $x_{IE},y_{IE}$ are the shape parameters. The infection efficiency depending on dew period at time t [11] can be described in Equation (9):

$${\displaystyle {\phi }_D\left(t\right)=1-e^{-d(D\left(t\right)-W_{min})},}$$

(9)

where $D\left(t\right)$ is the relative humidity at time t, $W_{min}$ is the minimum wetness period of the infection efficiency, and d is the shape parameter.

Many factors influence spore dispersion by rain splash, including leaf shape, inclination, drop size, initial position [2], and velocity of the raindrop on the leaf surface [12]. When rain falls on the leaf surface, it is fragmented and splashes onto a new susceptible host plant. A raindrop impacts a plant leaf such that large ejected droplets carry more pathogens, while small droplets contain fewer pathogens. In this study, we will focus on fluid fragmentation after rainfall only.

The probability that an individual pathogen migrates from a position to other places by wind can be described in Equation (4). The spore dispersion by wind and rain can be described in Equation (10). When the average flight duration, $1/\mu$, is very small compared with the average duration of the airflow in a fixed direction, $1/\gamma$, the spore dispersal distribution is approximated by the exponential distribution [3,13]. On the other hand, when the average time length during which a spore is wind-borne, $1/\mu$, is greater than the average duration of traveling in a fixed direction, $1/\gamma$, the spore dispersal distribution is approximated by the Bessel function,

$${\displaystyle J\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\mu }^2}{2\pi {(F{v}_1+v_2)}^2}\exp \left(-\frac{\mu }{F{v}_1+v_2}\left|x\right|\right)}\hfill & \mathrm{if}\mu >\gamma ,\hfill \\ {\displaystyle \frac{{\mu }^2}{8\pi {(F{v}_1+v_2)}^2}{K}_0\left(\frac{\mu }{(F{v}_1+v_2)\sqrt{2}}\left|x\right|\right)}\hfill & \mathrm{if}\mu <\gamma ,\hfill \end{array}\right.}$$

(10)

where $v_1$ is the fluid fragmentation velocity after rainfall on the host, $v_2$ is the wind velocity, and F is the probability of the rain during the day. If it rains, $F=1$, this means that spore dispersal is influenced by wind and rain. Furthermore, if there is no rain, $F=0$, spore dispersal is only dependent on the wind.

Therefore, we obtain the developed model as shown in the system (11):

$$\begin{gathered} \frac{\partial H(x,t)}{\partial t}=-\alpha \left(t\right)u(x,t)H(x,t),x\in \Omega ,t>0, \\ \frac{\partial u(x,t)}{\partial t}={\int }_{\Omega }J(x-y)[u(y,t)-u(x,t)]dy+f(u,H)(x,t),x\in \Omega ,t>0, \\ H(x,0)=H_0\left(x\right)\ge 0,x\in \Omega , \\ u(x,0)=u_0\left(x\right)\ge 0,x\in \Omega . \end{gathered}$$

(11)

2.2. Existence and Uniqueness

This section examines the existence and uniqueness of the developed model’s solution. The proof procedure was as follows: (i) find the developed model’s solutions, (ii) define the nonlinear mapping T that relates to the solutions, and (iii) use Banach’s fixed point theorem and Theorem 1 to verify the unique solution.

Step 1: Find the proposed model’s solutions. The results are given as:

$$\begin{array}{ccc}\hfill H(x,t)& =& H_0\left(x\right)exp\left\{-{\int }_0^t\alpha \left(s\right)u(x,s)ds\right\},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill u(x,t)& =& e^{-A\left(x\right)t}{u}_0\left(x\right)+{\int }_0^t{e}^{A\left(x\right)(s-t)}{\int }_{\Omega }J(x-y)u(y,s)dyds+{\int }_0^t{e}^{A\left(x\right)(s-t)}f(u,H)(x,s)ds,\hfill \end{array}$$

(13)

where $A\left(x\right)={\int }_{\Omega }J(x-y)dy$.

Step 2: Define the nonlinear mapping T that corresponding to the solutions.

Let $t_0>0$ and consider Banach space $W_{t_0}=C([0,t_0];L^2(\Omega ))$ with the norm:

$${∥w∥}_{w_{t_0}}=\underset{0\le t\le t_0}{max}{∥w_t∥}_{L^2(\Omega )}.$$

We set $W_{t_0}^M=\{w\in W_{t_0}\mid 0\le w\le M\mathrm{in}\Omega \times (0,\infty )\}$ to be closed in $W_{t_0}$ for every $M>0$ for every function $w_0\in L^2(\Omega )$ satisfying $0\le w_0\le M$ in $\Omega$. Therefore, we define the nonlinear mapping $T_{w_0}$ on $W_{t_0}$ through:

$$\begin{gathered} T_{w_0}\left(w\right)(x,t)={\int }_0^t{e}^{A\left(x\right)(s-t)}{\int }_{\Omega }J(x-y)w(y,s)dyds+e^{-A\left(x\right)t}{w}_0\left(x\right) \\ +{\int }_0^t{e}^{A\left(x\right)(s-t)}f\left(w\right)(x,s)ds. \end{gathered}$$

(14)

Lemma 1.

Let $w_0$ and $z_0$ be functions of $L^2(\Omega )$ satisfying $0\le w_0,z_0\le M$ in Ω. Then, for every $w,z\in W_{t_0}^M$, we have:

$${∥T_{w_0}\left(w\right)-T_{z_0}\left(z\right)∥}_{W_{t_0}}^2\le C(t_0,M){∥w-z∥}_{W_{t_0}}^2+(4M{t}_0+1){∥w_0-z_0∥}_{L^2(\Omega )}^2,$$

where

$$\begin{array}{cc}\hfill C(t_0,M)& =\frac{3}{2}{t}_0^2(1+\frac{2}{3}\kappa \underset{0\le t\le t_0}{max}{H}_0\left(x\right)\left(\frac{4}{3}M{t}_0+1\right)+\frac{8}{3}M\kappa \underset{0\le t\le t_0}{max}{H}_0\left(x\right)(M+1)\hfill \\ \hfill & +\frac{2}{3}{\kappa }^2\underset{0\le t\le t_0}{max}{H}_0^2\left(x\right)t_0\frac{{(2M{t}_0+1)}^3}{6M}).\hfill \end{array}$$

Proof. 

Consider

$$(T_{w_0}\left(w\right)-T_{z_0}\left(z\right))(x,t)={\int }_0^t{e}^{A\left(x\right)(s-t)}{\int }_{\Omega }J(x-y)(w-z)(y,s)dyds+e^{-A\left(x\right)t}(w_0-z_0)\left(x\right)$$

$$+{\int }_0^t{e}^{A\left(x\right)(s-t)}(f\left(w\right)-f\left(z\right))(x,s)ds.$$

(15)

After that, substituting Equation (12) into ${\displaystyle f\left(u\right)(x,t)=\kappa {\int }_0^t\left(uH\right)(x,t-s)\left(\alpha \beta \right)\left(s\right)ds}$, we get:

$$f\left(u\right)(x,t)=\kappa {\int }_0^t\left(\alpha \beta \right)\left(s\right)\left(u\right)(x,t-s)\left[H_0\left(x\right)exp\left\{-{\int }_0^s\alpha \left(\eta \right)u(x,\eta )d\eta \right\}\right]ds.$$

Next, consider

$$\begin{array}{cc}\hfill \left(f\right(w)-f(z\left)\right)(x,s)& =\kappa H_0\left(x\right){\int }_0^s\left(\alpha \beta \right)(s-\eta )[w(x,\eta )exp\left\{-{\int }_0^{\eta }\alpha \left(\xi \right)w(x,\xi )d\xi \right\}\hfill \\ \hfill & -z(x,\eta )exp\left\{-{\int }_0^{\eta }\alpha \left(\xi \right)z(x,\xi )d\xi \right\}]d\eta .\hfill \end{array}$$

Let ${\displaystyle \varphi \left(r\right)=[rw(x,\eta )+(1-r)z(x,\eta )]exp\left\{-{\int }_0^{\eta }\alpha \left(\xi \right)[rw(x,\xi )+(1-r)z(x,\xi )]d\xi \right\}}$. Then, for some $r_0\in (0,1)$ we have:

$$\begin{array}{cc}\hfill \left(f\right(w)-f(z\left)\right)(x,s)& =\kappa H_0\left(x\right){\int }_0^s\left(\alpha \beta \right)(s-\eta )[\varphi \left(1\right)-\varphi \left(0\right)]d\eta \hfill \\ \hfill & =\kappa H_0\left(x\right){\int }_0^s\left(\alpha \beta \right)(s-\eta ){\varphi }^{\prime }\left(r_0\right)d\eta .\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill {\varphi }^{\prime }\left(r\right)& =[rw(x,\eta )+(1-r)z(x,\eta )]exp\left\{-{\int }_0^{\eta }\alpha \left(\xi \right)[rw(x,\xi )+(1-r)z(x,\xi )]d\xi \right\}\hfill \\ \hfill & \times \left[-{\int }_0^{\eta }\alpha \left(\xi \right)\left[(w-z)(x,\xi )\right]d\xi \right]+exp\left\{-{\int }_0^{\eta }\alpha \left(\xi \right)[rw(x,\xi )+(1-r)z(x,\xi )]d\xi \right\}\left[(w-z)(x,\eta )\right].\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \left(f\right(w)-f(z\left)\right)(x,s)& =\kappa H_0\left(x\right){\int }_0^s\left(\alpha \beta \right)(s-\eta )\{[r_{0}w(x,\eta )+(1-r_0)z(x,\eta )]\hfill \\ \hfill & \times [exp\left\{-{\int }_0^{\eta }\alpha \left(\xi \right)[r_{0}w(x,\xi )+(1-r_0)z(x,\xi )]d\xi \right\}\left[-{\int }_0^{\eta }\alpha \left(\xi \right)\left[(w-z)(x,\xi )\right]d\xi \right]\hfill \\ \hfill & +exp\left\{-{\int }_0^{\eta }\alpha \left(\xi \right)[rw(x,\xi )+(1-r)z(x,\xi )]d\xi \right\}\left[(w-z)(x,\eta )\right]\}d\eta .\hfill \end{array}$$

Next, consider $\mid \left(f\right(w)-f(z\left)\right)(x,s)\mid$. Therefore, we obtain:

$$\mid (f\left(w\right)-f\left(z\right))(x,s)\mid \le \kappa H_0\left(x\right)\left[2M{\int }_0^{\eta }\alpha \left(\xi \right)d\xi +1\right]{\int }_0^s\left(\alpha \beta \right)(s-\eta )\left|(w-z)(x,\eta )\right|d\eta .$$

(16)

From Equation (15), using the Cauchy–Schwarz inequality:

$$\begin{array}{cc}\hfill |(T_{w_0}\left(w\right)-T_{z_0}\left(z\right)){(x,t)|}^2& =|{\int }_0^t{e}^{A\left(x\right)(s-t)}{\int }_{\Omega }J(x-y)(w-z)(y,s)dyds\hfill \\ \hfill & +e^{-A\left(x\right)t}(w_0-z_0)\left(x\right)+{\int }_0^t{e}^{A\left(x\right)(s-t)}(f\left(w\right)-f\left(z\right))(x,s)ds{|}^2\hfill \\ \hfill & \le {\int }_0^t{\int }_{\Omega }{J}^2(x-y){(w-z)}^2(y,s)dyds\hfill \\ \hfill & +2{\int }_0^t{\int }_{\Omega }J(x-y)|w-z|(y,s)dyds|w_0-z_0|\left(x\right)+{(w_0-z_0)}^2\left(x\right)\hfill \\ \hfill & +2{\int }_0^t{\int }_{\Omega }J(x-y)|w-z|(y,s)dyds{\int }_0^t|f\left(w\right)-f\left(z\right)|(x,s)ds\hfill \\ \hfill & +2|w_0-z_0|\left(x\right){\int }_0^t|f\left(w\right)-f\left(z\right)|(x,s)ds+{\int }_0^t{(f\left(w\right)-f\left(z\right))}^2(x,s)ds.\hfill \end{array}$$

After that, using inequality (16), we obtain:

$$\begin{array}{cc}\hfill |(T_{w_0}\left(w\right)-T_{z_0}\left(z\right)){(x,t)|}^2& \le \frac{3}{2}{t}_0^2(1+\frac{2}{3}\kappa H_0\left(x\right)\left(\frac{4}{3}M{t}_0+1\right)+\frac{8}{3}M\kappa H_0\left(x\right)(M+1)\hfill \\ \hfill & +\frac{2}{3}{\kappa }^2{H}_0^2\left(x\right)t_0\frac{{(2M{t}_0+1)}^3}{6M}){\int }_0^t{(w-z)}^2(y,s)ds+(4M{t}_0+1){(w_0-z_0)}^2\left(x\right).\hfill \end{array}$$

We integrate on $\Omega$ with respect to x, which implies that:

$${\int }_{\Omega }{\left|T_{w_0}\left(w\right)-T_{z_0}\left(z\right)\right|}^2(x,t)dx\le C(t_0,M)\underset{0\le t\le t_0}{max}{\int }_{\Omega }{(w-z)}^2(x,t)dx+(4M{t}_0+1){\int }_{\Omega }{(w_0-z_0)}^2\left(x\right)dx.$$

Therefore,

$${∥T_{w_0}\left(w\right)-T_{z_0}\left(z\right)∥}_{W_{t_0}}^2\le C(t_0,M){∥w-z∥}_{W_{t_0}}^2+(4M{t}_0+1){∥w_0-z_0∥}_{L^2(\Omega )}^2,$$

(17)

where

$$\begin{array}{cc}\hfill C(t_0,M)& =\frac{3}{2}{t}_0^2(1+\frac{2}{3}\kappa \underset{0\le t\le t_0}{max}{H}_0\left(x\right)\left(\frac{4}{3}M{t}_0+1\right)+\frac{8}{3}M\kappa \underset{0\le t\le t_0}{max}{H}_0\left(x\right)(M+1)\hfill \\ \hfill & +\frac{2}{3}{\kappa }^2\underset{0\le t\le t_0}{max}{H}_0^2\left(x\right)t_0\frac{{(2M{t}_0+1)}^3}{6M}).\hfill \end{array}$$

□

Step 3: Prove the uniqueness solution using Banach’s fixed point theorem [14] and using Theorem 1.

Theorem 1.

For every non-negative $u_0\in L^2(\Omega )$ and every non-negative function $H_0$ defined on Ω, there exists a unique solution $(u,H)$ to system (11) such that:

1.

$u\in C([0,\infty );L^2(\Omega )),0\le u\le M$ and H in Equation (12);

2.

u satisfies

$${\int }_{\Omega }\frac{\partial u(x,t)}{\partial t}dx={\int }_{\Omega }f(u,H)(x,t)dx,$$

(18)

$${\int }_{\Omega }u(x,t)dx={\int }_{\Omega }{u}_0\left(x\right)dx+{\int }_{\Omega }{\int }_0^{t}f(u,H)(x,s)dsdx.$$

(19)

Proof. 

(i) Taking $w_0=u_0,w=u$ and $z_0=z=0$ in Lemma 1, we get:

$${∥T_{u_0}\left(u\right)∥}_{W_{t_0}}^2\le C(t_0,M){∥u∥}_{W_{t_0}}^2+(4M{t}_0+1){∥u_0∥}_{L^2(\Omega )}^2,$$

hence, $T_{u_0}\left(u\right)\in W_{t_0}^M$ for a $t_0$ small enough.

Taking now $w_0=z_0=u_0$ in the above inequality, we obtain that $T_{u_0}$ is a contraction on $W_{t_0}^M$ for a $t_0$ that is small enough.

The existence and uniqueness of the theorem are provided by Banach’s fixed point theorem in the interval $[0,t_0]$. We can start with $u(x,t_0)\in L^2(\Omega )$ and extend the solution to $[0,\infty )$ to get a solution up to $[0,2t_0]$. Iterating this approach produces a solution in $[0,\infty )$.

(ii) From Equation (11), integrating over $\Omega$, we get:

$$\begin{array}{cc}\hfill \frac{\partial u(x,t)}{\partial t}& ={\int }_{\Omega }J(x-y)[u(y,t)-u(x,t)]dy+f(u,H)(x,t)\hfill \\ \hfill {\int }_{\Omega }\frac{\partial u(x,t)}{\partial t}dx& ={\int }_{\Omega }{\int }_{\Omega }J(x-y)u(y,t)dydx-{\int }_{\Omega }{\int }_{\Omega }J(x-y)u(x,t)dydx\hfill \\ \hfill & +{\int }_{\Omega }f(u,H)(x,t)dx.\hfill \end{array}$$

The two initial terms on the right-hand side cancel because J is symmetric. This implies Equation (18). After that, integrating Equation (18) with respect to time t, we obtain:

$$\begin{array}{cc}\hfill {\int }_{\Omega }\frac{\partial u(x,t)}{\partial t}dx& ={\int }_{\Omega }f(u,H)(x,t)dx\hfill \\ \hfill {\int }_0^t{\int }_{\Omega }\partial u(x,t)dx& ={\int }_0^t{\int }_{\Omega }f(u,H)(x,s)dxds\hfill \\ \hfill {\int }_{\Omega }{\int }_0^t\partial u(x,t)dx& ={\int }_{\Omega }{\int }_0^{t}f(u,H)(x,s)dsdx\hfill \\ \hfill {\int }_{\Omega }[u(x,t)-u(x,0)]dx& ={\int }_{\Omega }{\int }_0^{t}f(u,H)(x,s)dsdx.\hfill \end{array}$$

Which implies Equation (19):

$${\int }_{\Omega }u(x,t)dx={\int }_{\Omega }{u}_0\left(x\right)dx+{\int }_{\Omega }{\int }_0^{t}f(u,H)(x,s)dsdx.$$

□

2.3. Numerical Solutions

The finite difference method is solved using the model. Using the domains $\Omega =[0,1]$ and $t=[0,100]$, discretizing time into $N_t$ intervals and spatial into $N_x$ intervals as illustrated in Figure 1, $\delta x=1/N_x$ and $\delta t=1/N_t$ can be used to derive the spatial and time step sizes, respectively.

Consider Equation (11). The first term of the right-hand-side can approximate to:

$${\left({\int }_{\Omega }J(x-y)[u(y,t)-u(x,t)]dy\right)}_i^j\approx \sum _{p=1}^{N_x+1}J(x_i-x_p)(u_p^j-u_i^j)\delta x,$$

(20)

for every $i=1,2,3, \dots ,N_x+1$ and $j=1,2,3, \dots ,N_t+1$.

The second term of the right-hand side is called the spore production, which, as described in Equation (5), is approximated as follows:

$$f{(u,H)}_i^j\approx \kappa \sum _{l=1}^j{\alpha }^l\beta (t_j-t_l){\left(uH\right)}_i^j\delta t.$$

(21)

For every $i=1,2,3, \dots ,N_x+1$ and $j=1,2,3, \dots ,N_t+1$. After that, substituting Equations (20) and (21) into the Equation (11), we get the semi-discrete:

$$\frac{\partial u}{\partial t}=\sum _{p=1}^{N_x+1}J(x_i-x_p)(u_p^j-u_i^j)\delta x+\kappa \sum _{l=1}^j{\alpha }^l\beta (t_j-t_l){\left(uH\right)}_i^j\delta t.$$

(22)

After that, we apply the explicit rule to the left-hand-side of the Equation (22). Finally, we get the numerical solution:

$$\frac{u_i^{j+1}-u_i^j}{\delta t}=\sum _{p=1}^{N_x+1}J(x_i-x_p)(u_p^j-u_i^j)\delta x+\kappa \sum _{l=1}^j{\alpha }^l\beta (t_j-t_l){\left(uH\right)}_i^j\delta t.$$

Rearranging the equation above, we obtain:

$$u_i^{j+1}=u_i^j+\delta t\left(\sum _{p=1}^{N_x+1}J(x_i-x_p)(u_p^j-u_i^j)\delta x+\kappa \sum _{l=1}^j{\alpha }^l\beta (t_j-t_l){\left(uH\right)}_i^j\delta t\right).$$

From Equation (12), we approximate ${\displaystyle {\int }_0^t\alpha \left(s\right)u(x,s)ds}$ by using the multiple segment trapezoidal rule.

$$\begin{array}{cc}\hfill {\int }_0^t\alpha \left(s\right)u(x,s)ds& =\frac{\delta t}{2}\left(\alpha \left(t_0\right)u(x_i,t_0)+\alpha \left(t_j\right)u(x_i,t_j)+2\sum _{l=1}^{j-1}\alpha \left(t_l\right)u(x_i,t_l)\right)\hfill \\ \hfill & \approx \frac{\delta t}{2}\left({\alpha }^0{u}_i^0+{\alpha }^j{u}_i^j+2\sum _{l=1}^{j-1}{\alpha }^l{u}_i^l\right).\hfill \end{array}$$

Therefore, we get:

$${\int }_0^t\alpha \left(s\right)u(x,s)ds=\delta t\left(\frac{{\alpha }^0{u}_i^0+{\alpha }^j{u}_i^j}{2}+\sum _{l=1}^{j-1}{\alpha }^l{u}_i^l\right).$$

(23)

Substituting Equation (23) into Equation (12), we obtained the approximation of $H(x_i,t_j)$ as follows:

$$H_i^j\approx H_0\left(x_i\right)exp\left(-\delta t\left(\frac{{\alpha }^0{u}_i^0+{\alpha }^j{u}_i^j}{2}+\sum _{l=1}^{j-1}{\alpha }^l{u}_i^l\right)\right).$$

Therefore, we obtain that the numerical solutions of the developed model are:

$$\begin{array}{cc}\hfill H_i^{j+1}& =H_0\left(x_i\right)exp\left(-\delta t\left(\frac{{\alpha }^1{u}_i^1+{\alpha }^{j+1}{u}_i^{j+1}}{2}+\sum _{l=2}^j{\alpha }^l{u}_i^l\right)\right),\hfill \\ \hfill u_i^{j+1}& =u_i^j+\delta t\left(\sum _{p=1}^{N_x+1}J(x_i-x_p)(u_p^j-u_i^j)\delta x+\kappa \sum _{l=1}^j{\alpha }^l\beta (t_j-t_l){\left(uH\right)}_i^j\delta t\right),\hfill \\ \hfill H_i^1& =H_0\left(x_i\right)\ge 0,\hfill \\ \hfill u_i^1& =u_0\left(x_i\right)\ge 0.\hfill \end{array}$$

For every $i=1,2,3, \dots ,N_x+1$ and $j=1,2,3, \dots ,N_t$.

2.4. Parameters and Initial Data

The initial conditions $H_0\left(x\right)=1000{(1-x^2)}^2$ and $u_0\left(x\right)=100{(1-x^2)}^2$ where $x\in (0,1)$ are used for this simulation. All parameters used in the simulation are shown in

Table 1. Description and the value of variables and parameters.

Symbol	Description	Value	Source
$\mu$	The fraction of spore deposition on the crop	0.45	[3]
${\phi }_{max}$	The maximum infection efficiency	0.307	[15]
$T_{IEmax}$	The maximum temperature for bed 1	34.65	PRRC.
	The maximum temperature for bed 2	29.8	PRRC.
	The maximum temperature for bed 3	29.2	PRRC.
	The maximum temperature for bed 4	30.25	PRRC.
$T_{IEmin}$	The minimum temperature for bed 1	24.2	PRRC.
	The minimum temperature for bed 2	24.15	PRRC.
	The minimum temperature for bed 3	17.7	PRRC.
	The minimum temperature for bed 4	17.7	PRRC.
$k_{IE}$	The maximum number of infection efficiency	3	[15]
$x_{IE},y_{IE}$	The shape parameters of ${\phi }_T\left(t\right)$	1, 0.6	[15]
$W_{min}$	The minimum wetness period	68	PRRC.
d	The shape parameter of ${\phi }_D\left(t\right)$	0.1	[15]
$v_1$	The velocity of fluid fragmentation after rain falls on the host	7	[2]
$v_2$	The wind velocity	4	[5]
F	The probability of rain in the day	0, 1	-
$\tau$	The latent period	5	[16]
$\kappa$	The average number of pathogen particles produced per host	5000	[15]
	infected that reach the end of sporulation stage
$a,b$	The shape parameters	0.24, 2.86	[3]

.

2.5. Environmental Data

Temperature (T), relative humidity (D), and rain (R) were the three types of environmental data used in this study. We obtained all of our data from the PRRC. There were four beds. For bed 1, the mean temperature, relative humidity, and rain were 27.23 °C, 81.84%, and 6.53 mm, respectively. The temperature was between 24.2–34.65 °C. The relative humidity was between 68–91.5%. In addition, the maximum rain was 71 mm, and the minimum rain was 0 mm. The data are shown in Figure 2a. In bed 2, the mean temperature was 26.98 °C, and a temperature range was seen of 24.15–29.8 °C. The average relative humidity was 81.06%, while the relative humidity ranged from 69.5 to 91.5%. The average rainfall was 4.57 mm, with a high of 46.1 mm and a low of 0 mm. Figure 2b represents the results. The mean temperature in bed 3 was 26.02 °C, while the mean relative humidity and rain were 71.52% and 0.08 mm, respectively. The temperature was between 17.7 and 29.2 °C. The relative humidity ranged from 47 to 84.5%. In addition, the greatest rain was 5.8 mm, and the minimum was 0 mm. Figure 3a illustrates the data. In the last bed, the average temperature was 26.63 °C, with 17.7 to 30.25 °C. The mean relative humidity was 71%, while the relative humidity ranged from 47 to 84.5%. The mean rainfall was 0.16 mm, with a maximum of 9.6 mm and a minimum of 0 mm, as shown in Figure 3b.

3. Results

From the model development part, we received the created model for describing the behavior of spore dispersal by wind and rain. As shown in the model analysis section, the model has a unique solution with sufficient conditions. We calculated the proposed model for rice blast disease using MATLAB software and performed this for 100 days. Moreover, the values of all variables and parameters utilized in the simulation are provided in Table 1. We consider wind and rain, and both influence spore dispersal. This work chooses a wind velocity of 4 ms⁻¹ from Singh et al., 2019 [5], and a velocity of rain fragmentation after impact on the host of 7 ms⁻¹ from Gilet [2], 2015.

Moreover, we assumed that at each time t, only one drop of rain fell on the host with the same velocity. The results of simulations are shown in Figure 4. For bed 1, the density of the healthy host, H, and the density of spore dispersal, u, are shown in Figure 4a,b, respectively. Figure 4c,d, Figure 4e,f, and Figure 4g,h represented bed 2, bed 3, and bed 4, respectively. The x-axis represents time t, which ranges from 0 to 100 days, and the y-axis represents space x. We considered one unit, and the z-axis represents the densities of healthy and spore.

For all beds, we see that the healthy host density rapidly drops after the latent period. On the other hand, the spore density rises to a high level till the end of the process. The four beds of healthy hosts and spore curves tend to be in the same direction, which differs in the density change that is fast or slow, depending on environmental factors, including air temperature and relative humidity.

4. Conclusions

This study developed a model for rice blast disease affected by spore dispersion via wind and rain. Several terms of the model were developed, such as the probability that a spore on a susceptible host initiates infection, which was calculated based on three factors: (i) maximal infection efficiency, (ii) infection efficiency based on temperature, and (iii) infection efficiency based on dew period, and spore dispersion, which included a factor that accounted for spore dispersion due to rain splash. These terms influence the dynamics of daily disease. As a result, following the existence and uniqueness part, the developed model exists as a unique solution. Numerical simulations indicated that environmental data influenced the dynamics of host and spores.

Wind and rain influence the spread of spores, which leads to the rice blast disease epidemic. If the wind velocity and the velocity of rain fragmentation after the impact on the host are high, the likelihood of the spores spreading is high, which causes epidemics that lead to a continuous decline in rice yields.

Based on real scenarios, the current modeling did not divide the host population into healthy, latent, infectious, and removed groups. To improve this modeling, the model may separate the host population from the HLIR model and maybe use real experimental data to improve the parameters such as the fraction of spore deposited on the host because the parameter influences rice blast disease development.