Abstract
In this paper, an ecoepidemiological deterministic model for the transmission dynamics of maize streak virus (MSV) disease in maize plant is proposed and analysed qualitatively using the stability theory of differential equations.The basic reproduction number with respect to the MSV free equilibrium is obtained using next generation matrix approach. The conditions for local and global asymptotic stability of MSV free and endemic equilibria are established. The model exhibits forward bifurcation and the sensitivity indices of various embedded parameters with respect to the MSV eradication or spreading are determined. Numerical simulation is performed and dispalyed graphically to justify the analytical results.
1. Introduction
Maize (Zea mays L.) is grown globally across temperate and tropical zones, spanning all continents [1]. It is the most widely grown and consumed staple crop in Africa with more than 100 million Africans depending on it as their main food source which is annually planted over an area of 15.5 million hectares [2, 3]. In Ethiopia, maize is the most cereal gown which ranks first in yield per hectare and is grown in all 11 administrative regions [4]. It is the primary food, averaging slightly more than 20% of daily caloric intake. About 9 million smallholder farmers grow maize and 75% of the maize produced is consumed as food [5, 6]. The reports of Ethiopian Commodity Exchange show that three-fourth of maize produced is used for household expenditure; only about ten percent is marketed and the remaining is used for seed, in-kind expenses for labor and animal feed [6].
Maize production is constrained by many abiotic and biotic factors of which maize streak disease (MSD) is the major biotic threat in Ethiopia. It is the most destructive and devastating disease of maize in Sub-Saharan Africa which is caused by maize streak virus (MSV; genus Mastrevirus, family Geminiviridae) [3, 7] including Ethiopia [4]. MSV is a major constraint to maize and over 80 other crops species [8] including oats, wheat, sorghum, millet, finger millet, and sugarcane [2, 9]. MSD is a major threat to cereal crops amongst smallholder farmers in Sub-Saharan Africa causing up to US $ 480 million losses annually [10]. MSD is a viral disease which has single-component, circular, ssDNA [2, 3].
MSV has been reported to be the most economically significant causing 100% yield loss if infection occurs in the first three weeks of planting maize [3, 11]. It is irregular in nature and transmitted in a persistent manner by leafhoppers in the genus Cicadulina [2, 5]. Globally, 22 species of Cicadulina leafhoppers have been reported, of which 18 are found in Africa. Cicadulina mbila is the most predominant vector and the most important in the epidemiology of the virus [2] from the 8 known vectors of MSV in the genus. In Ethiopia, five of these 22 known species of Cicadulina have been recorded [4].
Mathematical modeling is an important tool used in analyzing the dynamics of infectious diseases. Several models have been formulated and analyzed to explain the dynamics of plant disease transmission. The authors in [12] investigated the impacts of foliar diseases on maize plant population dynamics from the developed epidemiological mathematical model. They also applied optimal control theory with chemical, cultural, and disease resistance as a control intervention. In [13], the authors derived a continuous epidemiological model of African cassava mosaic virus disease in which the dynamics are within a locality, of healthy and infected cassava and of infective and noninfective whitefly vectors. The authors in [14] reviewed a differential equation model on plants by adjusting to a specific plant disease model which was a general model of [15]. Optimal control theory to a continuous deterministic epidemiological Cassava brown streak disease model with chemical and uprooting as control measures was applied by the author [16]. In [17], the authors proposed and analyzed an SI-SEI maize disease model which is a combination of both the host and the vector population models and has been formulated to study and analyse the dynamics of maize lethal necrosis disease.
Motivated by references [12, 16, 17], in this paper, we present a deterministic model to study and analyze the dynamics of MSV in the maize plant population. We believe that the results of our research work will be useful indicating suitable means of controlling the disease transmission or rather eradicate it. This may ensure maximum maize harvest for farmers for food security. The organization of this paper is as follows. In Section 2, the mathematical analysis of the model including determining the invariant region where the model is mathematically and epidemiologically well posed is presented; basic reproduction number is obtained by using the next generation matrix method; the local stability of the equilibria is obtained by the Jacobian matrix method; and, by the method of Castillo- Chavez, the global stability of the disease-free equilibrium is investigated; we determine the endemic equilibrium point of the model and the local stability of the equilibria by the Jacobian matrix method; and, by the method of constructing a Lyapunov function, the global stability of the endemic equilibrium is investigated. At the end of Section 3, we determine the bifurcation and sensitivity analysis. In Section 4, we give some numerical simulations to prove our theoretical results with a brief discussion. In the final section, there is a conclusion.
2. Model Description and Formulation
The model we introduce consists of two populations: the maize population and the leafhopper vector population. Both populations have two subclasses: susceptible and infected. At time t, let S(t) denote the density of the susceptible maize and I(t) denote the density of the infected maize. The susceptible and infected leafhopper vector densities are denoted by H(t) and Y(t), respectively.
If there is no leafhopper predator population and no disease, the host population grows logistically with intrinsic growth rate and environmental carrying capacity . In the presence of the disease in maize population, the infected host population contributes to the susceptible host population growth towards the carrying capacity . The susceptible leafhopper vectors are recruited at rate q and move to infected leafhopper subgroup by eating infected maize plant at a rate and its natural death rate is . The infected leafhopper has a natural death rate . The disease spreads to the susceptible host when it comes in contact with the MSV pathogen infected leafhopper vector. Susceptible host plants move to the infected class following contact with infected leafhopper at a per capita rate . The host, once became infected, never recovers and gives no or very low yield of maize. The infected maize plants have death rate due to the disease. Furthermore, the disease can not be transmitted horizontally and vertically in both populations and it is not genetically inherited. The predation functional response of the leafhopper towards susceptible maize is assumed to follow Michaelis-Menten kinetics and is modelled using a Holling type II [18] functional form with predation and infection coefficients , and half saturation constants A and C. All the parameters and their descriptions are listed in Table 1.
With regard to the above considerations, we have the compartmental flow diagram shown in Figure 1. From the flow chart (Figure 1), the model will be governed by the following system of differential equation equations:With the initial condition
3. Model Analysis
3.1. Positivity of Solutions
For model (1) to be ecoepidemiologically meaningful and well posed, it is necessary to prove that all solutions of system with positive initial data will remain positive for all times . This will be established by the following theorem.
Theorem 1. Let . Then the solution set of system (1) is positive for all .
Proof. From the first equation of the system Then we have As t approachers , we obtain . By using the same procedure, we obtained Thus the model is meaningful and well posed. Therefore, it is sufficient to study the dynamics of the model in .
3.2. Invariant Region
Let us determine a region in which the solution of model(1) is bounded. For this model the total maize population is . Then, after differentiating with respect to time and substituting the expression of , we obtain where and . Then After solving Eq. (7) and evaluating it as , we got Similarly, for leafhopper population , we get Where . Then After solving Eq. (10) and evaluating it as , we got Therefore the feasible solution set for the MSV model given by is positively invariant, inside which the model is considered to be epidemiologically meaningful and mathematically well posed.
3.3. Disease-Free Equilibrium Point (DFE)
The disease-free equilibrium of model (1) is obtained by equating all equations of the model to zero and then letting and . Then we get
3.4. Basic Reproduction Number
We compute the basic reproduction number for the model, to analyze the stability of the equilibrium points. The basic reproduction number, , measures the expected number of secondary infections that result from one newly infected individual introduced into a susceptible population [16]. We calculate the basic reproduction number of the system by applying the next generation operator approach as laid out by [19] and so it is the spectral radius of the next-generation matrix. The first step to get is rewriting the model equations starting with newly infective classes:Then, by the principle of next-generation matrix, we obtained
Therefore, the basic reproduction number is given as is a threshold parameter that represents the average number of infected vectors and infected hosts caused by a cross-infection of one infectious maize plant and one infectious leafhopper vector when the other population consists of only susceptible population [19]. Two generations are required for transmission of MSV to take place in the maize field; that is why the square root is found in , that is, from an infectious maize plant to a susceptible leafhopper vector and then from an infectious leafhopper vector to susceptible maize [16]. It is too clear when is rewritten as follows: Where(i) is the maize plants contribution when they infect the leafhopper, and(ii) is the contribution of the leafhopper population when it infects maize plants.
3.5. Local Stability of DFE
Theorem 2. The DFE point is locally asymptotically stable if and unstable if .
Proof. To proof this theorem let us first find the Jacobian matrix of system (1): Evaluating Eq. (18) at the disease-free equilibrium , we get From the Jacobian matrix we obtained a characteristic polynomial as Where From Eq. (20), we see that From the last expression, that is we applied Routh-Hurwitz criteria, and by the principle Eq. (23) has strictly negative real root if and . Clearly we see that because it is the sum of positive parameters and also Hence the DFE is locally asymptotically stable if .
3.6. Global Stability of DFE
To investigate the global stability of DFE, we used technique implemented by Castillo-Chavez and Song [20] as done in the paper [16]. Thus we rewrite our model (1) in the form where denotes uninfected populations and denotes the infected population. represents the disease-free equilibrium of this system. is a globally asymptotically stable equilibrium for the model if it satisfies conditions (i) and (ii) below:(i)For is globally asymptotically stable.(ii)
where is an M-matrix (the off diagonal elements are nonnegative) and is also the Jacobian of G(X,Z) taken in (I, Y) and evaluated at . If system (25) satisfies the above conditions, then the following theorem holds.
Theorem 3. The equilibrium point of system (25) is globally asymptotically stable if and conditions (i) and (ii) are satisfied.
Proof. From system (1) we can get and :Now we consider the reduced system from condition (i) is a globally asymptotically stable equilibrium point for the reduced system . This can be verified from the solution of the first equation in Eq. (27); we obtain S(t) = which approaches K as and from the second equation of Eq. (27) we get which approaches as . We note that this asymptomatic dynamics is independent of the initial conditions in ; therefore the convergence of the solutions of the reduced system (27) is global in . Now we compute Then, can be written as and we want to show , which is obtained as Here and . Hence it is clear that for all (X,Z) . Therefore, this proves that DFE is globally asymptotically stable when .
3.7. The Endemic Equilibrium Point
In the presence of MSD, the model has an equilibrium point called endemic equilibrium point denoted by . is the steady state solution where MSD persist in the population of maize plants. It can be obtained by equating each equation of the model equal to zero; that is,Then we obtain and is the positive root of the equation where
3.8. Local Stability of Endemic Equilibrium
Theorem 4. The endemic equilibrium of system (1) is locally asymptotically stable in if the following conditions hold for :
Proof. Let us first obtain the Jacobian matrix of system (1): Evaluating this at the endemic equilibrium , we get where The characteristic equation of the Jacobian matrix is given bywhere The sufficient conditions for are as follows:Thus, according to the Routh Hurwitz criterion, the characteristic equation (39) will have negative roots or imaginary roots with negative real part for ; the endemic equilibrium is locally asymptotically stable.
3.9. Global Stability of Endemic Equilibrium
Theorem 5. If , the endemic equilibrium of the model (1) is globally stable.
Proof. To establish the global stability of the endemic equilibrium , we consider the following Lyapunov function:where are to be chosen properly such that and for all .
By direct calculation, the derivative of along the solution curve of system (1) yields Now substituting equations of model (1), we get By rearranging we obtainWe now choose Thus, and an endemic equilibrium point is globally stable. Also , if and only if . Therefore, the largest compact invariant set in is the singleton , where is the endemic equilibrium of the system (1). By LaSalle’s invariant principle [21], it implies that is globally asymptotically stable in .
3.10. Bifurcation Analysis
A bifurcation is a qualitative change in the nature of the solution trajectories due to a parameter change. The point at which this change takes place is called a bifurcation point. At the bifurcation point, a number of equilibrium points, or their stability properties, or both, change. We investigate the nature of the bifurcation by using the method introduced in [22], which is based on the use of the center manifold theory in [22].
Theorem 6 (Castillo-Chavez & Song [22]). Let us consider a general system of ODE’s with a parameter : where is an equilibrium point for the system in Eq. (47). That is for all . Assume the following.
: is the linearization matrix of the system given by (47) around the equilibrium 0 with evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts.
: Matrix A has a nonnegative right eigenvector and a left eigenvector corresponding to the zero eigenvalue. Let be the component of and The local dynamics of (52) around 0 are totally determined by and . (i). When with , 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when is unstable and there exists a negative, locally asymptotically stable equilibrium.(ii). When with , 0 is unstable; when is locally asymptotically stable equilibrium, and there exists a positive unstable equilibrium.(iii). When with is unstable, and there exists a locally asymptotically stable negative equilibrium; when is stable, and a positive unstable equilibrium appears.(iv). When changes from negative to positive, changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable. In particular, if and , then the bifurcation is forward; if and , then the bifurcation is backward. Using this approach, the following result may be obtained.
Theorem 7. The model in system (1) exhibits forward bifurcation at .
Proof. We prove, using center manifold theorem [22], the possibility of bifurcation at . Let , , and . In addition, using vector notation , and , with , then model in system (1) is rewritten in the formWe consider the predation and transmission rate as bifurcation parameters so that if The disease-free equilibrium is given by . Then the linearizion matrix of Eq. (49) at a disease-free Equilibrium is given by The right eigenvector, , associated with this simple zero eigenvalue can be obtained from . The system becomesFrom Eq. (52) we obtain Here we have taken into account the expression for . Next we compute the left eigenvector, ; associated with this simple zero eigenvalue can be obtained from and the system becomesFrom equation of Eq. (54), we obtain Here we have taken into account the expression for , where is calculated to ensure that the eigenvectors satisfy the condition . Since the first and third components of are zero, we do not need the derivatives of and . From the derivatives of and , the only ones that are nonzero are the following:and all the other partial derivatives of and are zero. The direction of the bifurcation at is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given, respectively, byHence As the coefficient is always positive and the sign of the coefficient is negative, MSD model exhibits a forward bifurcation and there exists at least one stable endemic equilibrium when . Using expression for in the endemic equilibrium, we plotted a forward bifurcation diagram in Figure 7. We used a set of estimated and assumed parameters in Table 3.
3.11. Sensitivity Analysis of Model Parameters
We carried out sensitivity analysis, on the basic parameters, to check and identify parameters that can impact the basic reproductive number. Sensitivity analysis notifies us on how significant each parameter is to disease transmission. To go through sensitivity analysis, we followed the approach defined by [23] like in [24]. This technique develops a formula to obtain the sensitivity index of all the basic parameters, defined as follows.
Definition 8. The normalized forward sensitivity index of a variable, , that depends differentiably on a parameter, , is defined as for represents all the basic parameters. Here we have . For example the sensitivity index of to is And we do this in a similar fashion for the remaining parameters.
3.12. Interpretation of Sensitivity Indices
The sensitivity indices of the basic reproductive number with respect to main parameters are found in Table 2. Those parameters that have positive indices show that they have great impact on expanding the disease in the community if their values are increasing. Due to the reason that the basic reproduction number increases as their values increase, it means that the average number of secondary cases of infection increases in the community. And also those parameters in which their sensitivity indices are negative have an effect of minimizing the burden of the disease in the community as their values increase while the others are left constant. And also as their values increase, the basic reproduction number decreases, which leads to minimizing the endemicity of the disease in the community.
4. Numerical Simulation
Numerical simulations of the model (1) are carried out, in order to illustrate some of the analytical results of the study. A set of reasonable parameter values is given in Table 3. These parameter values were obtained from literature and some of them were assumed. We used S(0) = 1000, I(0) = 20, H(0) = 100, Y(0) = 0, as initial values and parameter values in Table 3 for simulation of MSV model in addition to parameter values in Table 3.
From the left-hand side of Figure 2, the susceptible maize population decelerates exponentially to acquire endemic equilibrium level as they die due to infected leafhopper vector population. The infected maize population assumes parabolic curve as it increases exponentially to a certain maximum point before exponential deceleration to the certain endemic level.
From the right-hand side of Figure 2, the susceptible vectors decrease exponentially due to natural death and acquisition of infestation from severely infected maize and MSV from the environment and finally acquire the endemic equilibrium level and the infected vectors form a parabolic curve as they do raise and drop exponentially to the endemicity level.
Figure 3 shows the simulation of infected maize and susceptible maize for different value of . We can see from the figure that, increasing the infection and predation rate of infected leafhopper on susceptible maize , the basic reproduction number increases, which leads to an increase in the number of infective maize and on the other hand the number of susceptible maize population decrease.
Figure 4 shows the simulation of infected maize and infected leafhopper for different value of . We can see from the figure that, increasing the infection and predation rate of susceptible leafhopper on infected maize , the basic reproduction number increases, which will lead to an increase in the number of the infected maizes as well as the number of infected leafhoppers.
Figure 5 shows simulation of infected leafhopper and susceptible leafhopper for a different value of and . We can see that increasing the death rate of the leafhopper population reduces the basic reproduction number. Due to the indirect relation of and and it leads to a decrease in the population of the leafhopper.
Figure 6 shows simulation of infected maize for different value of . We can see that increasing the death rate of infected and infectious maize reduces the reproduction number. This leads to a decrease in the infection rate of maize.
5. Discussions and Conclusions
In this paper, we have proposed and analysed an ecoepidemiological mathematical model of MSV. We considered a Holling type II functional response which is biologically realistic. We showed that the system was uniformly bounded and positive. We found the disease-free and endemic equilibrium points and their local and global stability analysis has been investigated. The bifurcation analysis of the model is shown. The model analysis also shows the sensitivity of parameters to the disease persistence and dying out.
Finally, analytical results were confirmed by numerical simulation with realistic parameter values. We showed that increasing the infection and predation rates, and , makes an increase of basic reproduction number which leads to the increase of the number of infected maize population. However, increasing death rate of infected maize and leafhopper population decreases the reproduction number which in turn means that the disease dies out from the maize population. Thus, from the results of this paper, control intervention strategies reduces the disease infection of maize population. The model shows that the spread of the disease largely depends on the infection and predation rates and ; therefore efforts should be made to minimize the contact of infected maize and susceptible leafhopper and MSV infected maize should be treated either using insecticide chemical to reduce the infection rate of leafhoppers and it should be done before the arrival of leafhopper or uprooting and burning the infected maize from the field. This implies that, to get the best and cost-effective control strategy, we should apply optimal control theory. Thus, we come next with a paper applying the optimal and cost-effective strategies to identify the best and cost-effective strategy for this model.
Data Availability
The data supporting this deterministic model are from previous published articles and they have been duly cited in this paper. These published articles are cited in Table 3 and relevant places in this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by Pan African University Institute of Basic Sciences Technology and Innovation (PAUSTI). We would like to express our appreciation for the support.