Based on the sensitivity analysis of the basic reproduction numbers, the model 1 extended to an optimal control model to hinder CSD transmission and investigate the ways that are effective in controlling the disease.We apply control strategies on the model with the following assumptions. The first strategy is prevention strategy \((u_{1})\) that protect susceptible from contacting the disease using cultural, biological or disease resistance seeds. Hence, it reduces or eradicate infected -susceptible faba bean contacts by a factor \((1-u_{1}(t))\). The control function \(u_{2}(t)\) represents the control effort on the uprooting and burning of infectious faba bean individuals. When infected faba bean individuals in the field are uprooted and burned, it reduces pathogen -susceptible faba bean contacts by a factor \((1-u_{2}(t))\) and it increases their removal rate by \(u_{2}(t)\). The infected and exposed population removed from the area by \(u_{2}(t)\). Due to this infected and exposed faba bean the pathogen in the environment is removed by a factor (1- \(u_{2}(t))\).A third control variable \(u_{3}(t)\) for the chocolate spot pathogen population. The insecticide chemical harms the entire population of by raising their mortality rate by \(u_{3}(t)\). On the time interval \([0, t_f]\), the control functions are performed. We used Pontryagin’€™s Maximum Principle to determine the situations under which disease eradication can be attained in a finite moment. After incorporating the assumptions of the controls \(u_{1}\), \(u_{2}\) and \(u_{3}\) in CSD model (1) , we obtain the optimal control model:$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ccl} \frac{dS}{dt} & = & rS(1-\frac{N}{K})-(1-u_{2})\tau _{2}S-(1-u_{1})\tau _{1}S \\ \frac{dE}{dt} & = & (1-u_{2})\tau _{2}S-(1-u_{1})\tau _{1}S-(\mu +\eta +\alpha +u_{2})E \\ \frac{dI}{dt} & = & \alpha E-(\mu +\eta +u_{2})I \\ \frac{dP}{dt} & =& \eta (E+ I)-(\delta +u_{3})P. \end{array} \end{array}\right. } \end{aligned}$$
(8)
The ultimate aim is to find the optimal level of the intervention targeted to minimize infection and cost of the controls. In order to achieve this, the following objective functional is considered:$$\begin{aligned} J= \int \limits _{0}^{t_f}\left[ a_{1}E+a_{2}I+a_{3}P+\frac{1}{2}(w_{1}u_{1}^2+w_{2}u_{2}^2+w_{3}u_{3}^2)\right] dt, \end{aligned}$$
(9)
where; \(t_{f}\) is the final time, \(a_{1}\), \(a_{2}\) and \(a_{3}\) are balancing constants of E, I and P respectively, while \(w_{1}\), \(w_{2}\) and \(w_{3}\) are weight constants for each individual control measure. The cost control functions is considered a quadratic form due to the fact that cost is nonlinear in its nature. Thus, we are looking optimal controls \(u^*_ {1}\),\(u^*_ {2}\) and \(u^*_ {3}\) such that:$$\begin{aligned} J(u^*_ {1},u^*_ {2}, u^*_ {3}) = min\{{J(u_{1},u_{2}, u_{3}) | (u_{1},u_{2},u_ {3}) \in U }\}. \end{aligned}$$
(10)
where; \(U = \{(u_{1},u_{2},u_{3})~|~u_{i}(t)\) are measurable with \(0 \le u_{i}(t) \le 1,i= 1,2,3\) for \(t\in [0,t_ {f}]\}\) is the closed set.Existence of an optimal control
Theorem 2
Given the objective functional J in (9),defined on the control set U, and subject to the state system(1)with non-negative initial conditions at t=0,there exists an optimal control triple \(u^{*}=(u^{*}_i)\) such that \(J(u^{*}) = min\{{J(u_{i}) | u_{i} \in U }\}(i=1,2,3)\).
Proof
Let the control set \(U=[0, 1]^3\), \(V=(u_{1},u_{2},u_{3}) \in U\) and \(X=(S,E,I, P)\). The proof of Theorem 1 is based on satisfying the following properties28,29:
(1)
Convexity and closure of the control set U.
(2)
Boundedness of the state system by a linear function in the state and control variables.
(3)
Convexity of the integrand of the objective functional with respect to the control.
(4)
There exist constants \(d_{1},d_{2}>0\) and \(d_3>1\) such that the Lagrangian is bounded below \(c_{1}\left( \sum _{i=0}^{3} u_{i}^2\right) ^{\frac{c_{3}}{2}}- c_{2}.\)
Thus to show the above properties, lets start:
(1)
Given that the control set \(U=[0, 1]^3\) ,then U is closed by definition. Further, for any two arbitrary points \(x,q\in U\),where \(x=(x_1,x_2,x_3)\)and \(q=(q_1,q_2,q_3)\). It follows, by the definition of a convex set [?], that $$\vartheta x_{i}+(1-\vartheta )q_{i} \in [0, 1]^{3},\forall \vartheta \in [0, 1],i=1,2,3$$ Thus \(\vartheta x_{i}+(1-\vartheta )q_{i} \in U\) implying that U is convex.
(2)
By definition, each right hand side of system (8) is continuous. All variables are bounded on [0, 1]. By definition, each right hand side of system (8) is continuous and can be written as a linear function ofuwith coefficients depending on time and state. Furthermore, the fact that all variables S, E, I, P and u are bounded on [0, 1] implies the rest of the hypothesis holds.
(3)
The integrand of the objective functional (9) is the Lagrangian of the form $$\begin{aligned} L(t,X,V)&= a_{1}E+a_{2}I+a_{3}P+\frac{1}{2}\sum _{i=0}^{3} (w_{i}u_{i}^2\\ &=g(t,x)+h(t,V) \end{aligned}$$ It suffices to show that \(h(t,V)=\frac{1}{2}\sum _{i=0}^{3} (w_{i}u_{i}^2\) is convex on the control variable V. Clearly,h(t, V)is a finite linear combination with positive coefficients of the functions \(z_i(u)=\frac{1}{2}u_{i}^2\). Hence, it is more convenient to show that the function \(z:U\longrightarrow \mathbb {R}\) defined by \(z(u)=\frac{1}{2}u^2\) is convex. To do this,let \(x,q \in U\) and \(\vartheta \in [0, 1]\). Then, $$\begin{aligned} z(\vartheta x+(1-\vartheta )q)-(\vartheta z(x)+(1-\vartheta )z(q ))&\\=\frac{1}{2}(\vartheta x+(1-\vartheta )q)^{2}-\frac{1}{2}(\vartheta x^{2}+(1-\vartheta )q^2)&\\= \frac{1}{2}(\vartheta ^{2}-\vartheta )(x-q)^{2}\le 0 \end{aligned}$$ Hence, \(z(\vartheta x+(1-\vartheta )q)\le \vartheta z(x)+(1-\vartheta )z(q )\), satisfying the definition of a convex function [?].
(4)
$$\begin{aligned} L(t,X,V)&= a_{1}E+a_{2}I+a_{3}P+\frac{1}{2}\sum _{i=0}^{3} (w_{i}u_{i}^2\\ &\ge \frac{1}{2}\sum _{i=0}^{3} (w_{i}u_{i}^2 \ge c_{1}\left( \sum _{i=0}^{3} u_{i}^2\right) ^{\frac{c_{3}}{2}}- c_{2} \end{aligned}$$
where \(c_1= \frac{1}{2}min\{w_{i},i=1,2,3\},c_2>0\) and \(c_{3}=2\). This completes the proof and the optimal control u exists. \(\square\) Hamiltonian and optimality systemThe necessary conditions that the optimal control should satisfy are derived from Pontryagin’s maximum principle27. This principle translates the system in equation (8) and equation (9) into a problem of minimizing point-wise Hamiltonian, with regard to \(u_{1}(t), u_{2}(t)\) and \(u_{3}(t)\) and it is given by:$$\begin{aligned} H&=a_{{1}}E+a_{{2}}I+a_{{3}}P+\frac{1}{2}\,w_{{1}}{u_{{1}}}^{2}+\frac{1}{2}\,w_{{2}}{u_{{ 2}}}^{2}+\frac{1}{2}\,w_{{3}}{u_{{3}}}^{2}\\ &\quad +\lambda _{{1}}\left[ rS \left( 1-{\frac{N}{K}} \right) -(1-u_{2})\tau _{2}S-(1-u_{1})\tau _{1}S \right] \\ &\quad +\lambda _ {{2}}\left[ (1-u_{2})\tau _{2}S+(1-u_{1})\tau _{1}S – \left( \mu +\alpha + \eta +u_{{2}} \right) E\right] \\ &\quad +\lambda _{{3}}\left[ \alpha \,E- \left( \mu +\eta +u_{{2 }} \right) I\right] +\lambda _{{4}}\left[ \eta \, \left( E+I \right) – \left( \delta + u_{{3}} \right) P\right] . \end{aligned}$$where; \(~~\lambda _{i}, i =1, 2, 3, 4\) are the adjoint variable functions to be determined. Because of the convexity of the integrand of J with respect to \(u_1\), \(u_2\) and \(u_3\), a priori boundaries of the state solutions and the Lipschitz property of the state system relating to the state variables, the existence of the optimal control triple was proved using woke of30.
Theorem 3
Suppose we have an optimal controls \(u^{*}_{1}\), \(u^{*}_{2}\), \(u^{*}_{3}\) and S, E, I, P solutions of the respective state system that minimizes J over U, there exist adjoint variables, \(\lambda _{1},…,\lambda _{4}\) such that:$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d\lambda _{1}}{dt}& =\lambda _{{1}}[{\frac{rS}{k}}+{\frac{ \left( 1-u_{{2}} \right) \beta _ {{1}}P}{A+P}}+{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}I}{N}}-r \left( 1-{\frac{N}{K}} \right) -{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{{N}^{2}}}]+\lambda _{{2}}[{\frac{ \left( 1-u_{ {1}} \right) \beta _{{2}}SI}{{N}^{2}}}-{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}P}{A+P}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{ {2}}I}{N}}] , \\ \frac{d\lambda _{2}}{dt} & =-c_{1}+\lambda _{{1}}[{\frac{rS}{K}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{ N ^{2}}}]+\lambda _{{2}}[{ \frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{ N ^{2}}}+\alpha +\eta +u_{{2}}+\mu ]-\lambda _{{3}}(\alpha )-\lambda _{{4}}(\eta ),\\ \frac{d\lambda _{3}}{dt}& =-c_{2}+\lambda _{{1}}[{\frac{rS}{K}}+{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}S}{N}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{{2 }}SI}{{N}^{2}}}]+\lambda _{{2}}[{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{{N}^{2}}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}S}{N} }]+\lambda _{{3}}(\mu +\eta +u_{{2}})+\lambda _{{4}}(\eta ) ,\\ \frac{d\lambda _{4}}{dt}& =-c_{3}+\lambda _{{1}}[{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}S }{A+P}}-{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}SP}{ \left( A+P \right) ^{2}}}]+\lambda _{{2}}[-{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}S}{A+P}}+{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}SP}{ \left( A+P \right) ^{2}}}]+\lambda _{{4}}(\delta +u_{{3}}), \end{array}\right. } \end{aligned}$$with transversality conditions, \(\lambda _{i}(t_{f} )= 0, i = 1,. . . ,4\). Furthermore, the optimal control is \((u^*_{1}, u^*_{2}, u^*_{3})\) characterized by$$\begin{aligned} u^*_{1}&= max\{0, min(1,\frac{\beta _{2}SI}{w_{1}N}(\lambda _{2}-\lambda _{1}))\},\\ u^*_{2}&= max\{0, min(1,\frac{\beta _{1}SP(\lambda _{2}-\lambda _{1})+(A+P)(\lambda _{2}E+\lambda _{3}I)}{w_{2}(A+P)})\},\\ u^*_{3}&= max\{0, min(1,\frac{\lambda _{4}P}{w_{2}})\}. \end{aligned}$$
Proof
The adjoint equation and transversality conditions are standard results obtained from the principle27. We differentiated the Hamiltonian in equation above with respect to states S, E, I and P respectively as,$$\begin{aligned} \frac{d\lambda _{1}}{dt}&=-\frac{dH}{dS}\\ \frac{d\lambda _{2}}{dt}&=-\frac{dH}{dE}\\ \frac{d\lambda _{3}}{dt}&=-\frac{dH}{dI}\\ \frac{d\lambda _{4}}{dt}&=-\frac{dH}{dP} \end{aligned}$$Then, we got the adjoint equations as:$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d\lambda _{1}}{dt}& =\lambda _{{1}}[{\frac{rS}{k}}+{\frac{ \left( 1-u_{{2}} \right) \beta _ {{1}}P}{A+P}}+{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}I}{N}}-r \left( 1-{\frac{N}{K}} \right) -{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{{N}^{2}}}]+\lambda _{{2}}[{\frac{ \left( 1-u_{ {1}} \right) \beta _{{2}}SI}{{N}^{2}}}-{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}P}{A+P}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{ {2}}I}{N}}] , \\ \frac{d\lambda _{2}}{dt} & =-c_{1}+\lambda _{{1}}[{\frac{rS}{K}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{ N ^{2}}}]+\lambda _{{2}}[{ \frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{ N ^{2}}}+\alpha +\eta +u_{{2}}+\mu ]-\lambda _{{3}}(\alpha )-\lambda _{{4}}(\eta ),\\ \frac{d\lambda _{3}}{dt}& =-c_{2}+\lambda _{{1}}[{\frac{rS}{K}}+{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}S}{N}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{{2 }}SI}{{N}^{2}}}]+\lambda _{{2}}[{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}SI}{{N}^{2}}}-{\frac{ \left( 1-u_{{1}} \right) \beta _{{2}}S}{N} }]+\lambda _{{3}}(\mu +\eta +u_{{2}})+\lambda _{{4}}(\eta ) ,\\ \frac{d\lambda _{4}}{dt}& =-c_{3}+\lambda _{{1}}[{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}S }{A+P}}-{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}SP}{ \left( A+P \right) ^{2}}}]+\lambda _{{2}}[-{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}S}{A+P}}+{\frac{ \left( 1-u_{{2}} \right) \beta _{{1}}SP}{ \left( A+P \right) ^{2}}}]+\lambda _{{4}}(\delta +u_{{3}}), \end{array}\right. } \end{aligned}$$with transversality conditions, \(\lambda _{i}(t_{f} )= 0, i = 1,…,4\).
Moreover, the optimal control characterization is determined by solving the following partial differential equations:$$\begin{aligned} \frac{\partial H}{\partial u_{i}}&=0, \end{aligned}$$for \(u^*_{i}\) where \(i=1,2,3\). Thus, we got the characteristic equation in standard control arguments form involving the bounds on the controls as follows:$$\begin{aligned} u_{i}^*= {\left\{ \begin{array}{ll} 0, & \text { if }\psi _{i}\le 0\\ \psi _{i}, & \text { if } 0<\psi _{i}<1\\ 1, & \text { if }\psi _{i}\ge 1, \end{array}\right. } \end{aligned}$$for \(i =1, 2, 3\) and where$$\begin{aligned} \psi _{1}&=\frac{\beta _{2}SI}{w_{1}N}(\lambda _{2}-\lambda _{1})\\ \psi _{2}&=\frac{E\lambda _{2}+(\alpha E+I)\lambda _{3}+\eta (E+I)\lambda _{4}}{w_{2}}\\ \psi _{3}&=\frac{\beta _{1}SP(\lambda _{2}-\lambda _{1})+P(A+P)\lambda _{4}}{w_{3}(A+P)} \end{aligned}$$In compacted notation:$$\begin{aligned} \begin{array}{ccl} u^*_{1} & =& max\{0, min(1,\psi _{1})\}, \\ u^*_{2} & =& max\{0, min(1,\psi _{2})\}, \\ u^*_{3} & =& max\{0, min(1,\psi _{3})\}. \end{array} \end{aligned}$$
(11)
This completes the proof. Next, we check the optimal control and we find that it is indeed a minimum one by checking the condition \(\frac{\partial ^2H}{\partial u^2}>0\). The second derivative of the Hamiltonian is:$$\frac{\partial ^2H}{\partial u^2} = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\\ \end{pmatrix}$$Since this matrix is positive definite the optimal control is a minimizer.
The optimality system is formed from the optimal control system (the state system) and the adjoint variable system by incorporating the characterized control set and initial & transversal condition:$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{dS}{dt} = rS(1-\frac{N}{K})-(1-u^{*}_{2})\frac{\beta _{1}SP}{A+P}-(1-u^{*}_{1})\frac{\beta _{2}SI}{N} \\ \frac{dE}{dt} = (1-u^{*}_{2})\frac{\beta _{1}SP}{A+P}+(1-u^{*}_{1})\frac{\beta _{2}SI}{N}-(\mu +\eta +\alpha +u^{*}_{2})E \\ \frac{dI}{dt} = \alpha E-(\mu +\eta +u^{*}_{2})I \\ \frac{dP}{dt} =\eta (E+ I)-(\delta +u^{*}_{3})P,\\ \frac{d\lambda _{1}}{dt}=\lambda _{{1}}[{\frac{rS}{k}}+{\frac{ \left( 1-u^{*}_{{2}} \right) \beta _{{1}}P}{A+P}}+{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2}}I}{N}}-r \left( 1-{\frac{N}{K}} \right) -{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2}}SI}{{N}^{2}}}]+\lambda _{{2}}[{\frac{ \left( 1-u^{*}_{ {1}} \right) \beta _{{2}}SI}{{N}^{2}}}-{\frac{ \left( 1-u^{*}_{{2}} \right) \beta _{{1}}P}{A+P}}-{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{ {2}}I}{N}}],\\ \frac{d\lambda _{2}}{dt} =-c_{1}+\lambda _{{1}}[{\frac{rS}{K}}-{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2}}SI}{ N ^{2}}}]+\lambda _{{2}}[{ \frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2}}SI}{ N ^{2}}}+\alpha +\eta +u^{*}_{{2}}+\mu ]-\lambda _{{3}}(\alpha )-\lambda _{{4}}(\eta ),\\ \frac{d\lambda _{3}}{dt}=-c_{2}+\lambda _{{1}}[{\frac{rS}{K}}+{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2}}S}{N}}-{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2 }}SI}{{N}^{2}}}]+\lambda _{{2}}[{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2}}SI}{{N}^{2}}}-{\frac{ \left( 1-u^{*}_{{1}} \right) \beta _{{2}}S}{N} }]+\lambda _{{3}}(\mu +\eta +u^{*}_{{2}})+\lambda _{{4}}(\eta ),\\ \frac{d\lambda _{4}}{dt}=-c_{3}+\lambda _{{1}}[{\frac{ \left( 1-u^{*}_{{2}} \right) \beta _{{1}}S }{A+P}}-{\frac{ \left( 1-u^{*}_{{2}} \right) \beta _{{1}}SP}{ \left( A+P \right) ^{2}}}]+\lambda _{{2}}[-{\frac{ \left( 1-u^{*}_{{2}} \right) \beta _{{1}}S}{A+P}}+{\frac{ \left( 1-u^{*}_{{2}} \right) \beta _{{1}}SP}{ \left( A+P \right) ^{2}}}]+\lambda _{{4}}(\delta +u^{*}_{{3}}),\\ u^{*}_{1} = max\{0, min(1,\frac{\beta _{2}SI}{w_{1}N}(\lambda _{2}-\lambda _{1}))\},\\ u^*_{2}= max\{0, min(1,\frac{\beta _{1}SP(\lambda _{2}-\lambda _{1})+(A+P)(\lambda _{2}E+\lambda _{3}I)}{w_{2}(A+P)})\},\\ u^*_{3} = max\{0, min(1,\frac{\lambda _{4}P}{w_{2}})\},\\ \lambda _{i}(t_{f} )= 0, i = 1,…,4~~~~ S(0)=S_0 ~~,E(0)=E_0 ~~,I(0)=I_0 ~~,P(0)=P_{0}. \end{array}\right. } \end{aligned}$$
(12)
\(\square\)