We introduced six control variables in the NiV model presented in (11) for the population-level mitigation of the infection. These six control measures, namely \(u_1, u_2, u_3, u_4, u_5,\) and \(u_6\), have been implemented in the model (11) to ensure its effectiveness. Each control measure plays a significant role in achieving our goals, and we have provided a brief overview of their roles below: The control variables implemented in the NiV model serve specific purposes to control the disease incidence at the population level. The variable \(u_1\) minimizes the transmission of viruses between susceptible flying foxes by restricting their movement with protective coverings. The variable \(u_2\) aims to increase the mortality rate of infected Pteropus flying foxes by targeted culling particularly in areas where outbreaks have been identified. The variable \(u_3\) manages the zoonotic transmission of viruses from infected flying foxes to humans through food contamination. The variable \(u_4\) controls the transmission of infectious diseases by managing human contact with infected individuals and contaminated cadavers through personal protections. The variable \(u_5\) focuses on implementing strategies to increase the number of individuals who have received NiV vaccinations. The variable \(u_6\) measures the effort required to provide health treatment to those who are infected.The structure of system (11) by incorporating the above control variables is given below:$$\begin{aligned} {\left\{ \begin{array}{ll} S_{f}^{‘}\left( t\right) =\Lambda _{f}-\frac{\beta _{1}V(1-u_{1})}{N_{f}}S_{f}-d_{f}S_{f},\\ I_{f}^{‘}\left( t\right) =\frac{\beta _{1}V(1-u_{1})}{N_{f}}S_{f}-(d_{f}+u_{2})I_{f},\\ V^{‘}\left( t\right) =pI_{f}-\theta V,\\ S_{h}^{‘}\left( t\right) =\Lambda _{h}-\frac{\beta _{2}V(1-u_{3})}{N_{h}}S_{h}-\frac{(\beta _{3}I_{h}+\beta _{4}\kappa D_{h})(1-u_{4})}{N_{h}}S_{h}-(d_{h}+u_{5})S_{h}+\gamma R_{h}+\zeta V_{h},\\ V_{h}^{‘}\left( t\right) =u_{5}S_{h}-(\zeta +d_{h})V_{h},\\ I_{h}^{‘}\left( t\right) =\frac{\beta _{2}V(1-u_{3})}{N_{h}}S_{h}+\frac{(\beta _{3}I_{h}+\beta _{4}\kappa D_{h})(1-u_{4})}{N_{h}}S_{h}-\left( u_{6}+\alpha _{1}+d_{1}+d_{h}\right) I_{h},\\ T_{h}^{‘}\left( t\right) =u_{6}I_{h}-\left( \alpha _{2}+d_{h}\right) T_{h},\\ R_{h}^{‘}\left( t\right) =\alpha _{1}I_{h}+\alpha _{2}T_{h}-\left( \gamma +d_{h}\right) R_{h},\\ D_{h}^{‘}\left( t\right) =d_{1}I_{h}-\nu D_{h}, \end{array}\right. } \end{aligned}$$
(26)
with the ICs stated in (12). The aim of optimal control problem is to reduce the density of infected flying foxes and infected humans in class \(I_f\) and \(I_h\), respectively, while maximizing the the vaccinated and under treatment humans. In order to accomplish this, we have formulated the following objective functional:$$\begin{aligned}&J(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})=\int ^{T}_{0}(A_{1}I_{f}+A_{2}V+A_{3}I_{h}\nonumber \\&\quad +\frac{1}{2}(a_{1}u_{1}^{2}+a_{2}u_{2}^{2}+a_{3}u_{3}^{2}+a_{4}u_{4}^{2}+a_{5}u_{5}^{2}+a_{6}u_{6}^{2}))dt, \end{aligned}$$
(27)
subject to problem (26). In the objective functional (27). The constants \(A_1, A_2, A_3\) are associated weight constants while \(a_1, a_2, a_3, a_4, a_5,\) and \(a_6\) represent the associated cost factors. The final step size is represented by T41. Our main goal is to find the optimal controls for \(u_{1}^{*},u_{2}^{*},u_{3}^{*},u_{4}^{*},u_{5}^{*}\) and \(u_{6}^{*}\), so that we can achieve our objective.$$\begin{aligned} J(u_{1}^{*}, u_{2}^{*}, u_{3}^{*},u_{4}^{*},u_{5}^{*},u_{6}^{*})=\text {min}\{J(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}),(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}){\in }U\}. \end{aligned}$$
(28)
The system described by Eq. (26) must satisfy certain constraints, and the control set is stated as follows:$$\begin{aligned} U=\{(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})|u_{i}(t)\text { is Lebesgue’ Measurable on }[0,1],i=1,2,3,4,5,6\}. \end{aligned}$$
(29)
To solve an optimum control problem, we use Pontryagin’s Principle42. This principle helps us formulate the necessary conditions and solutions for the problem. To begin solving the optimal problem, we focus on the Lagrangian and Hamiltonian for Eqs. (26) to (28). The Lagrangian for the optimal problem is defined as follows:$$\begin{aligned} L=A_{1}I_{f}+A_{2}V+A_{3}I_{h}+\frac{1}{2}(a_{1}u_{1}^{2}+a_{2}u_{2}^{2}+a_{3}u_{3}^{2}+a_{4}u_{4}^{2}+a_{5}u_{5}^{2}+a_{6}u_{6}^{2}). \end{aligned}$$
(30)
Our goal is to find the minimum value of the Lagrangian L mentioned above. To achieve this, we introduce the Hamiltonian function \(\mathbb {H}\) defined as follows:$$\begin{aligned} \mathbb {H}=L+\lambda _{1}\frac{dS_{f}}{dt}+\lambda _{2}\frac{dI_{f}}{dt}+\lambda _{3}\frac{dV}{dt}+\lambda _{4}\frac{dS_{h}}{dt}+\lambda _{5}\frac{dV_{h}}{dt}+\lambda _{6}\frac{dI_{h}}{dt}+\lambda _{7}\frac{dT_{h}}{dt}+\lambda _{8}\frac{dR_{h}}{dt}+\lambda _{9}\frac{dD_{h}}{dt}. \end{aligned}$$
(31)
\(\mathbb {H}\) is associated with the adjoint variables \(\lambda _{1},\ldots ,\lambda _{9}\). In the next step, we will demonstrate that an optimal control exists for system (26).Existence of the optimal control problemTo establish the existence of the optimal control problem, we utilize the procedure outlined in43, along with the method employed by Ngina et al. in44. We then present the following result.
Theorem 6
Consider the objective function as define above$$\begin{aligned} J(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})=\int _{0}^{T}(A_{1}I_{f}+A_{2}V+A_{3}I_{h}+\frac{1}{2}(a_{1}u_{1}^{2}+a_{2}u_{2}^{2}+a_{3}u_{3}^{2}+a_{4}u_{4}^{2}+a_{5}u_{5}^{2}+a_{6}u_{6}^{2}))dt, \end{aligned}$$corresponding to state and controls variables mentioned in the system (26) and satisfies the ICs$$\begin{aligned} {\left\{ \begin{array}{ll} S_{f}(0)=S_{f}{}_{0}\ge 0,I_{f}(0)=I_{f}{}_{0}\ge 0,V(0)=V_{0}\ge 0,S_{h}(0)=S_{h}{}_{0}\ge 0,V_{h}(0)=V_{h0}\ge 0,\\ I_{h}(0)=I_{h}{}_{0}\ge 0,T_{h}(0)=T_{h}{}_{0}\ge 0,R_{h}(0)=R_{h}{}_{0}\ge 0,D_{h}(0)=D_{h}{}_{0}\ge 0. \end{array}\right. } \end{aligned}$$then there exists optimal controls \((u_{1}^{*},u_{2}^{*},u_{3}^{*},u_{4}^{*},u_{5}^{*},u_{6}^{*})\) such that$$\begin{aligned} J(u_{1}^{*},u_{2}^{*},u_{3}^{*},u_{4}^{*},u_{5}^{*},u_{6}^{*})=\text {min}\{J(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})|(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}){\in }U\}. \end{aligned}$$
Proof
In order to establish the existence of the said problem, we need to satisfy the following steps:
(i)
The control set U paired with each state variable equation is not empty. This condition is met as all control and state variables within U are non-empty and non-negative, and \(u_{i},i=1,2,3,4,5,6\), is a Lebesque integrable function on the interval [0, T].
(ii)
The control set is both convex as well as closed. This has been demonstrated using the methodology outlined by Ngina et al.44). Initially, we represent the elements U in the following vector form. $$\begin{aligned} \hat{U}=(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}), for 0 \le u_{1},u_{2},u_{3},u_{4},u_{5},u_{6} \le 1. \end{aligned}$$
(32)
To prove the convexity of set U, let’s consider constants \(\varepsilon =\varepsilon _{1},\varepsilon _{2},\varepsilon _{3},\varepsilon _{4},\varepsilon _{5},\varepsilon _{6}\) that belong to the control set U, with the condition \(0 \le \varepsilon _{1},\varepsilon _{2},\varepsilon _{3},\varepsilon _{4},\varepsilon _{5},\varepsilon _{6} \le 1\).
We aim to prove that the equation$$\begin{aligned} \omega =\phi U+(1-\phi )\varepsilon , for 0\le \phi \le 1 \end{aligned}$$is a part of the control set U. Therefore, our objective is to demonstrate that$$\begin{aligned} \omega&=\phi (u_{1},u_{2},u_{3},u_{4},u_{4},u_{5},u_{6})+(1-\phi )(\varepsilon _{1},\varepsilon _{2},\varepsilon _{3},\varepsilon _{4},\varepsilon _{5},\varepsilon _{6}),\nonumber \\&=(\phi u_{1}+(1-\phi )\varepsilon _{1},\phi u_{2}+(1-\phi )\varepsilon _{2},\phi u_{3}+(1-\phi )\varepsilon _{3},\phi u_{4}+(1-\phi )\varepsilon _{4},\nonumber \\&\quad \phi u_{5}+(1-\phi )\varepsilon _{5},\phi u_{6}+(1-\phi )\varepsilon _{6}),\nonumber \\&=(\omega _{1},\omega _{2},\omega _{3},\omega _{4},\omega _{5},\omega _{6}), \end{aligned}$$
(33)
where, \(\omega _{1}=\phi u_{1}+(1-\phi )\varepsilon _{1}\) , is within the interval [0, 1], hence, \(0\le \omega _{1}\le 1\).
The same applies for all \(\omega _{2},\omega _{3},\omega _{4},\omega _{5},\omega _{6}\). Hence, the vector \(\omega =(\omega _{1},\omega _{2},\omega _{3},\omega _{4},\omega _{5},\omega _{6})\) fulfills condition (32) for convexity, establishing the convex and closed nature of the control set U.
(iii)
The boundedness of each right-hand side of (26) is a linear function of U varying with time and state variables. To establish this, we are employing the approach introduced by44,45. Let $$\begin{aligned} \dot{\Upsilon }=W\Upsilon +N(\Upsilon ), \end{aligned}$$
(34)
where \(\Upsilon =(S_{f},I_{f},V,S_{h},V_{h},I_{h},T_{h},R_{h},D_{h})^{T},\)$$\begin{aligned} W=\left[ \begin{array}{ccccccccc} -d_{f} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} -d_{f} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -\theta &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -d_{h} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} -d_{h} &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -d_{h} &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -d_{h} &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -d_{h} &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -\nu \end{array}\right] \end{aligned}$$
(35)
and $$\begin{aligned} N(\Upsilon )=\left[ \begin{array}{c} \Lambda _{f}-\frac{\beta _{1}V(1-u_{1})}{N_{f}}S_{f}\\ \\ \frac{\beta _{1}V(1-u_{1})}{N_{f}}S_{f}-u_{2}I_{f}\\ \\ pI_{f}\\ \\ \Lambda _{h}-\frac{\beta _{2}V(1-u_{3})}{N_{h}}S_{h}-\frac{(\beta _{3}I_{h}+\beta _{4}\kappa D_{h})(1-u_{4})}{N_{h}}S_{h}-u_{5}S_{h}+\gamma R_{h}+\zeta V_{h}\\ \\ u_{5}S_{h}-\zeta V_{h}\\ \\ \frac{\beta _{2}V(1-u_{3})}{N_{h}}S_{h}+\frac{(\beta _{3}I_{h}+\beta _{4}\kappa D_{h})(1-u_{4})}{N_{h}}S_{h}-\left( u_{6}+\alpha _{1}+d_{1}\right) I_{h}\\ \\ u_{6}I_{h}-\alpha _{2}T_{h}\\ \\ \alpha _{1}I_{h}+\alpha _{2}T_{h}-\gamma R_{h}\\ \\ d_{1}I_{h} \end{array}\right] . \end{aligned}$$
(36)
Representing $$\begin{aligned} G(\Upsilon )=W\Upsilon +N(\Upsilon ), \end{aligned}$$
(37)
the second term, \(N(\Upsilon )\) in Eq. (37) satisfies $$\begin{aligned} \begin{array}{ll} |N(\Upsilon _{1})-N(\Upsilon _{2})| &{} \le g_{1}|S_{f1}-S_{f2}|+g_{2}|I_{f1}-I_{f2}|+g_{3}|V_{1}-V_{2}|+g_{4}|S_{h1}-S_{h2}|+g_{5}|V_{h1}-V_{h2}|\\ \\ &{}\quad +g_{6}|I_{h1}-I_{h2}|+g_{7}|T_{h1}-T_{h2}|+g_{8}|R_{h1}-R_{h2}|+g_{9}|D_{h1}-D_{h2}|\\ \\ &{} \le g[|S_{f1}-S_{f2}|+|I_{f1}-I_{f2}|+|V_{1}-V_{2}|+|S_{h1}-S_{h2}|+|V_{h1}-V_{h2}|\\ \\ &{}\quad +|I_{h1}-I_{h2}|+|T_{h1}-T_{h2}|+|R_{h1}-R_{h2}|+|D_{h1}-D_{h2}|], \end{array} \end{aligned}$$
(38)
where $$\begin{aligned} \Upsilon _{1}=(S_{f1},I_{f1},V_{1},S_{h1},V_{h1},I_{h1},T_{h1},R_{h1},D_{h1}), \Upsilon _{2}=(S_{f2},I_{f2},V_{2},S_{h2},V_{h2},I_{h2},T_{h2},R_{h2},D_{h2}) \end{aligned}$$ and g is an independent constant such that
$$\begin{aligned} g=\text {max}(g_{i},i=1,\cdots ,9). \end{aligned}$$Also,$$\begin{aligned}{}[G(\Upsilon _{1})-G(\Upsilon _{2})]\le g|\Upsilon _{1}-\Upsilon _{2}|, where \;\;g=\sum _{i=1}^{9}g_{i}+||G||\le \infty . \end{aligned}$$The function \(G(\Upsilon )\) is uniformly Lipschitz continuous, given the definitions of the control variables \(u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t),u_{5}(t),u_{6}(t)\) and the constraints on the state variables;$$\begin{aligned} (S_{f}>0,I_{f}\ge 0,V\ge 0,S_{h}\ge 0,V_{h}\ge 0,I_{h}\ge 0,T_{h}\ge 0,R_{h}\ge 0,D_{h}\ge 0), \end{aligned}$$Consequently, the solution of the system (26) exists, thereby concluding the proof third condition.
(iv)
The integrand in (27) provided by $$\begin{aligned} \mathcal {P}=A_{1}I_{f}+A_{2}V+A_{3}I_{h}+\frac{1}{2}(a_{1}u_{1}^{2}+a_{2}u_{2}^{2}+a_{3}u_{3}^{2}+a_{4}u_{4}^{2}+a_{5}u_{5}^{2}+a_{9}u_{6}^{2}) \end{aligned}$$
(39)
exhibits convexity with respect to U. To establish this property, we employ the Hessian matrix method, defined as below:
Definition 6.1
Given a function \(\mathcal {V}(y_{1},y_{2},y_{3},\ldots ,y_{n})\), of several variables, then \(\mathcal {V}\), will be convex if and only if the Hessian matrix, \(H_{m}\) is such that$$\begin{aligned} H_{m}(y)=\left[ \frac{\partial ^{2}\mathcal {V}}{\partial y_{i}\partial y_{j}}\right] \ge 0,\forall y\ne 0, \end{aligned}$$and it is considered to be concave if and only if$$\begin{aligned} H_{m}(y)=\left[ \frac{\partial ^{2}\mathcal {V}}{\partial y_{i}\partial y_{j}}\right] \le 0,\forall y\ne 0. \end{aligned}$$So let$$\begin{aligned} \mathcal {P}_{i}=\frac{1}{2}(a_{1}u_{1}^{2}+a_{2}u_{2}^{2}+a_{3}u_{3}^{2}+a_{4}u_{4}^{2}+a_{5}u_{5}^{2}+a_{6}u_{6}^{2}), \end{aligned}$$where \(\mathcal {P}_{i}\in \mathcal {P},\) the Hessian Matrix for \(\mathcal {P}_{i}\) is given by$$\begin{aligned} H_{m}=\left[ \begin{array}{cccccc} a_{1} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} a_{2} &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} a_{3} &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} a_{4} &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} a_{5} &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} a_{6} \end{array}\right] \ge 0,\text {for} \;a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\ge 0. \end{aligned}$$
(40)
It can be observed that \(\mathcal {P}_{i}\) is a convex function in the domain U. If \(\mathcal {P}_{i}\in \mathcal {P}\) is convex in U, then it implies that \(\mathcal {P}\) is also convex in U and satisfies condition (iv).
(xxii)
Additionally, there exist some constants \(n_{1}\), \(n_{2}> 0\), and \(n_{3}> 1\), such that the integrand of (39) is both convex and bounded by $$\begin{aligned} \mathcal {P}(t,S_{f},I_{f},V,S_{h},V_{h},I_{h},T_{h},R_{h},D_{h},u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\le n_{2}-\frac{n_{1}}{2}\left( \sum _{i=1}^{6}|u_{i}|^{2}\right) ^{\frac{n_{3}}{2}}. \end{aligned}$$
From (27), we proceed as$$\begin{aligned} J\left( u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t),u_{5}(t),u_{6}(t)\right) =A_{1}I_{f}+A_{2}V+A_{3}I_{h}+\frac{1}{2}\left( \sum _{i=1}^{6}a_{i}u_{i}^{2}\right) , \end{aligned}$$then,$$\begin{aligned} J\left( u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t),u_{5}(t),u_{6}(t)\right) \le A_{1}I_{f}+A_{2}V+A_{3}I_{h}+\sum _{i=1}^{6}a_{i}u_{i}^{2}. \end{aligned}$$
(41)
Letting \(a=\text {max}(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\) such that$$\begin{aligned} J\left( u_{1}(t),u_{2}(t),u_{3}(t),u_{4}(t),u_{5}(t),u_{6}(t)\right) \le A_{1}I_{f}+A_{2}V+A_{3}I_{h}+a\sum _{i=1}^{6}u_{i}^{2}, \end{aligned}$$
(42)
this means that$$\begin{aligned} A_{1}I_{f}+A_{2}V+A_{3}I_{h}+\sum _{i=1}^{6}a_{i}u_{i}^{2}\le n_{2}+n_{1}\sum _{i=1}^{6}|u_{i}|^{2}, \end{aligned}$$
(43)
where \(n_{2}\) based on the upper bound on \(I_{f},V,I_{h}\) and \(n_{1}>0\) since \(a_{i}>0\) for \(i=1,\cdots ,6\). Therefore, Eq. (42) can be revised as$$\begin{aligned} J(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})\le n_{2}+n_{1}(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})^{2}. \end{aligned}$$
(44)
Now, from Eq. (44), it is observed that \(n_{1},n_{2}>0\), and \(n_{3}=2>1\). Finally, we conclude that there must exists an optimal control variables \(u_{1}^{*},u_{2}^{*},u_{3}^{*},u_{4}^{*},u_{5}^{*},u_{6}^{*}\). This completes the proof. \(\square\)
$$\begin{aligned} {\left\{ \begin{array}{ll} z^{‘}=\frac{\partial {\mathbb {H}}(t,z,u,\lambda )}{\partial \lambda },\\ 0=\frac{\partial {\mathbb {H}}(t,z,u,\lambda )}{\partial u},\\ \lambda ^{‘}=\frac{\partial {\mathbb {H}}(t,z,u,\lambda )}{\partial z}. \end{array}\right. } \end{aligned}$$
(45)
To obtain the necessary results, we use the above described conditions as follow.
Theorem 7
Let \(S_{f}^{*}\), \(I_{f}^{*}\), \(V^{*}\), \(S_{h}^{*}\), \(V_{h}^{*}\), \(I_{h}^{*}\), \(T_{h}^{*}\), \(R_{h}^{*}\), and \(D_{h}^{*}\) represent the state solutions associated with the optimal control measures \(u_{1}^{*}\), \(u_{2}^{*}\), \(u_{3}^{*}\), \(u_{4}^{*}\), \(u_{5}^{*}\), and \(u_{6}^{*}\) for the optimum control system stated in (26) and (27). Then, we find the adjoint variables \(\lambda _{1}, \ldots , \lambda _{9}\) that satisfy:$$\begin{aligned} {\left\{ \begin{array}{ll} \lambda _{1}^{‘} =d_f\lambda _{1}+(\lambda _{2}-\lambda _{1})\frac{S_f V\beta _1(1-u_1)}{N_f^2}+(\lambda _{1}-\lambda _{2})\frac{V\beta _1(1-u_1)}{N_f},\\ \lambda _{2}^{‘}=-A_3+(\lambda _6-\lambda _7)u_6+(\lambda _6-\lambda _8)\alpha _1-\lambda _9(d_1+d_h)+(\lambda _6-\lambda _4) \frac{S_hI_h\beta _2(1-u_3)}{N_h^2}\\ +(\lambda _4-\lambda _6)\frac{(S_hI_h)\beta _2(1-u_3)}{N_h} +(\lambda _4-\lambda _6)\frac{S_h\beta _3(1-u_4)}{N_h}+(\lambda _6-\lambda _4)\frac{S_h(\beta _3I_h+\beta _4\kappa D_h)(1-u_4)}{N_h^2},\\ \lambda _{3}^{‘}=\theta \lambda _{3}+(\lambda _{1}-\lambda _{2})\frac{S_f\beta _1(1-u_1)}{N_f},\\ \lambda _{4}^{‘}=d_h\lambda _4+(\lambda _4-\lambda _5)u_5+(\lambda _6-\lambda _4)\frac{\beta _2 S_h I_h (1-u_3)}{N_h^2}+(\lambda _4-\lambda _6)\frac{\beta _2I_h(1-u_3)}{N_h}\\ +(\lambda _6-\lambda _4)\frac{S_h(\beta _3I_h+\beta _4 \kappa D_h)}{N_h^2}+(\lambda _4-\lambda _6)\frac{(\beta _3I_h+\beta _4 \kappa D_h)(1-u_4)}{N_h},\\ \lambda _{5}^{‘}=-A_1+\lambda _5 d_h+(\lambda _5-\lambda _4)\zeta +(\lambda _6-\lambda _4)\frac{\beta _2 S_hI_h(1-u_3)}{N_h^2} +(\lambda _6-\lambda _4)\frac{\beta _2S_hI_h(I_h\beta _3+\kappa d_h\beta _4)(1-u_4)}{N_h^2},\\ \lambda _{6}^{‘}=-A_3+(\lambda _6-\lambda _7)u_6+\alpha _1(\lambda _6-\lambda _8)+(\lambda _6-\lambda _9)d_1+\lambda _6d_h+(\lambda _6-\lambda _4)\frac{\beta _2S_hI_h(1-u_3)}{N_h^2}\\ +(\lambda _4-\lambda _6)\frac{S_h\beta _2(1-u_3)}{N_h}+(\lambda _4-\lambda _6)\frac{S_h\beta _3(1-u_4))}{N_h}+(\lambda _6-\lambda _4)\frac{S_h(\beta _3 I_h+\beta _4 \kappa D_h )(1-u_4)}{N_h^2},\\ \dot{\lambda _{7}}=-A_2+d_h\lambda _7+\alpha _2(\lambda _7-\lambda _8)+(\lambda _6-\lambda _4)\frac{\beta _2 S_hI_h(1-u_3)}{N_h^2}+(\lambda _6-\lambda _4)\frac{S_h(\beta _3I_h+\beta _4\kappa D_h)(1-u_4)}{N_h^2},\\ \lambda _{8}^{‘}=d_h\lambda _8+(\lambda _8-\lambda _4)\gamma +(\lambda _6-\lambda _4)\frac{\beta _2S_hI_h(1-u_3)}{N_h^2}+(\lambda _6-\lambda _4)\frac{S_h(\beta _3I_h+\beta _4\kappa D_h)(1-u_4)}{N_h^2},\\ \lambda _{9}^{‘}=\nu \lambda _9+(\lambda _4-\lambda _6)\frac{\beta _4\kappa S_h(1-u_4)}{N_h},\\ \end{array}\right. } \end{aligned}$$
(46)
with boundary conditions or transversality conditions$$\begin{aligned} \lambda _{i}(T)=0,i=1,{{\dots }},9. \end{aligned}$$Moreover, the control measures \(u_{1}^{*}\), \(u_{2}^{*}\),\(u_{3}^{*}\), \(u_{4}^{*}\),\(u_{5}^{*}\) and \(u_{6}^{*}\) are given by$$\begin{aligned} {\left\{ \begin{array}{ll} u_{1}^{*}=\text {max }\{\text {min}(\frac{\beta _{1}S_{f}V(\lambda _{3}-\lambda _{2})}{N_{f}a_{1}},1),0\},\\ u_{2}^{*}=\text {max }\{\text {min}(\frac{\lambda _{3}I_{f}}{a_{2}},1),0\},\\ u_{3}^{*}=\text {max }\{\text {min}(\frac{(\lambda _{6}-\lambda _{4})\beta _{2}I_{h}S_{h}}{N_{h}a_{3}},1),0\},\\ u_{4}^{*}=\text {max }\{\text {min}(\frac{(\lambda _{6}-\lambda _{4})(\beta _{3}I_{h}+\beta _{4}\kappa D_{h})S_{h}}{N_{h}a_{4}},1),0\},\\ u_{5}^{*}=\text {max }\{\text {min}(\frac{(\lambda _{4}-\lambda _{5})S_{h}}{a_{5}},1),0\},\\ u_{6}^{*}=\text {max }\{\text {min}(\frac{(\lambda _{6}-\lambda _{7})I_{h}}{a_{6}},1),0\}. \end{array}\right. } \end{aligned}$$
(47)
Proof
In order to obtain the transversality and adjoint system conditions with the correct boundary values, we utilize the Hamiltonian function \(\mathbb {H}\), as shown in Eq. (31). By employing Pontryagin’s maximum principle, we can derive the adjoint equations in a methodical and effective way.$$\begin{aligned} \frac{d\lambda _{1}(t)}{dt}=-\frac{\partial {\mathbb {H}}}{\partial S_f}, \frac{d\lambda _{2}(t)}{dt}=-\frac{\partial {\mathbb {H}}}{\partial I_f}, \cdots , \frac{d\lambda _{8}(t)}{dt}=-\frac{\partial {\mathbb {H}}}{\partial D_h}, \end{aligned}$$subject to boundary time conditions (i.e. final) \(\lambda _{i}(T)=0\) for \(i=1,2,\ldots ,9\). In order to achieve the desired problem (47), we utilize the following equations:$$\begin{aligned} \frac{\partial {\mathbb {H}}}{\partial u_{i}}=0, ~~i=1,\cdots ,6. \end{aligned}$$By utilizing the property of the control space U being in the interior of the control set, we obtain the desired result. \(\square\)
Simulation and discussionIn order to examine the effectiveness of the aforementioned control measures, we conducted a detailed numerical analysis of the NiV epidemic models with variable and constant controls given (26) and (11) respectively. Utilizing the backward fourth-order Runge-Kutta method, the model was numerically solved. The parameter’s values used in simulation are mentioned in Table 2. The balance constants and weights were adjusted to \(A_1=01,~ A_2=A_3=10, ~ a_1=a_2=15,~ a_3=10,~ a_4=15, ~a_5=10\) and \(a_6=50\). The control profile for this scenario is represented by Fig. j in all figures. The red solid plots indicate the dynamics without control, while the black dashed trajectories depict the dynamics with optimal control. Each and every section of graphs have nine classes as: (a) Virus class (b) susceptible flying foxes (c) infected flying foxes (d) susceptible human (e) vaccinated human (f) infected human (g) treated human (h) recovered human (i) deceased human and (j) all controls.Applying all controls simultaneously, i.e., \(u_i \ne 0,\) where, \(i=1,2,3,4,5,6\)
In order to analyze the effectiveness of optimal control measures in reducing infections in the NiV control model, we conducted simulations that included all suggested control interventions. The simulation demonstrated a notable decrease in the virus and the number of infected individuals among flying foxes, as well as among the susceptible, infected, recovered, and deceased human populations when all control measures were implemented. Conversely, the population of susceptible flying foxes, vaccinated humans, and treated human groups increased with the implementation of optimal controls. For a visual representation of the simulation dynamics, see Fig. 4. It is worth mentioning here that with the application of optimal controls, most of the population in the susceptible class are either vaccinated or protected from the virus (with other non-pharmaceutical measures) as shown in Fig. 4d,e,g, subsequently there are fewer people at risk of infection (see Fig. 4f). This reduction in risk results in fewer infection cases and, consequently, a decrease in the number of recovery cases (see Fig. 4h). The same scenario is observed in all subsequent strategies. This type of behavior can also be found in previous studies, such as the one described in46.Figure 4Usage of all control interventions.Second strategy: \(u_1 = 0,\) and \(u_i \ne 0,\) where, \(i=2,3,4,5,6\)
In the second strategy, we simulate the NiV model by activating all control interventions simultaneously except \(u_1\) to investigate their combined impact on the disease dynamics. The graphical representation of this scenario is illustrated in Fig. 5, which shows the dynamics of both human and flying fox populations. According to the system (26) with control measures, there was a decrease in the virus and the susceptible and infected populations of flying foxes. Similarly, there was a significant decrease in the susceptible, infected, recovered, and deceased human populations compared to the system (11) without control measures. In contrast, the number of vaccinated and treated individuals showed a significant increase. Figure 5j displays the control profile for this scenario. The graphical findings of this strategy, both with and without control, revealed more noteworthy outcomes for the human population. Thus, control interventions can be effectively used to reduce infection.Figure 5Usage of all controls except \(u_1\).Third strategy: \(u_2 = 0,\) and \(u_i \ne 0,\) where, \(i=1,3,4,5,6\)
In the third strategy, we simulate the NiV model by activating all control interventions except \(u_2\) simultaneously. This allows us to examine how these interventions work together to affect the disease dynamics. Figure 6 provides a graphical interpretation of this scenario, showing the dynamics of the human and flying fox populations. The results from system (26) with control measures show a decrease in the infected population of flying foxes, but an increase in susceptible flying foxes. Comparing the susceptible, recovered, infectious and deceased human individuals to the system (11) without control measures, a significant drop was observed. However, there was a notable increase in the number of people who received treatment and vaccinations. Figure 6j displays the control profile corresponding to this case. When comparing the results with and without control, it is evident that the control intervention has a more significant impact on the human population in this strategy. Therefore, control interventions can be used to effectively minimize the infection.Figure 6Impact of using all controls except \(u_2\).Fourth strategy: \(u_3 = 0,\) and \(u_i \ne 0,\) where, \(i=1,2,4,5,6\)
In the fourth strategy, we examine the effects of simultaneously activating all control interventions except \(u_3\) on the dynamics of the NiV model. This helps us understand the combined impact of control measures on the disease dynamics. Figure 7 provides a graphical illustration of this scenario, showing the dynamics of the flying fox and human populations. The results obtained from system (26) with control measures indicate a decrease in the infected population of flying foxes, but an increase in the susceptible flying foxes. Additionally, the population density in susceptible, recovered, infectious and deceased human show a significant decrease compared to the uncontrolled system (11). On the other hand, the density of vaccinated and treated classes observe a substantial increase. The corresponding control profile is depicted in Fig. 7j. The simulation in both cases reveal more significant findings for the human population, indicating that control methods can be effectively employed to reduce the infection.Figure 7Impact of using all controls except \(u_3\).Fifth strategy: \(u_4 = 0,\) and \(u_i \ne 0,\) where, \(i=1,2,3,5,6\)
In the fifth strategy, we examine the NiV model by activating all control interventions except \(u_4\) simultaneously to observe their effects on disease dynamics. Figure 8 provides a graphical interpretation of this scenario, illustrating the interactions between human and flying fox populations. The results from system (26) with control measures indicate that the virus and the infected population of flying foxes decreased, while the susceptible flying fox population increased. Additionally, the number of susceptible, infected, recovered, and deceased humans notably declined compared to the without variable controls. Conversely, the density of individuals who have received vaccinations and treatments significantly increased. Figure 8j illustrates the profile of all controls corresponding to this scenario. The simulation in both cases show more significant outcomes for the human population in this strategy, indicating that this control interventions can effectively minimize the infection.Figure 8Impact of using all controls except \(u_4\).Sixth strategy: \(u_5 = 0,\) and \(u_i \ne 0,\) where, \(i=1,2,3,4,6\)
As part of strategy 6, a simulation was conducted on the NiV model by activating all control interventions except for \(u_5\) simultaneously. This was done to observe the collective impacts on the disease dynamics. Figure 9 provides a graphical interpretation of this scenario, outlining the interactions between flying fox and human populations. The control results derived from system (26) indicate a decrease in the virus and the population of infected flying foxes, while the susceptible flying fox population increased. Similarly, in humans, the susceptible, infected, recovered, and deceased populations all saw a significant decrease compared to (11) results without control. Interestingly, the treated class dramatically enhanced, but there was no change in the number of vaccinated individuals. The control profile for this particular instance can be found in Fig. 9j. The simulation with control intervention depicts a reasonable impact on the human population in this strategy, indicating that control measures can effectively minimize the spread of the infection.Figure 9Impact of using all controls except \(u_5\).Seventh strategy: \(u_6 = 0,\) and \(u_i \ne 0,\) where, \(i=2,3,4,5\)
In the seventh strategy, an analysis was conducted to simulate the NiV model and examine the combined effects of all control interventions except \(u_6\) on the disease dynamics. The simulation results, illustrated in Fig. 10, showed that the infected population of flying foxes decreased, while the susceptible flying fox population increased. In humans, the susceptible, infected, recovered, and deceased populations showed a significant decrease compared to (11) results without control. Moreover, the vaccinated individuals density increased but there was no change observed in the treated individuals. The analysis of both cases i.e., with constant and time-varying strategies demonstrated a higher degree of significance in the human population under the variable strategy. These findings suggest that variable control interventions can be effectively employed to mitigate infection rates and reduce the spread of the disease.Figure 10Impact of using all controls except \(u_6\).To get a better grasp of the entire process, refer to the following table 3.
Table 3 Description of the suggested strategies.