# American Institute of Mathematical Sciences

## Optimal control of an avian influenza model with multiple time delays in state and control variables

 1 School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China 2 Xinhua College, Ningxia University, Yinchuan, 750021, China 3 School of Mathematical and Natural Sciences, Arizona State University, AZ, USA

* Corresponding author: Qimin Zhang

Received  December 2019 Revised  August 2020 Published  September 2020

Fund Project: Ting Kang and Qimin Zhang are supported by the Natural Science Foundation of China (11661064), Ningxia Natural Science Foundation Project (2019AAC03069) and the Funds for Improving the International Education Capacity of Ningxia University (030900001921)

In this paper, we consider an optimal control model governed by a class of delay differential equation, which describe the spread of avian influenza virus from the poultry to human. We take three control variables into the optimal control model, namely: slaughtering to the susceptible and infected poultry ($u_{1}(t)$), educational campaign to the susceptible human population ($u_{2}(t)$) and treatment to infected population ($u_{3}(t)$). The model involves two time delays that stand for the incubation periods of avian influenza virus in the infective poultry and human populations. We derive first order necessary conditions for existence of the optimal control and perform several numerical simulations. Numerical results show that different control strategies have different effects on controlling the outbreak of avian influenza. At the same time, we discuss the influence of time delays on objective function and conclude that the spread of avian influenza will slow down as the time delays increase.

Citation: Ting Kang, Qimin Zhang, Haiyan Wang. Optimal control of an avian influenza model with multiple time delays in state and control variables. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020278
##### References:

show all references

##### References:
Schematic diagram of the model with delay
The optimal states of $I_a(t)$ and $I_h(t)$, and optimal controls under all of control
The optimal states of $I_a(t)$ and $I_h(t)$ with one control and without control
The optimal states of $I_a(t)$ and $I_h(t)$ with two controls and without control
Values of objective function under different time delays for model (6)
Effect of $\mathscr{R}_0$
Effect of $\alpha$
Effect of $\alpha_1$
Effect of $\alpha_2$
Effect of $\beta_1$
Effect of $\beta_1$
Algorithm
 Step 1: for $k = -m, -(m-1), ..., 0$ do: $S_a^k = S_a(0); I_a^k = I_a(0); S_h^k = S_h(0); I_h^k = I_h(0); R_h^k = R_h(0)$ end for for $k = n, n+1, ..., n+m$ do: $\lambda_1^k = 0; \lambda_2^k = 0; \lambda_3^k = 0; \lambda_4^k = 0; \lambda_5^k = 0$ end for $m_1 = \lfloor\tau_1/\Delta\rfloor$; $m_2 = \lfloor\tau_2/\Delta\rfloor$ Step 2: for $k = 0, 1, ..., n-1$ do: $S_a^{k+1} = S_a^{k} + \Delta\left[\Lambda_{a} -\frac{\beta_{a} S_{a}^k I_{a}^k}{1 +\alpha_{1} S_{a}^k+\alpha_{2} I_{a}^k}-(\mu_{a} +u_{1}(t)) S_{a}^k \right]$ $I_a^{k+1} = I_a^{k} + \Delta\left[ \frac{\beta_{a} e^{-\mu_{a} \tau_{1}}S_{a}^{k-m_1} I_{a}^{k-m_1}}{1 +\alpha_{1}S_{a}^{k-m_1} +\alpha_{2} I_{a}^{k-m_1}} -(\mu_{a} +\delta_{a} +u_{1}^k) I_{a}^k \right]$ $S_h^{k+1} = S_h^{k} + \Delta\left[ \Lambda_{h} -(1-u_{2}^k) \frac{\beta_{h} S_{h}^k I_{a}^k}{1 +\beta_{1}S_{h}^k +\beta_{2}I_{a}^k} -\mu_{h}S_{h}^k \right]$ $I_h^{k+1} = I_h^{k} + \Delta\Big[ (1-u_{2}^{k-m_2}) \frac{\beta_{h}e^{ -\mu_{h}\tau_{2}} S_{h}^{k-m_2} I_{a}^{k-m_2}}{1 +\beta_{1} S_{h}^{k-m_2} +\beta_{2} I_{a}^{k-m_2}}$ $-(\mu_{h} +\delta_{h} +\gamma) I_{h}^k -\frac{c u_{3}^k I_{h}^k}{1+\alpha I_{h}^k} \Big]$ $R_h^{k+1} = R_h^{k} + \Delta\left[ \gamma I_{h}^k -\mu_{h}R_{h}^k +\frac{c u_{3}^k I_{h}^k}{1 +\alpha I_{h}^k} \right]$ for $j = 1, 2, 3, 4, 5$ do: $\lambda_j^{n-k-1} = \lambda_j^{n-k} - \Delta\times\text{Temp}_j$ end for $D_1^{k+1} = [(\lambda_{1}^{n-k}-B_{1})S_{a}^k +(\lambda_{2}^{n-k} -B_{1})I_{a}^k]/C_{1}$; $D_2^{k+1} = \text{Temp}_6/C_2$ $D_3^{k+1} = \left[(\lambda_{4}^{n-k} -\lambda_{5}^{n-k}) \frac{c I_{h}^k}{1+\alpha I_{h}^k} -B_{3}I_{h}^k \right] /C_3$ $u_1^{k+1} = \min\{\max(0, D_1^{k+1}), 1\}$; $u_2^{k+1} = \min\{\max(0, D_2^{k+1}), 1\}$ $u_3^{k+1} = \min\{\max(0, D_3^{k+1}), 1\}$ end for Step 3: for $k = 1, 2, ..., n$ do: $S_a^*(t_k) = S_a^k; I_a^*(t_k) = I_a^k; S_h^*(t_k) = S_h^k; I_h^*(t_k) = I_h^k; R_h^*(t_k) = R_h^k$ $u_1^*(t_k) = u_1^k; u_2^*(t_k) = u_2^k; u_3^*(t_k) = u_3^k$ end for $\dagger$ The $\text{Temp}_i (1\leq i\leq 6)$ is defined in C.
 Step 1: for $k = -m, -(m-1), ..., 0$ do: $S_a^k = S_a(0); I_a^k = I_a(0); S_h^k = S_h(0); I_h^k = I_h(0); R_h^k = R_h(0)$ end for for $k = n, n+1, ..., n+m$ do: $\lambda_1^k = 0; \lambda_2^k = 0; \lambda_3^k = 0; \lambda_4^k = 0; \lambda_5^k = 0$ end for $m_1 = \lfloor\tau_1/\Delta\rfloor$; $m_2 = \lfloor\tau_2/\Delta\rfloor$ Step 2: for $k = 0, 1, ..., n-1$ do: $S_a^{k+1} = S_a^{k} + \Delta\left[\Lambda_{a} -\frac{\beta_{a} S_{a}^k I_{a}^k}{1 +\alpha_{1} S_{a}^k+\alpha_{2} I_{a}^k}-(\mu_{a} +u_{1}(t)) S_{a}^k \right]$ $I_a^{k+1} = I_a^{k} + \Delta\left[ \frac{\beta_{a} e^{-\mu_{a} \tau_{1}}S_{a}^{k-m_1} I_{a}^{k-m_1}}{1 +\alpha_{1}S_{a}^{k-m_1} +\alpha_{2} I_{a}^{k-m_1}} -(\mu_{a} +\delta_{a} +u_{1}^k) I_{a}^k \right]$ $S_h^{k+1} = S_h^{k} + \Delta\left[ \Lambda_{h} -(1-u_{2}^k) \frac{\beta_{h} S_{h}^k I_{a}^k}{1 +\beta_{1}S_{h}^k +\beta_{2}I_{a}^k} -\mu_{h}S_{h}^k \right]$ $I_h^{k+1} = I_h^{k} + \Delta\Big[ (1-u_{2}^{k-m_2}) \frac{\beta_{h}e^{ -\mu_{h}\tau_{2}} S_{h}^{k-m_2} I_{a}^{k-m_2}}{1 +\beta_{1} S_{h}^{k-m_2} +\beta_{2} I_{a}^{k-m_2}}$ $-(\mu_{h} +\delta_{h} +\gamma) I_{h}^k -\frac{c u_{3}^k I_{h}^k}{1+\alpha I_{h}^k} \Big]$ $R_h^{k+1} = R_h^{k} + \Delta\left[ \gamma I_{h}^k -\mu_{h}R_{h}^k +\frac{c u_{3}^k I_{h}^k}{1 +\alpha I_{h}^k} \right]$ for $j = 1, 2, 3, 4, 5$ do: $\lambda_j^{n-k-1} = \lambda_j^{n-k} - \Delta\times\text{Temp}_j$ end for $D_1^{k+1} = [(\lambda_{1}^{n-k}-B_{1})S_{a}^k +(\lambda_{2}^{n-k} -B_{1})I_{a}^k]/C_{1}$; $D_2^{k+1} = \text{Temp}_6/C_2$ $D_3^{k+1} = \left[(\lambda_{4}^{n-k} -\lambda_{5}^{n-k}) \frac{c I_{h}^k}{1+\alpha I_{h}^k} -B_{3}I_{h}^k \right] /C_3$ $u_1^{k+1} = \min\{\max(0, D_1^{k+1}), 1\}$; $u_2^{k+1} = \min\{\max(0, D_2^{k+1}), 1\}$ $u_3^{k+1} = \min\{\max(0, D_3^{k+1}), 1\}$ end for Step 3: for $k = 1, 2, ..., n$ do: $S_a^*(t_k) = S_a^k; I_a^*(t_k) = I_a^k; S_h^*(t_k) = S_h^k; I_h^*(t_k) = I_h^k; R_h^*(t_k) = R_h^k$ $u_1^*(t_k) = u_1^k; u_2^*(t_k) = u_2^k; u_3^*(t_k) = u_3^k$ end for $\dagger$ The $\text{Temp}_i (1\leq i\leq 6)$ is defined in C.
Parameter values of numerical experiments for model (2)
 Parameter Value Source of data $\Lambda_a$ $1000/245$ per day [5,9] $\beta_a$ $5.1\times10^{-4}$ per day [5], $\mu_a$ $1/245$ per day [5,9] $\delta_a$ $1/400$ per day [5] $\Lambda_h$ $2000/36500$ per day [5] $\beta_h$ $2\times10^{-6}$ per day [5] $\mu_h$ $5.48\times10^{-5}$ per day [26,37] $\delta_h$ 0.001 per day [26,37] $\gamma$ 0.1 per day [26,37] $c$ 0.5 Assumed $\alpha$ 0.1 Assumed $\alpha_1$ 0.01 Assumed $\alpha_2$ 0.03 Assumed $\beta_1$ 0.01 Assumed $\beta_2$ 0.01 Assumed
 Parameter Value Source of data $\Lambda_a$ $1000/245$ per day [5,9] $\beta_a$ $5.1\times10^{-4}$ per day [5], $\mu_a$ $1/245$ per day [5,9] $\delta_a$ $1/400$ per day [5] $\Lambda_h$ $2000/36500$ per day [5] $\beta_h$ $2\times10^{-6}$ per day [5] $\mu_h$ $5.48\times10^{-5}$ per day [26,37] $\delta_h$ 0.001 per day [26,37] $\gamma$ 0.1 per day [26,37] $c$ 0.5 Assumed $\alpha$ 0.1 Assumed $\alpha_1$ 0.01 Assumed $\alpha_2$ 0.03 Assumed $\beta_1$ 0.01 Assumed $\beta_2$ 0.01 Assumed
Values of objective function under different control variables for model (2)
 Value of control $\mathbf{u(t)}$ Value of objective function ($\times10^4$) $u_1(t), u_2(t), u_3(t)\equiv0$ (Without control) $1.4681$ $u_1(t) \neq 0, u_2(t), u_3(t)\equiv0$ $1.2038$ $u_2(t) \neq 0, u_1(t), u_3(t)\equiv0$ $1.4692$ $u_3(t) \neq 0, u_1(t), u_2(t)\equiv0$ $1.4684$ $u_1(t), u_2(t) \neq 0, u_3(t)\equiv0$ $1.2039$ $u_1(t), u_3(t) \neq 0, u_2(t)\equiv0$ $1.2041$ $u_2(t), u_3(t) \neq 0, u_1(t)\equiv0$ $1.4692$ $u_1(t), u_2(t), u_3(t) \neq 0$ (With all of controls) $1.2043$
 Value of control $\mathbf{u(t)}$ Value of objective function ($\times10^4$) $u_1(t), u_2(t), u_3(t)\equiv0$ (Without control) $1.4681$ $u_1(t) \neq 0, u_2(t), u_3(t)\equiv0$ $1.2038$ $u_2(t) \neq 0, u_1(t), u_3(t)\equiv0$ $1.4692$ $u_3(t) \neq 0, u_1(t), u_2(t)\equiv0$ $1.4684$ $u_1(t), u_2(t) \neq 0, u_3(t)\equiv0$ $1.2039$ $u_1(t), u_3(t) \neq 0, u_2(t)\equiv0$ $1.2041$ $u_2(t), u_3(t) \neq 0, u_1(t)\equiv0$ $1.4692$ $u_1(t), u_2(t), u_3(t) \neq 0$ (With all of controls) $1.2043$
 [1] Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 [2] John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026 [3] Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 [4] Demou Luo, Qiru Wang. Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3427-3453. doi: 10.3934/dcdsb.2020238 [5] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [6] Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021076 [7] Xiaozhong Yang, Xinlong Liu. Numerical analysis of two new finite difference methods for time-fractional telegraph equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3921-3942. doi: 10.3934/dcdsb.2020269 [8] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003 [9] Jaouad Danane. Optimal control of viral infection model with saturated infection rate. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031 [10] Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021021 [11] Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013 [12] Akio Matsumoto, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021069 [13] Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022 [14] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014 [15] Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056 [16] Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021035 [17] Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012 [18] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2021, 13 (1) : 1-23. doi: 10.3934/jgm.2020032 [19] Raghda A. M. Attia, Dumitru Baleanu, Dianchen Lu, Mostafa M. A. Khater, El-Sayed Ahmed. Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021018 [20] Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

2019 Impact Factor: 1.27