# American Institute of Mathematical Sciences

August  2021, 26(8): 4147-4171. doi: 10.3934/dcdsb.2020278

## 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  August 2021 Early access  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, 2021, 26 (8) : 4147-4171. doi: 10.3934/dcdsb.2020278
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##### 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) $1.2043$
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