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

• * Corresponding author: Qimin Zhang

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.

Mathematics Subject Classification: Primary: 34K00; Secondary: 92B10, 93D02.

 Citation: • • Figure 1.  Schematic diagram of the model with delay

Figure 2.  The optimal states of $I_a(t)$ and $I_h(t)$, and optimal controls under all of control

Figure 3.  The optimal states of $I_a(t)$ and $I_h(t)$ with one control and without control

Figure 4.  The optimal states of $I_a(t)$ and $I_h(t)$ with two controls and without control

Figure 5.  Values of objective function under different time delays for model (6)

Figure 6.  Effect of $\mathscr{R}_0$

Figure 7.  Effect of $\alpha$

Figure 8.  Effect of $\alpha_1$

Figure 9.  Effect of $\alpha_2$

Figure 10.  Effect of $\beta_1$

Figure 11.  Effect of $\beta_1$

Table 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.

Table 2.  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 , $\mu_a$ $1/245$ per day [5,9] $\delta_a$ $1/400$ per day  $\Lambda_h$ $2000/36500$ per day  $\beta_h$ $2\times10^{-6}$ per day  $\mu_h$ $5.48\times10^{-5}$ 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

Table 3.  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) $1.2043$
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