American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020278

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

Ting Kang 1,2, , Qimin Zhang 1,, and  Haiyan Wang 3,

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.

Keywords: Avian influenza model, multiple time delays, stability analysis, optimal control, numerical results.
Mathematics Subject Classification: Primary: 34K00; Secondary: 92B10, 93D02.
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:
[1]

A. Abta, A. Kaddar and H. T. Alaoui, Global stability for delay SIR and SEIR epidemic models with saturated incidence rates, Electronic Journal of Differential Equations, 2012 (2012), 1-13.   Google Scholar

[2]

C. Bao, L. Cui and M. Zhou et al., Live-animal markets and influenza a (H7N9) virus infection, New England Journal of Medicine, 368 (2013), 2337-2339.  doi: 10.1056/NEJMc1306100.  Google Scholar

[3]

E. B. M. Bashier and K. C. Patidar, Optimal control of an epidemiological model with multiple time delays, Applied Mathematics and Computation, 292 (2017), 47-56.  doi: 10.1016/j.amc.2016.07.009.  Google Scholar

[4]

L. Bourouiba, S. A. Gourley, R. Liu and J. Wu, The interaction of migratory birds and domestic poultry and its role in sustaining avian influenza, SIAM Journal on Applied Mathematics, 71 (2011), 487-516.  doi: 10.1137/100803110.  Google Scholar

[5]

F. Chen and J. Cui, Cross-species epidemic dynamic model of influenza, in 2016 9th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI), IEEE, 2016. doi: 10.1109/CISP-BMEI.2016.7852965.  Google Scholar

[6]

Z. Chen and Z. Xu, A delayed diffusive influenza model with two-strain and two vaccinations, Applied Mathematics and Computation, 349 (2019), 439-453.  doi: 10.1016/j.amc.2018.12.065.  Google Scholar

[7]

N. S. Chong, J. M. Tchuenche and R. J. Smith, A mathematical model of avian influenza with half-saturated incidence, Theory in Biosciences, 133 (2014), 23-38.  doi: 10.1007/s12064-013-0183-6.  Google Scholar

[8]

E. Claas, A. Osterhaus and R. Van-Beek, Human influenza A H5N1 virus related to a highly pathogenic avian influenza virus, The Lancet, 351 (1998), 472-477.  doi: 10.1016/S0140-6736(97)11212-0.  Google Scholar

[9]

C. A. Y. E. Committee, China agriculture yearbook, 2012. Google Scholar

[10]

W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, Springer Verlag, New York, 1975.  Google Scholar

[11]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Mathematical Biosciences and Engineering, 6 (2009), 469-492.  doi: 10.3934/mbe.2009.6.469.  Google Scholar

[12]

N. Gao, Z. Lu and B. Cao, Clinical findings in 111 cases of influenza A (H7N9) virus infection, New England Journal of Medicine, 369 (2013), 1869-1869.  doi: 10.1056/NEJMoa1305584.  Google Scholar

[13]

R. Gao, B. Cao and Y. Hu, Human infection with a novel avian-origin influenza A (H7N9) virus, New England Journal of Medicine, 368 (2013), 1888-1897.  doi: 10.1056/NEJMoa1304459.  Google Scholar

[14]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.  Google Scholar

[15]

S. Iwami, Y. Takeuchi, A. Korobeinikov and X. Liu, Prevention of avian influenza epidemic: what policy should we choose?, Journal of Theoretical Biology, 252 (2008), 732-741.  doi: 10.1016/j.jtbi.2008.02.020.  Google Scholar

[16]

S. Iwami, Y. Takeuchi and X. Liu, Avian-human influenza epidemic model, Mathematical Biosciences, 207 (2007), 1-25.  doi: 10.1016/j.mbs.2006.08.001.  Google Scholar

[17]

S. Iwami, Y. Takeuchi and X. Liu, Avian flu pandemic: Can we prevent it?, Journal of Theoretical Biology, 257 (2009), 181-190.  doi: 10.1016/j.jtbi.2008.11.011.  Google Scholar

[18]

S. Iwami, Y. Takeuchi, X. Liu and S. Nakaoka, A geographical spread of vaccine-resistance in avian influenza epidemics, Journal of Theoretical Biology, 259 (2009), 219-228.  doi: 10.1016/j.jtbi.2009.03.040.  Google Scholar

[19]

E. Jung, S. Iwami and Y. Takeuchi, Optimal control strategy for prevention of avian influenza pandemic, Journal of Theoretical Biology, 260 (2009), 220-229.  doi: 10.1016/j.jtbi.2009.05.031.  Google Scholar

[20]

T. Kang, Q. Zhang and L. Rong, A delayed avian influenza model with avian slaughter: Stability analysis and optimal control, Physica A: Statistical Mechanics and its Applications, 529 (2019), 121544. doi: 10.1016/j.physa.2019.121544.  Google Scholar

[21]

M. J. Keeling and P. Rohani, Modeling infectious diseases in humans and animals, Princeton University Press, 2008.  Google Scholar

[22]

A. Lahrouz, H. El Mahjour and A. Settati, Dynamics and optimal control of a non-linear epidemic model with relapse and cure, Physica A: Statistical Mechanics and its Applications, 496 (2018), 299-317.  doi: 10.1016/j.physa.2018.01.007.  Google Scholar

[23]

D. Liu, W. Shi and Y. Shi, Origin and diversity of novel avian influenza A H7N9 viruses causing human infection: Phylogenetic, structural, and coalescent analyses, The Lancet, 381 (2013), 1926-1932.  doi: 10.1016/S0140-6736(13)60938-1.  Google Scholar

[24]

S. Liu, L. Pang, S. Ruan and X. Zhang, Global dynamics of avian influenza epidemic models with psychological effect, Computational and Mathematical Methods in Medicine, 2015, Article ID 913726. doi: 10.1155/2015/913726.  Google Scholar

[25]

S. Liu, S. Ruan and X. Zhang, On avian influenza epidemic models with time delay, Theory in Biosciences, 134 (2015), 75-82.  doi: 10.1007/s12064-015-0212-8.  Google Scholar

[26]

S. Liu, S. Ruan and X. Zhang, Nonlinear dynamics of avian influenza epidemic models, Mathematical Biosciences, 283 (2017), 118-135.  doi: 10.1016/j.mbs.2016.11.014.  Google Scholar

[27]

F. K. Mbabazi, J. Y. T. Mugisha and M. Kimathi, Modeling the within-host co-infection of influenza {A} virus and pneumococcus, Applied Mathematics and Computation, 339 (2018), 488-506.  doi: 10.1016/j.amc.2018.07.031.  Google Scholar

[28]

O. P. Misra and D. K. Mishra, Spread and control of influenza in two groups: A model, Applied Mathematics and Computation, 219 (2013), 7982-7996.  doi: 10.1016/j.amc.2013.02.050.  Google Scholar

[29]

G. P. Samanta, Permanence and extinction for a nonautonomous avian-human influenza epidemic model with distributed time delay, Mathematical and Computer Modelling, 52 (2010), 1794-1811.  doi: 10.1016/j.mcm.2010.07.006.  Google Scholar

[30]

S. Sharma, A. Mondal and A. K. Pal, Stability analysis and optimal control of avian influenza virus A with time delays, International Journal of Dynamics and Control, 6 (2018), 1351-1366.  doi: 10.1007/s40435-017-0379-6.  Google Scholar

[31]

Z. Shi, X. Zhang and D. Jiang, Dynamics of an avian influenza model with half-saturated incidence, Applied Mathematics and Computation, 355 (2019), 399-416.  doi: 10.1016/j.amc.2019.02.070.  Google Scholar

[32]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[33]

A. Wang and Y. Xiao, A filippov system describing media effects on the spread of infectious diseases, Nonlinear Analysis: Hybrid Systems, 11 (2014), 84-97.  doi: 10.1016/j.nahs.2013.06.005.  Google Scholar

[34]

World Health Organization, WHO risk assessments of human infection with avian influenza A(H7N9) virus, 2017, https://www.who.int/influenza/human_animal_interface/influenza_h7n9/Risk_Assessment/en/. Google Scholar

[35]

Y. Xiao, X. Sun and S. Tang, Transmission potential of the novel avian influenza A(H7N9) infection in mainland China, Journal of Theoretical Biology, 352 (2014), 1-5.  doi: 10.1016/j.jtbi.2014.02.038.  Google Scholar

[36]

H. Yoko, K. Yoshinari and Y. Takehisa, Potential risk associated with animal culling and disposal during the foot-and-mouth disease epidemic in japan in 2010, Research in Veterinary Science, 102 (2015), 228-230.  doi: 10.1016/j.rvsc.2015.08.017.  Google Scholar

[37]

X. Zhang, Global dynamics of a stochastic avian-human influenza epidemic model with logistic growth for avian population, Nonlinear Dynamics, 90 (2017), 2331-2343.  doi: 10.1007/s11071-017-3806-5.  Google Scholar

[38]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, Journal of Mathematical Analysis and Applications, 348 (2008), 433-443.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

show all references

References:
[1]

A. Abta, A. Kaddar and H. T. Alaoui, Global stability for delay SIR and SEIR epidemic models with saturated incidence rates, Electronic Journal of Differential Equations, 2012 (2012), 1-13.   Google Scholar

[2]

C. Bao, L. Cui and M. Zhou et al., Live-animal markets and influenza a (H7N9) virus infection, New England Journal of Medicine, 368 (2013), 2337-2339.  doi: 10.1056/NEJMc1306100.  Google Scholar

[3]

E. B. M. Bashier and K. C. Patidar, Optimal control of an epidemiological model with multiple time delays, Applied Mathematics and Computation, 292 (2017), 47-56.  doi: 10.1016/j.amc.2016.07.009.  Google Scholar

[4]

L. Bourouiba, S. A. Gourley, R. Liu and J. Wu, The interaction of migratory birds and domestic poultry and its role in sustaining avian influenza, SIAM Journal on Applied Mathematics, 71 (2011), 487-516.  doi: 10.1137/100803110.  Google Scholar

[5]

F. Chen and J. Cui, Cross-species epidemic dynamic model of influenza, in 2016 9th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI), IEEE, 2016. doi: 10.1109/CISP-BMEI.2016.7852965.  Google Scholar

[6]

Z. Chen and Z. Xu, A delayed diffusive influenza model with two-strain and two vaccinations, Applied Mathematics and Computation, 349 (2019), 439-453.  doi: 10.1016/j.amc.2018.12.065.  Google Scholar

[7]

N. S. Chong, J. M. Tchuenche and R. J. Smith, A mathematical model of avian influenza with half-saturated incidence, Theory in Biosciences, 133 (2014), 23-38.  doi: 10.1007/s12064-013-0183-6.  Google Scholar

[8]

E. Claas, A. Osterhaus and R. Van-Beek, Human influenza A H5N1 virus related to a highly pathogenic avian influenza virus, The Lancet, 351 (1998), 472-477.  doi: 10.1016/S0140-6736(97)11212-0.  Google Scholar

[9]

C. A. Y. E. Committee, China agriculture yearbook, 2012. Google Scholar

[10]

W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, Springer Verlag, New York, 1975.  Google Scholar

[11]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Mathematical Biosciences and Engineering, 6 (2009), 469-492.  doi: 10.3934/mbe.2009.6.469.  Google Scholar

[12]

N. Gao, Z. Lu and B. Cao, Clinical findings in 111 cases of influenza A (H7N9) virus infection, New England Journal of Medicine, 369 (2013), 1869-1869.  doi: 10.1056/NEJMoa1305584.  Google Scholar

[13]

R. Gao, B. Cao and Y. Hu, Human infection with a novel avian-origin influenza A (H7N9) virus, New England Journal of Medicine, 368 (2013), 1888-1897.  doi: 10.1056/NEJMoa1304459.  Google Scholar

[14]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.  Google Scholar

[15]

S. Iwami, Y. Takeuchi, A. Korobeinikov and X. Liu, Prevention of avian influenza epidemic: what policy should we choose?, Journal of Theoretical Biology, 252 (2008), 732-741.  doi: 10.1016/j.jtbi.2008.02.020.  Google Scholar

[16]

S. Iwami, Y. Takeuchi and X. Liu, Avian-human influenza epidemic model, Mathematical Biosciences, 207 (2007), 1-25.  doi: 10.1016/j.mbs.2006.08.001.  Google Scholar

[17]

S. Iwami, Y. Takeuchi and X. Liu, Avian flu pandemic: Can we prevent it?, Journal of Theoretical Biology, 257 (2009), 181-190.  doi: 10.1016/j.jtbi.2008.11.011.  Google Scholar

[18]

S. Iwami, Y. Takeuchi, X. Liu and S. Nakaoka, A geographical spread of vaccine-resistance in avian influenza epidemics, Journal of Theoretical Biology, 259 (2009), 219-228.  doi: 10.1016/j.jtbi.2009.03.040.  Google Scholar

[19]

E. Jung, S. Iwami and Y. Takeuchi, Optimal control strategy for prevention of avian influenza pandemic, Journal of Theoretical Biology, 260 (2009), 220-229.  doi: 10.1016/j.jtbi.2009.05.031.  Google Scholar

[20]

T. Kang, Q. Zhang and L. Rong, A delayed avian influenza model with avian slaughter: Stability analysis and optimal control, Physica A: Statistical Mechanics and its Applications, 529 (2019), 121544. doi: 10.1016/j.physa.2019.121544.  Google Scholar

[21]

M. J. Keeling and P. Rohani, Modeling infectious diseases in humans and animals, Princeton University Press, 2008.  Google Scholar

[22]

A. Lahrouz, H. El Mahjour and A. Settati, Dynamics and optimal control of a non-linear epidemic model with relapse and cure, Physica A: Statistical Mechanics and its Applications, 496 (2018), 299-317.  doi: 10.1016/j.physa.2018.01.007.  Google Scholar

[23]

D. Liu, W. Shi and Y. Shi, Origin and diversity of novel avian influenza A H7N9 viruses causing human infection: Phylogenetic, structural, and coalescent analyses, The Lancet, 381 (2013), 1926-1932.  doi: 10.1016/S0140-6736(13)60938-1.  Google Scholar

[24]

S. Liu, L. Pang, S. Ruan and X. Zhang, Global dynamics of avian influenza epidemic models with psychological effect, Computational and Mathematical Methods in Medicine, 2015, Article ID 913726. doi: 10.1155/2015/913726.  Google Scholar

[25]

S. Liu, S. Ruan and X. Zhang, On avian influenza epidemic models with time delay, Theory in Biosciences, 134 (2015), 75-82.  doi: 10.1007/s12064-015-0212-8.  Google Scholar

[26]

S. Liu, S. Ruan and X. Zhang, Nonlinear dynamics of avian influenza epidemic models, Mathematical Biosciences, 283 (2017), 118-135.  doi: 10.1016/j.mbs.2016.11.014.  Google Scholar

[27]

F. K. Mbabazi, J. Y. T. Mugisha and M. Kimathi, Modeling the within-host co-infection of influenza {A} virus and pneumococcus, Applied Mathematics and Computation, 339 (2018), 488-506.  doi: 10.1016/j.amc.2018.07.031.  Google Scholar

[28]

O. P. Misra and D. K. Mishra, Spread and control of influenza in two groups: A model, Applied Mathematics and Computation, 219 (2013), 7982-7996.  doi: 10.1016/j.amc.2013.02.050.  Google Scholar

[29]

G. P. Samanta, Permanence and extinction for a nonautonomous avian-human influenza epidemic model with distributed time delay, Mathematical and Computer Modelling, 52 (2010), 1794-1811.  doi: 10.1016/j.mcm.2010.07.006.  Google Scholar

[30]

S. Sharma, A. Mondal and A. K. Pal, Stability analysis and optimal control of avian influenza virus A with time delays, International Journal of Dynamics and Control, 6 (2018), 1351-1366.  doi: 10.1007/s40435-017-0379-6.  Google Scholar

[31]

Z. Shi, X. Zhang and D. Jiang, Dynamics of an avian influenza model with half-saturated incidence, Applied Mathematics and Computation, 355 (2019), 399-416.  doi: 10.1016/j.amc.2019.02.070.  Google Scholar

[32]

P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[33]

A. Wang and Y. Xiao, A filippov system describing media effects on the spread of infectious diseases, Nonlinear Analysis: Hybrid Systems, 11 (2014), 84-97.  doi: 10.1016/j.nahs.2013.06.005.  Google Scholar

[34]

World Health Organization, WHO risk assessments of human infection with avian influenza A(H7N9) virus, 2017, https://www.who.int/influenza/human_animal_interface/influenza_h7n9/Risk_Assessment/en/. Google Scholar

[35]

Y. Xiao, X. Sun and S. Tang, Transmission potential of the novel avian influenza A(H7N9) infection in mainland China, Journal of Theoretical Biology, 352 (2014), 1-5.  doi: 10.1016/j.jtbi.2014.02.038.  Google Scholar

[36]

H. Yoko, K. Yoshinari and Y. Takehisa, Potential risk associated with animal culling and disposal during the foot-and-mouth disease epidemic in japan in 2010, Research in Veterinary Science, 102 (2015), 228-230.  doi: 10.1016/j.rvsc.2015.08.017.  Google Scholar

[37]

X. Zhang, Global dynamics of a stochastic avian-human influenza epidemic model with logistic growth for avian population, Nonlinear Dynamics, 90 (2017), 2331-2343.  doi: 10.1007/s11071-017-3806-5.  Google Scholar

[38]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, Journal of Mathematical Analysis and Applications, 348 (2008), 433-443.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

Figure 1.  Schematic diagram of the model with delay
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Figure 2.  The optimal states of $ I_a(t) $ and $ I_h(t) $, and optimal controls under all of control
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Figure 3.  The optimal states of $ I_a(t) $ and $ I_h(t) $ with one control and without control
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Figure 4.  The optimal states of $ I_a(t) $ and $ I_h(t) $ with two controls and without control
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Figure 5.  Values of objective function under different time delays for model (6)
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Figure 6.  Effect of $ \mathscr{R}_0 $
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Figure 7.  Effect of $ \alpha $
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Figure 8.  Effect of $ \alpha_1 $
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Figure 9.  Effect of $ \alpha_2 $
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Figure 10.  Effect of $ \beta_1 $
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Figure 11.  Effect of $ \beta_1 $
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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.
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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 [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
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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
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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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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