In this work, the concept of the fractional derivative is used to improve a mathematical model for the transmission dynamics of pneumonia in the Al-Baha region of the Kingdom of Saudi Arabia. We establish a dynamics model to predict the transmission of pneumonia in some local sheep and goat herds. The proposed model is a generalization of a system of five ordinary differential equations of the first order, regarding five unknowns, which are the numbers of certain groups of animals (susceptible, vaccinated, carrier, infected, and recovered). This consists of investigating the equilibrium, basic reproduction number, stability analysis, and bifurcation analysis. It is observed that the free equilibrium point is local and global asymptotic stable if the basic reproduction number is less than one, and the endemic equilibrium is local and global asymptotic stable if the basic reproduction number is greater than one. The optimal control problem is formulated using Pontryagin's maximum principle, with three control strategies: Disease prevention through education, treatment, and screening. The most cost-effective intervention strategy to combat the pneumonia pandemic is a combination of prevention and treatment, according to the cost-effectiveness analysis of the adopted control techniques. A numerical simulation is performed, and the significant data are graphically displayed. The results predicted by the model show a good agreement with the actual reported data.
Citation: Sayed Saber, Azza M. Alghamdi, Ghada A. Ahmed, Khulud M. Alshehri. Mathematical Modelling and optimal control of pneumonia disease in sheep and goats in Al-Baha region with cost-effective strategies[J]. AIMS Mathematics, 2022, 7(7): 12011-12049. doi: 10.3934/math.2022669
In this work, the concept of the fractional derivative is used to improve a mathematical model for the transmission dynamics of pneumonia in the Al-Baha region of the Kingdom of Saudi Arabia. We establish a dynamics model to predict the transmission of pneumonia in some local sheep and goat herds. The proposed model is a generalization of a system of five ordinary differential equations of the first order, regarding five unknowns, which are the numbers of certain groups of animals (susceptible, vaccinated, carrier, infected, and recovered). This consists of investigating the equilibrium, basic reproduction number, stability analysis, and bifurcation analysis. It is observed that the free equilibrium point is local and global asymptotic stable if the basic reproduction number is less than one, and the endemic equilibrium is local and global asymptotic stable if the basic reproduction number is greater than one. The optimal control problem is formulated using Pontryagin's maximum principle, with three control strategies: Disease prevention through education, treatment, and screening. The most cost-effective intervention strategy to combat the pneumonia pandemic is a combination of prevention and treatment, according to the cost-effectiveness analysis of the adopted control techniques. A numerical simulation is performed, and the significant data are graphically displayed. The results predicted by the model show a good agreement with the actual reported data.
| [1] |
Z. Yener, F. Ilhan, Z. Ilhan, Y. S. Saglam, Immunohistochemical detection of Mannheimia (Pasteurella) haemolytica antigens in goats with natural pneumonia, Vet. Res. Commun., 33 (2009), 305–313. https://doi.org/10.1007/s11259-008-9178-z doi: 10.1007/s11259-008-9178-z
|
| [2] | D. M. West, A. N. Bruere, A. L. Ridler, The sheep: Health, disease & production: Written for veterinarians and farmers, Veterinary Continuing Education, Massey University, 2002. |
| [3] | G. Salah, A. R. Ferial, R. A. Mohamed, S. Ihab, Small ruminant breeds of Egypt, In: Characterization of small ruminant breeds in West Asia and North Africa, Aleppo: ICARDA, 2 (2005), 141–193. |
| [4] | R. L. Baker, G. D. Gray, Appropriate breeds and breeding schemes for sheep and goats in the tropics, Worm Control Small Ruminants Trop. Asia, 63 (2004), 63–95. |
| [5] | T. U. Obi, Clinical and epidemiological studies on PPR in sheep and goats in Southern Nigeria, Ph.D Thesis, Univ. Ibadan, 1984. |
| [6] |
B. O. Emikpe, S. O. Akpavie, Clinicopathologic effects of peste des petit ruminant virus infection in West African dwarf goats, Small Ruminant Res., 95 (2011), 168–173. https://doi.org/10.1016/j.smallrumres.2010.09.009 doi: 10.1016/j.smallrumres.2010.09.009
|
| [7] | M. Naveed, M. T. Javed, A. Khan, R. Kausar, Haematological and bacteriological studies in neonatal lambs with reference to neonatal lamb mortality, Pakistan Vet. J., 19 (1999), 127–131. |
| [8] | G. Ayelet, L. Yigezu, E. Gelaye, S. Tariku, K. Asmare, Epidemiologic and serologic investigation of multifactorial respiratory disease of sheep in the central highland of Ethiopia, Int. J. Appl. Res. Vet. Med., 2 (2004), 275–278. |
| [9] | Manual of diagnostic tests and vaccines for terrestrial animals, Paris: Office Intl Des Epizooties, 2008, 1092–1106. |
| [10] |
K. A. Goodwin, R. Jackson, C. Brown, P. R. Davies, R. S. Morris, N. R. Perkins, Pneumonic lesions in lambs in New Zealand: Patterns of prevalence and effects on production, New Zeal. Vet. J., 52 (2004), 175–179. https://doi.org/10.1080/00480169.2004.36425 doi: 10.1080/00480169.2004.36425
|
| [11] | Ministry of Environment, Water and Agriculture, 2022. Available from: https://www.mewa.gov.sa/en/Pages/default.aspx. |
| [12] |
A. Melegaro, N. J. Gay, G. F. Medley, Estimating the transmission parameters of pneumococcal carriage in households, Epidemiol. Infect., 132 (2004), 433–441. https://doi.org/10.1017/S0950268804001980 doi: 10.1017/S0950268804001980
|
| [13] | E. Joseph, Mathematical analysis of prevention and control strategies of pneumonia in adults and children, Diss. Univ. Dar es Salaam, Tanzania, 2012. |
| [14] | D. Ssebuliba, Mathematical modelling of the effectiveness of two training interventions on infectious diseases in Uganda, Ph.D Stellenbosch Univ., 2013. |
| [15] |
K. O. Okosun, O. D. Makinde, Modelling the impact of drug resistance in malaria transmission and its optimal control analysis, Int. J. Phys. Sci., 6 (2011), 6479–6487. https://doi.org/10.5897/IJPS10.542 doi: 10.5897/IJPS10.542
|
| [16] |
K. O. Okosun, O. D. Makinde, On a drug-resistant malaria model with susceptible individuals without access to basic amenities, J. Biol. Phys., 38 (2012), 507–530. https://doi.org/10.1007/s10867-012-9269-5 doi: 10.1007/s10867-012-9269-5
|
| [17] | K. O. Okosun, O. D. Makinde, Optimal control analysis of Malaria in the presence of nonlinear incidence rate, Appl. Comput. Math., 12 (2013), 20–32. |
| [18] |
K. O. Okosun, O. D. Makinde, A co-infection model of malaria and cholera diseases with optimal control, Math. Biosci., 258 (2014), 19–32. https://doi.org/10.1016/j.mbs.2014.09.008 doi: 10.1016/j.mbs.2014.09.008
|
| [19] |
C. A. Okaka, J. Y. T. Mugisha, A. Manyonge, C. Ouma, Modelling the impact of misdiagnosis and treatment on the dynamics of malaria concurrent and co-infection with pneumonia, Appl. Math. Sci., 7 (2013), 6275–6296. https://doi.org/10.12988/ams.2013.39521 doi: 10.12988/ams.2013.39521
|
| [20] |
J. Ong'ala, J. Y. T Mugisha, P. Oleche, Mathematical model for pneumonia dynamics with carriers, Int. J. Math. Anal., 7 (2013), 2457–2473. https://doi.org/10.12988/ijma.2013.35109 doi: 10.12988/ijma.2013.35109
|
| [21] |
G. T. Tilahun, O. D. Makinde, D. Malonza, Modelling and optimal control of pneumonia disease with cost-effective strategies, J Biol Dyn., 11 (2017), 400–426. https://doi.org/10.1080/17513758.2017.1337245 doi: 10.1080/17513758.2017.1337245
|
| [22] |
G. T. Tilahun, O. D. Makinde, D. Malonza, Co-dynamics of pneumonia and typhoid fever diseases with cost effective optimal control analysis, Appl. Math. Comput., 316 (2018), 438–459. https://doi.org/10.1016/j.amc.2017.07.063 doi: 10.1016/j.amc.2017.07.063
|
| [23] |
A. Akgul, E. Karatas, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos: Interdiscip. J. Nonlinear Sci., 29 (2019), 023108. https://doi.org/10.1063/1.5084035 doi: 10.1063/1.5084035
|
| [24] |
M. H. Alshehri, F. Z. Duraihem, A. Ahmad, S. Saber, A Caputo (discretization) fractional-order model of glucose-insulin interaction: Numerical solution and comparisons with experimental data, J. Taibah Univ. Sci., 15 (2021), 26–36. https://doi.org/10.1080/16583655.2021.1872197 doi: 10.1080/16583655.2021.1872197
|
| [25] | S. M. Al-Zahrani, F. E. I. Elsmih, K. S. Al-Zahrani, S. Saber, A fractional order SITR model for forecasting of transmission of COVID-19: Sensitivity statistical analysis, Malays. J. Math. Sci., in press. |
| [26] |
M. H. Alshehri, S. Saber, F. Z. Duraihem, Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction, Int. J. Nonlinear Sci. Numer. Simul., 2021. https://doi.org/10.1515/ijnsns-2020-0201 doi: 10.1515/ijnsns-2020-0201
|
| [27] | A. Ahmad, S. Saber, Stability analysis and numerical simulations of the fractional COVID-19 pandemic model, Int. J. Nonlinear Sci. Numer. Simul., in press. |
| [28] |
M. A. Dokuyucu, E. Celik, H. Bulut, H. M. Baskonus, Cancer treatment model with the Caputo-Fabrizio fractional derivative, Eur. Phys. J. Plus, 133 (2018), 92. https://doi.org/10.1140/epjp/i2018-11950-y doi: 10.1140/epjp/i2018-11950-y
|
| [29] |
M. A. Dokuyucu, H. Dutta, A fractional order model for Ebola virus with the new Caputo fractional derivative without singular kernel, Chaos, Solitons Fract., 134 (2020), 109717. https://doi.org/10.1016/j.chaos.2020.109717 doi: 10.1016/j.chaos.2020.109717
|
| [30] |
D. Baleanu, A. Fernandez, A. Akgül, On a fractional operator combining proportional and classical differintegrals, Mathematics, 8 (2020), 1–13. https://doi.org/10.3390/math8030360 doi: 10.3390/math8030360
|
| [31] |
C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404. https://doi.org/10.3934/mbe.2004.1.361 doi: 10.3934/mbe.2004.1.361
|
| [32] |
H. Delavari, D. Baleanu, J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dyn., 67 (2012), 2433–2439. https://doi.org/10.1007/s11071-011-0157-5 doi: 10.1007/s11071-011-0157-5
|
| [33] |
M. Caputo, Linear model of dissipation whose Q is almost frequency independent-Ⅱ, Geophys. J. Int., 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
|
| [34] |
A. Al-Aklabi, A. W. Al-Khulaidi, A. Hussain, N. Al-Sagheer, Main vegetation types and plant species diversity along an altitudinal gradient of Al Baha region, Saudi Arabia, Saudi J. Biol. Sci., 23 (2016), 687–697. https://doi.org/10.1016/j.sjbs.2016.02.007 doi: 10.1016/j.sjbs.2016.02.007
|
| [35] | R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific Publishing, 2000. https://doi.org/10.1142/3779 |
| [36] |
M. H. Al-Smadi, G. N. Gumah, On the homotopy analysis method for fractional SEIR epidemic model, Res. J. Appl. Sci. Eng. Technol., 7 (2014), 3809–3820. http://dx.doi.org/10.19026/rjaset.7.738 doi: 10.19026/rjaset.7.738
|
| [37] | H. Sherief, A. M. A. El-Sayed, S. Behiry, W. E. Raslan, Using fractional derivatives to generalize the Hodgkin-Huxley model, In: Fractional dynamics and control, Springer, 2012,275–282. https://doi.org/10.1007/978-1-4614-0457-6_23 |
| [38] |
M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134–147. https://doi.org/10.1007/BF00879562 doi: 10.1007/BF00879562
|
| [39] |
M. Caputo, F. Mainardi, Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento, 1 (1971), 161–198. https://doi.org/10.1007/BF02820620 doi: 10.1007/BF02820620
|
| [40] |
M. Caputo, Vibrations of an infinite viscoelastic layer with a dissipative memory, J. Acoust. Soc. Am., 56 (1974), 897–904. https://doi.org/10.1121/1.1903344 doi: 10.1121/1.1903344
|
| [41] |
H. Sherief, A. M. A. El-Sayed, A. M. Abd El-Latief, Fractional order theory of thermoelasticity, Int. J. Solids Struct., 47 (2010) 269–275. https://doi.org/10.1016/j.ijsolstr.2009.09.034 doi: 10.1016/j.ijsolstr.2009.09.034
|
| [42] | W. E. Raslan, Application of fractional order theory of thermoelasticity to a 1D problem for a cylindrical cavity, Arch. Mech., 66 (2014), 257–267. |
| [43] |
W. E. Raslan, Application of fractional order theory of thermoelasticity in a thick plate under axisymmetric temperature distribution, J. Therm. Stresses, 38 (2015), 733–743. https://doi.org/10.1080/01495739.2015.1040307 doi: 10.1080/01495739.2015.1040307
|
| [44] | L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishechenko, The mathematical theory of optimal processes, JohnWiley & Sons, New York, 1962. |
| [45] |
A. Akgul, A novel method for a fractional derivative with non-local and nonsingular kernel, Chaos Soliton Fract., 114 (2018), 478–482. https://doi.org/10.1016/j.chaos.2018.07.032 doi: 10.1016/j.chaos.2018.07.032
|
| [46] |
A. Akgul, M. Modanli, Crank-Nicholson difference method and reproducing kernel function for third order fractional differential equations in the sense of Atangana-Baleanu Caputo derivative, Chaos Soliton Fract., 127 (2019), 10–16. https://doi.org/10.1016/j.chaos.2019.06.011 doi: 10.1016/j.chaos.2019.06.011
|
| [47] | I. Podlubny, Fractional differential equations, New York: Academic Press, 1998. |
| [48] |
V. Daftardar-Gejji, H. Jafari, Analysis of a system of non autonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026–1033. https://doi.org/10.1016/j.jmaa.2006.06.007 doi: 10.1016/j.jmaa.2006.06.007
|
| [49] |
W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709–726. https://doi.org/10.1016/j.jmaa.2006.10.040 doi: 10.1016/j.jmaa.2006.10.040
|
| [50] |
H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, Z. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435–449. https://doi.org/10.1007/s12190-016-1017-8 doi: 10.1007/s12190-016-1017-8
|
| [51] |
C. Vargas-De-Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 75–85. https://doi.org/10.1016/j.cnsns.2014.12.013 doi: 10.1016/j.cnsns.2014.12.013
|
| [52] |
J. P. C. dos Santos, E. Monteiro, G. B. Vieira, Global stability of fractional SIR epidemic model, Proc. Ser. Braz. Soc. Appl. Comput. Math., 5 (2017), 1–7. https://doi.org/10.5540/03.2017.005.01.0019 doi: 10.5540/03.2017.005.01.0019
|
| [53] | J. P. LaSalle, The stability of dynamics systems, In: CBMS-NSF regional conference series in applied mathematics, SIAM, Philadelphia, 1976. https://doi.org/10.1137/1.9781611970432 |
| [54] |
J. LaSalle, Some extensions of Liapunov's second method, IRE Trans. Circuit Theory, 7 (1960), 520–527. https://doi.org/10.1109/TCT.1960.1086720 doi: 10.1109/TCT.1960.1086720
|
| [55] |
C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404. https://doi.org/10.3934/mbe.2004.1.361 doi: 10.3934/mbe.2004.1.361
|
| [56] |
A. Boukhouima, K. Hattaf, N. Yousfi, Dynamics of a fractional order HIV infection model with specific functional response and cure rate, Int. J. Differ. Equ., 2017 (2017), 8372140. https://doi.org/10.1155/2017/8372140 doi: 10.1155/2017/8372140
|
| [57] |
S. K. Choi, B. Kang, N. Koo, Stability for Caputo fractional differential systems, Abstr. Appl. Anal., 2014 (2014), 631419. https://doi.org/10.1155/2014/631419 doi: 10.1155/2014/631419
|
| [58] |
P. Van den Driessche, J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
|
| [59] |
S. Baba, O. D. Makinde, Optimal control of HIV/AIDS in the workplace in the presence of careless individuals, Comput. Math. Methods Med., 2014 (2014), 1–19. https://doi.org/10.1155/2014/831506 doi: 10.1155/2014/831506
|
| [60] | W. H. Fleming, R. W. Rishel, Deterministic and stochastic optimal control, New york: Springer, 1975. https://doi.org/10.1007/978-1-4612-6380-7 |
| [61] | D. L. Lukes, Differential equations: Classical to controlled, Academic Press, 1982. |
| [62] | W. H. Fleming, R. W. Rishel, Deterministic and stochastic optimal control (stochastic modelling and applied probability, 1), Springer, 1975. |
| [63] | Z. M. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inform., 26 (2008), 15–27. |
| [64] |
Z. Odibat, N. Shawagfeh, Generalized Taylor's formula, Appl. Math. Comput., 186 (2007), 286–293. https://doi.org/10.1016/j.amc.2006.07.102 doi: 10.1016/j.amc.2006.07.102
|
Figures(10) / Tables(1)
Sayed Saber, Azza M. Alghamdi, Ghada A. Ahmed, Khulud M. Alshehri. Mathematical Modelling and optimal control of pneumonia disease in sheep and goats in Al-Baha region with cost-effective strategies[J]. AIMS Mathematics, 2022, 7(7): 12011-12049. doi: 10.3934/math.2022669
DownLoad: