Integer and fractional order models for rabies: a theoretical approach

Document Type : Original Article


Department of Mathematics, Faculty of Science, University of Lagos, Lagos, Nigeria


The rabies virus usually transmits either from animal to man or animal to animal or human to human as the case may be but the major transmitters of the virus are dogs globally. In this work, an integer order SEIR epidemic model was formulated to quantify and give an insight into the spread of rabies within the dog population. The integer order model was studied qualitatively and its results were obtained theoretically before it was modified into the non-integer order counterpart to cater for the data agreement deficiency associated with the integer order epidemic models. As rabies prevalence varies from localities to localities, the theoretical outcomes of the analysis could reveal the genuine picture of rabies dynamics in a particular locality if the actual rabies data of the locality is available.


[1] K.M. Addo, An SEIR mathematical model for dog rabies case study: Bongo district, Ghana, M. Phil thesis, Kwame Nkrumah University of Science and Technology, 2012.
[2] J.O. Aiyedun, B.O. Olugasa, Use of aerial photograph to enhance dog population census in Ilorin, Nigeria, SJVS, 10(1) 2012, 22-27.
[3] T. Ameen, The effect of vaccination and treatment of measles disease described by a fractional order model, World Journal of Modelling and Simulation, 14(1) 2018, 30-38.
[4] J.K.K. Asamoah, F.T. Oduro, E. Bonyah, B. Seidu, Modelling of rabbies transmission dynamics using optimal control analysis, Hindawi Journal of Applied Mathematics, 2017.
[5] A.A. Ayoade, O.J. Peter, T.A. Ayoola, S. Amadiegwu, A.A. Victor, A saturated treatment model for the transmission dynamics of rabies, Malaysian Journal of Computing, 4(1) 2019, 201-213.
[6] A. A. Ayoade, M. O. Ibrahim, O. Odetunde, Analytical solution of a fractional order cholera model, Nigerian Journal of Scienti c Research, 17(2) 2018, 158-164.
[7] A.A. Ayoade, S.O. Agboola, M.O. Ibrahim, Mathematical analysis of the effect of maternal immunity on the global eradication of measles, Anale. Seria Informatica, 17(1) 2019, 235-241.
[8] A.A. Ayoade, P. I. Farayola, T.O. Lamidi, Dynamics and stability analysis of party switching in politics of Nigeria: a mathematical approach, Daffoldil International University Journal of Science and Technology, 14(1) 2019, 53-60.
[9] N.N. Chidumayo, System dynamics modelling approach to explore the effect of dog demography on rabies vaccination coverage in Africa, PLoS One, 13(10) 2018, e0205884.
[10] A. Conan, O. Akerele, G. Simpson, B. Reininghaus, J. van Rooyen, D. Knobel, Population dynamics of owned, free-roaming dogs: implication for rabies control, PLoS Neglected Tropical Diseases, 9(11) 2015, e0004177.
[11] A.M. Czupryna, J.S. Brown, M.A. Bigambo, C.J. Whelan, S.D. Mehta, R.M. Santymire, et al., Ecology and demography of free-roaming domestic dogs in rural villages near Serengeti National Park, Tanzania, PLos One, 11(11) 2016, e0167092.
[12] S. Darkaoui, O. Fassi Fihri, J.L. Schereffer, N. Aboul daa, M. Wasniewski, K. Zouine, et al., Immunogenicity and efficacy of rabivac vaccine for animals rabies control in Morocco, Clin Exp Vaccine Res., 5(1) 2016, 60-69.
[13] R. Darzi, B. Agheli, An analytic approach for system of fractional differential equations by means of innovative homotopy perturbation method, Mathematica Moravica, 22(1) 2018, 93-105.
[14] E. Dirmirci, A new mathematical approach for rabies endemic, Applied Mathematical Sciences, 8(2) 2014, 59-67.
[15] K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynamics, 71(4) 2013, 613-619.
[16] E. Donkoh, K. Kabo-Bah, S. Aseidu, The trend of dog rabies with vaccination using SEIR model case study: Bolgatanga district, Ghana, International Journal of Applied Mathematical Research, 3(4) 2014, 380-389.
[17] B. Ebenezer, On fractional order in uenza A epidemic model, Applied and Computational Mathematics, 4(2) 2015, 77-82.
[18] T.T. Ega, L.S. Luboobi, D. Kuznestsov, Modelling the dynamics of rabies transmission with vaccination and stability analysis, Applied and Computational Mathematics, 4(6) 2015, 409-419.
[19] M. El-Shahed, Fractional order model for the spread of Leptospirosis, International Journal of Mathematical Analysis, 8(54) 2014, 2651-2667.
[20] H.A.A. El-Saka, E. Ahmed, A fractional order network model for Zika, BioRxiv, 2016, 039917.
[21] A. Garba, A. Dzikwi, H. Kazeem, O. Makanju, F. Hambagba, N. Abduazeez, et al., Dog ecology and management in Niger State, Nigeria: a basic tools for rabies control, JAERI, 12(1) 2017, 1-9.
[22] A.D. Gibson, I.G. Handel, K. Shervell, T. Roux, D. Mayer, S. Muyila, et al., The vaccination of 35 000 dogs in 20 working days using combined static point and door-to-door methods in Blantyre, Malawi, PLoS Neglected Tropical Diseases, 10(7) 2018, e0004824.
[23] D. Guo, W. Yin, H. Yu, J-C. Thill, W. Yang, F. Chen, et al., The role of socioeconomic and climate factors in the spatiotemporal variation of human rabies in China, BMC Infectious Diseases, 18 2018, 1-13.
[24] S.E. Hambolu, A.A. Dzikwi, J.K. Kwaga, H.M. Kazeem, J.U. Umoh, D.A. Hambolu, Dog ecology and population studies in Lagos State, Nigeria, Global Journal Health Science, 6(2) 2014, 209-220.
[25] E.H. Hikufe, Rabies sero-survey in vaccinated domestic dogs and knowledge assessment of rabies among dog owners, Ohangwena region, Namibia, M.Sc. thesis, University of Namibia, 2016.
[26] D. Hyeroba, S. Frianti, J. Acon, J. Okwee-Acai, T.L. Goldberg, Demography and health of village dogs in rural Western Uganda, Prev Vet Med., 137 2017, 24-27.
[27] R. Jan, A. Jan, MSGDTM for solution of fractional order dengue disease model, International Journal of Science and Research, 6(3) 2017, 1140-1144.
[28] H. Kadowaki, K. Hampson, K. Tojinbara, A. Yamada, K. Makita, The risks of rabies spread in Japan: a mathematical modelling assessment, Epidemiology and Infection, 146 2018, 1245-1252.
[29] D.L. Knobel, S. Arega, B. Reininghaus, G.J. Simpson, B.D. Gessner, H. Stryhn, et al., Rabies vaccines is associated with decrease all-cause mortality in dogs, Vaccine, 35(1) 2017, 3844-3849.
[30] M. Laager, C. Mbilo, E.A. Madaye, A. Naminou, M. Lechenne, A. Tschoop, et al., The importance of dog population contact network structures in rabies transmission, PLoS Neglected Tropical Diseases, 12(8) 2018, e0006680.
[31] K. LeRoux, D. Stewart, K. Perrett, L.H. Nel, J. Kesssels, B. Abela-Ridder, Rabies control in KwaZulu-Natal, South Africa, Bull World Health Organisation, 96(5) 2018, 360-371.
[32] S. Mauti, A. Traore, A. Sery, W. Bryssinckx, J. Hattendorf, J. Zinsstag,First study on domestic dog ecology, demographic structure and dynamics in Bamako, Mali. Prev Vet Med., 146 2017, 44-51.
[33] E. Okyere, F. T. Oduro, S. K. Amponsah, T. K. Dontwi, N. K. Frempong, Fractional order SIR model with constant population, British Journal of Mathematics and Computer Science, 14(2) 2016, 1-12.
[34] G.R. Otolorin, J. U. Umoh, A.A. Dzikwi, Demographic and ecological survey of dog population in Aba, Abia State, Nigeria, International Scholarly Research Notices, 2014 2014.
[35] S. Ruan, Modelling the transmission dynamics and control of rabies in China, Mathematical Biosciences, 286 2017, 65-93.
[36] E.D. Salahot, Mathematical modelling of rabies transmission and the impact of prevention strategies: a case study of Tanzania, M.Sc. dissertation, African Institute for Mathematical Sciences (AIMS)
[37] M. Sambo, S. Cleaveland, H. Ferguson, T. Lembo, C. Simon, H. Urassa, K. Hampson, The burden of rabies in Tanzania and its impact on local communities, PLoS Neglected Tropical Diseases, 7(11) 2013, e2510.
[38] O. Sharomi, T. Malik, Optimal control in epidemiology, Annals Operation Research, 25(12) 2017, 55-71.
[39] R. Tschoop, S. Bekele, A. Aseffa, Dog demography, animal bite management and rabies knowledge attitude and practices in the Awash Basin, Eastern Ethiopia, PLos Neglected Tropical Diseases, 10(2) 2016, e0004471.
[40] A.M. Tulu, P.R. Koya, The impact of infective immigrants on spread of dog rabies, American Journal of Applied Mathematics, 5(3)2017, 68-77.
Volume 3, Issue 1
February 2022
Pages 1-9
  • Receive Date: 17 October 2021
  • Revise Date: 30 November 2021
  • Accept Date: 03 December 2021
  • First Publish Date: 13 January 2022