# A Fractional-order mathematical model to analyze the stability and develop a sterilization strategy for the habitat of stray dogs

## Authors

• Zafer Öztürk Institute of Science, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Turkiye
• Ali Yousef Department of Natural Sciences and Mathematics, College of Engineering, International University of Science and Technology in Kuwait, 92400 Al-Ardiya, Kuwait
• Halis Bilgil Faculty of Engineering, Architecture and Design, Kayseri University, 38280, Kayseri, Turkiye
• Sezer Sorgun Department of Mathematics, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Türkiye

## Keywords:

Fractional-order Differential Equation, Euler Method, Stability Analysis, Stray Dog Population and Sterilization, Strategy, Semi-cycle Solution

## Abstract

Today, the socio-cultural lack of some countries with increased urbanization has led to the unconscious breeding of stray dogs. The failure to care for the offspring of possessive dogs or ignoring the responsibility to find a suitable family for the offspring increased the dog population on the streets and in the shelters. In this study, our main target is to analyze the habitat of stray dogs and the strategy of how to control the population without damaging the ecosystem of the species. For this aim, we establish a fractional-order differential equation system to investigate the fractal dimension with long-term memory that invovles two compartments; the non-sterilized dog population (x(t)) and the sterilized one (y(t)). Firstly, we analyze the stability of the equilibrium points using the Routh-Hurwitz criteria to discuss cases that should not affect the ecosystem of the dog population, but control the stray dog population in the habitat. Since the intervention to the stray dog population occurs at discrete time impulses, we use the Euler method's discretization process to analyse the local and global stability around the equilibrium points. Besides this, we show that the solutions of the system represent semi-cycle behaviors. At the end of the study, we use accurate data to demonstrate the sterilization rate of stray dogs in their habitat.

## Author Biographies

### Zafer Öztürk, Institute of Science, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Turkiye

Zafer Öztürk currently a doctorate student of mathematics in the Department of Mathematics, Nevsehir Haci Bektas Veli University. His interests include fractional modeling and bifurcations.

### Ali Yousef, Department of Natural Sciences and Mathematics, College of Engineering, International University of Science and Technology in Kuwait, 92400 Al-Ardiya, Kuwait

Ali Yousef obtained his Ph.D. from the School of Mathematics, the University of Southampton, the UK. His main research area is sequential estimation and biological mathematics. He established the Department of Basic Sciences at Middle East University MEU in Jordan and was the statistical consultation center manager. In February 2016, he established and coordinated the mathematics program at Kuwait College of Science and Technology until September 2020. He published papers in sequential estimation and biomathematics related to HIV, breast cancer, Monoclonal Tumors, and Covid-19 in high-impact journals. Recently, he has been the chair of the Department of Natural Sciences and Mathematics at the International University of Science and Technology in Kuwait. In addition, a member of several committees in the university. He teaches mathematics, probability, and statistics to science and engineering students.

### Halis Bilgil, Faculty of Engineering, Architecture and Design, Kayseri University, 38280, Kayseri, Turkiye

Halis Bilgil received Ph.D. in Mathematics from Erciyes University, Turkiye in 2010. He is a Professor with the Department of engineering basic sciences, Kayseri University, Turkiye. His current research interests include fractional differential systems, nonlinear evaluation equations, grey modeling, fluid mechanics and numerical analysis.

### Sezer Sorgun, Department of Mathematics, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Türkiye

Sezer Sorgun is an associate professor at Department of Mathematics, Science and Letter Faculty, Nevsehir Haci Bektas Veli University. He received his PhD degree from the Erciyes University, Kayseri, Turkey. His research interests include ODEs, graf theory and matrix theory.

## References

Amaku, M., Dias, R.A. & Ferreira, F. (2010). Dynamics and control of stray dog populations. Mathematical Population Studies, 17 (2), 69-78. https://doi.org/10.1080/08898481003689452

Fournier, A. & Geller, E. (2004). Behavior analysis of companion-animal overpopulation: a conceptualization of the problem and suggestions for intervention. Behavior and Social Issues, 68, 51- 68. https://doi.org/10.5210/bsi.v13i1.35

Thirthar, A. A., Majeed, S. J., Shah, K., & Abdeljawad, T. (2022). The dynamics of an aquatic ecological model with aggregation, fear and harvesting effect. AIMS Mathematics, 7(10), 18532-18552. https://doi.org/10.3934/math.20221018

Thirthar, A. A., Panja, P., Khan, A., Alqudah, M. A., & Abdeljawad, T. (2023). An ecosystem model with memory effect considering global warming phenomena and an exponential fear function. Fractals, 31(10), 1-19. https://doi.org/10.1142/S0218348X2340162X

Thirthar, A. A. (2023). A mathematical modelling of a plant-herbivore community with additional effects of food on the environment. Iraqi Journal of Science, 64(7), 3551-3566.

Yousef, A. & Bozkurt Yousef, F. (2019). Bifurcation and stability analysis of a system of fractional-order differential equations for a plant- herbivore model with Allee effect. Mathematics, 7 (454), 1-18. https://doi.org/10.3390/math7050454

Morters, M. K., McKinley, T. J., Restif, O., Conlan, A. J. K., Cleaveland, S., Hampson, K., Whay, H. R., Damriyasa, I. M. & Wood, J.L.N. (2014). The demography free-roaming dog populations and applications to disease and population control. Journal of Applied Ecology, 51, 1096-1106. https://doi.org/10.1111/1365-2664.12279

Sene, N. (2022). Theory and applications of new fractional-order chaotic system under Caputo operator. An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 12(1), 20-38. https://doi.org/10.11121/ijocta.2022.1108

Kashyap, A.J., Bhattacharjee, D. & Sarmah, H.K. (2021). A fractional model in exploring the role of fear in mass mortality of pelicans in the Salton Sea. An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 11(3), 28-51.

Podlubny, I. (1999). Fractional Differential Equations. Academy Press, San Diego CA.

Nunes, C. M., de Lima, V. M. F., de Paula, H. B., Perri, S. H. V., de Andrade, A. M., Dias, F. E. F., & Burattini, M. N. (2008). Dog culling and replacement in an area endemic for visceral leishmaniasis in Brazil. Veterinary Parasitology, 153, 19-23. https://doi.org/10.1016/j.vetpar.2008.01.005

Linda, J.S.A. (2007). An Introduction to Mathematical Biology. Pearson Education Ltd., USA, 123-127.

Bilgil, H., Yousef, A., Erciyes, A., Erdinc, U. & Ozturk, Z. (2022). A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event. Journal of Computational and Applied Mathematics, 115015. https://doi.org/10.1016/j.cam.2022.115015

Hethcote, H., Zhien, M., & Shengbing, L. (2002). Effects of quarantine in six endemic models for infectious diseases. Mathematical Biosciences, 180, 141160. https://doi.org/10.1016/S0025-5564(02)00111-6

Ozturk, Z., Bilgil, H., & Sorgun, E. (2023). Application of fractional SIQRV model for SARS-CoV- 2 and stability analysis. Symmetry, 15(5), 1-13. https://doi.org/10.3390/sym15051048

Slater, M. R. (2001). The role of veterinary epidemiology in the study of free-roaming dogs and cats. Preventive Veterinary Medicine, 48, 273- 286. https://doi.org/10.1016/S0167-5877(00)00201-4

Allen L. J. S. (2007). An Introduction to Mathematical Biology. Department of Mathematics and Statistics, Texas Tech University, Pearson Education., 348.

Santos Baquero, O., Amaku, M., & Ferreira, F. (2015). capm: An R package for Companion Animal Population Management.

Ozturk, Z. , Bilgil, H., & Erdinc, U. (2022). An optimized continuous fractional grey model for forecasting of the time dependent real world cases. Hacettepe Journal of Mathematics and Statistics, 51 (1), 308-326. https://doi.org/10.15672/hujms.939543

Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics, in: Proceedings of the Royal Society of London, Series A, Containing Papers of a Mathematical and Physical Character, 115 (772), 700-721. https://doi.org/10.1098/rspa.1927.0118

Yousef, A., Bozkurt, F., & Abdeljawad, A. (2020). Qualitative analysis of a fractional pandemic spread model of the novel coronavirus (COVID- 19). Comput Materials Continua, 66, 843-869. https://doi.org/10.32604/cmc.2020.012060

Bozkurt Yousef, F., Yousef, A., Abdeljawad, T., & Kalinli, A. (2020). Mathematical modeling of breast cancer in a mixed immune-chemotherapy treatment considering the effect of ketogenic diet. The European Physical Journal Plus, 135(12), 1- 23. https://doi.org/10.1140/epjp/s13360-020-00991-8

Li, L., & Liu, J. G. (2016). A generalized definition of Caputo derivatives and its application to fractional ODEs. SIAM Journal on Mathematical Analysis, 50(3), 2867-2900.

Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations, v.204, Elsevier.

Yaro, D., Omari-Sasu, S. K., Harvim, P., Saviour, A. W. & Obeng, B. A. (2015). Generalized Euler method for modeling measles with fractional differential equations. The International Journal of Innovative Research and Development, 4(4), 380- 384.

Nazir, G., Zeb, A., Shah, K., Saeed, T., Khan, R. A. & Khan, S. I. U. (2021). Study of COVID-19 mathematical model of fractional order via modified Euler method. Alexandria Engineering Journal, 60(6), 5287-5296. https://doi.org/10.1016/j.aej.2021.04.032

Gibbons, C., Kulenovic, M. R. S., Ladas, G. (2000). On the recursive sequence xn+1=(alpha+betaxn-1)/(y+xn). Mathematical Sciences Research Hot-line, 4(2), 1-11.

Cunningham, K., Kulenovic, M.R.S., Ladas, G. & Valicenti, S.V. (2001). On the recursive sequence xn+1=(alpha+betaxn-1)/(Bxn+Cxn-1). Non- linear Analysis: Theory, Methods and Applications, 47, 4603-4614. https://doi.org/10.1016/S0362-546X(01)00573-9

https://www.bbc.com, [Date Accessed 19 June 2023]

Garcia, R.D.C.M., Calderon, N., & Ferreira, F. (2012). Consolidation of international guidelines for the management of canine populations in urban areas and proposal of indicators for their management. Revista Panamericana de Salud Publica, 32 , 140-144. https://doi.org/10.1590/S1020-49892012000800008

Baquero, O. S., Akamine, L. A., Amaku, M., Ferreira, F. (2016). Dening priorities for dog population management through mathematical modeling. Preventive Veterinary Medicine, 123, 121-127. https://doi.org/10.1016/j.prevetmed.2015.11.009

## Published

2024-03-30
CITATION
DOI: 10.11121/ijocta.1418
Published: 2024-03-30

## How to Cite

Öztürk, Z., Yousef, A., Bilgil, H., & Sorgun, S. (2024). A Fractional-order mathematical model to analyze the stability and develop a sterilization strategy for the habitat of stray dogs. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(2), 134–146. https://doi.org/10.11121/ijocta.1418

## Section

Research Articles