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Date Published:
24/06/2023
Online Published:
26/06/2023
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Views pdf: 95
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Natural Sciences Issue
How to Cite
Jose, S. A., & Raja, R. (2023). Changing perception through the art of mathematical modeling. Dong Thap University Journal of Science, 12(5), 9-20. https://doi.org/10.52714/dthu.12.5.2023.1066
Changing perception through the art of mathematical modeling
Main Article Content
Abstract
Nowadays, infectious diseases are disorders caused by organisms — such as bacteria, viruses, fungi or parasites. Many organisms live in and on our bodies. They’re normally harmless or even helpful. But certain microbes have the potential to cause disease in specific situations. It is possible for some infectious diseases to spread from person to person. Others are spread by animals or insects. In this study, we build certain fundamental models, such as SI, SIR, SIRS, and SEIR, and in numerical simulations, we take into account random parameters to determine the dynamics of the model’s behavior. Finally, we present several studies in mathematical modeling of real situation relevant to epidemiology and population dynamic systems.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Keywords
Art of Mathematical Modeling, Changing Perception, infectious diseases
References
Aba Oud, M. A., Ali, A., Alrabaiah, H., Ullah, S., Khan, M. A., & Islam, S. (2021). A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load. Advances in Difference Equations, 2021(1), 1-19.
Abdulrahman, S. (2014). Stability analysis of the transmission dynamics and control of corruption. Pacific Journal of Science Technology, 15(1), 99-113.
Adu, I. K., Osman, M., & Yang, C. (2017). Mathematical model of drinking epidemic. Br. J. Math. Computer Sci, 22(5).
Agrawal, A., Tenguria, A., & Modi, G. (2018). Role of epidemic model to control drinking problem. Int. J. Sci. Res. in Mathematical Statistical Sciences, 5, 4.
Aguilar-Canto, F. J., de León, U. A.-P., & Avila-Vales, E. (2022). Sensitivity theorems of a model of multiple imperfect vaccines for COVID-19. Chaos, Solitons Fractals, 156, 111844.
Alzahrani, A. K., Alshomrani, A. S., Pal, N., & Samanta, S. (2018). Study of an eco-epidemiological model with Z-type control. Chaos, Solitons Fractals, 113, 197-208.
Athithan, S., Ghosh, M., & Li, X.-Z. (2018). Mathematical modeling and optimal control of corruption dynamics. Asian-European Journal of Mathematics, 11(06), 1850090.
Balatif, O., Labzai, A., & Rachik, M. (2018). A discrete mathematical modeling and optimal control of the electoral behavior with regard to a political party. Discrete Dynamics in Nature Society, 2018, 1-14.
Brauer, F., & Castillo-Chavez, C. (2012). Mathematical models in population biology and epidemiology (Vol. 2): Springer.
Braun, M. (1993). Differential Equation and Their Application (4 ed.): Springer.
Brianzoni, S., Coppier, R., & Michetti, E. (2011). Complex dynamics in a growth model with corruption in public procurement. Discrete Dynamics in Nature Society, 2011.
Costa, M., & dos Anjos, L. (2015). Integrated pest management in a predator-prey system with Allee effects. Neotropical entomology, 44, 385-391.
Cuervo-Cazurra, A. (2016). Corruption in international business. Journal of World Business, 51(1), 35-49.
Dianavinnarasi, J., Cao, Y., Raja, R., Rajchakit, G., & Lim, C. P. (2020). Delay-dependent stability criteria of delayed positive systems with uncertain control inputs: Application in mosquito-borne morbidities control. Applied Mathematics Computational and Theoretical Chemistry, 382, 125210.
Dianavinnarasi, J., Raja, R., Alzabut, J., Cao, J., Niezabitowski, M., & Bagdasar, O. (2022). Application of Caputo–Fabrizio operator to suppress the Aedes Aegypti mosquitoes via Wolbachia: An LMI approach. Mathematics Computers in Simulation, 201, 462-485.
Dianavinnarasi, J., Raja, R., Alzabut, J., Niezabitowski, M., & Bagdasar, O. (2021). Controlling Wolbachia transmission and invasion dynamics among aedes aegypti population via impulsive control strategy. Symmetry, 13(3), 434.
Dianavinnarasi, J., Raja, R., Alzabut, J., Niezabitowski, M., Selvam, G., & Bagdasar, O. (2021). An LMI Approach-Based Mathematical Model to Control Aedes aegypti Mosquitoes Population via Biological Control. Mathematical Problems in Engineering, 2021, 1-18.
Dutta, P., Sahoo, D., Mondal, S., & Samanta, G. (2022). Dynamical complexity of a delay-induced eco-epidemic model with Beddington–DeAngelis incidence rate. Mathematics Computers in Simulation, 197, 45-90.
Ginoux, J.-M., Naeck, R., Ruhomally, Y. B., Dauhoo, M. Z., & Perc, M. (2019). Chaos in a predator–prey-based mathematical model for illicit drug consumption. Applied Mathematics Computation, 347, 502-513.
Giordano, G., Blanchini, F., Bruno, R., Colaneri, P., Di Filippo, A., Di Matteo, A., & Colaneri, M. (2020). Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nature medicine, 26(6), 855-860.
Gonzalez-Parra, G., Martínez-Rodríguez, D., & Villanueva-Micó, R. (2021). Impact of a new SARS-CoV-2 variant on the population: A mathematical modeling approach. Mathematical Computational Applications, 26(2), 25.
Helbing, D., Brockmann, D., Chadefaux, T., Donnay, K., Blanke, U., Woolley-Meza, O., Moussaid, M., Johansson, A., Krause, J., & Schutte, S. (2015). Saving human lives: What complexity science and information systems can contribute. Journal of statistical physics, 158, 735-781.
Huo, H.-F., Chen, Y.-L., & Xiang, H. (2017). Stability of a binge drinking model with delay. Journal of Biological Dynamics, 11(1), 210-225.
Huo, H.-F., Huang, S.-R., Wang, X.-Y., & Xiang, H. (2017). Optimal control of a social epidemic model with media coverage. Journal of Biological Dynamics, 11(1), 226-243.
Ivorra, B., Ferrández, M. R., Vela-Pérez, M., & Ramos, A. M. (2020). Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China. Communications in Nonlinear Science and Numerical Simulation, 88, 105303.
Jana, C., Maiti, A. P., & Maiti, D. K. (2022). Complex dynamical behavior of a ratio-dependent eco-epidemic model with Holling type-II incidence rate in the presence of two delays. Communications in Nonlinear Science Numerical Simulation, 110, 106380.
Jose, S. A., Raja, R., Alzabut, J., Rajchakit, G., Cao, J., & Balas, V. E. (2022). Mathematical modeling on transmission and optimal control strategies of corruption dynamics. Nonlinear Dynamics, 109(4), 3169-3187.
Jose, S. A., Raja, R., Dianavinnarasi, J., Baleanu, D., & Jirawattanapanit, A. (2023). Mathematical modeling of chickenpox in Phuket: Efficacy of precautionary measures and bifurcation analysis. Biomedical Signal Processing Control, 84, 104714.
Jose, S. A., Raja, R., Zhu, Q., Alzabut, J., Niezabitowski, M., & Balas, V. E. (2022a). Impact of strong determination and awareness on substance addictions: A mathematical modeling approach. Mathematical Methods in the Applied Sciences, 45(8), 4140-4160.
Jose, S. A., Raja, R., Zhu, Q., Alzabut, J., Niezabitowski, M., & Balas, V. E. (2022b). An Integrated Eco-Epidemiological Plant Pest Natural Enemy Differential Equation Model with Various Impulsive Strategies. Mathematical Problems in Engineering, 2022.
Jose, S. A., Ramachandran, R., Cao, J., Alzabut, J., Niezabitowski, M., & Balas, V. E. (2022). Stability analysis and comparative study on different eco‐epidemiological models: stage structure for prey and predator concerning impulsive control. Optimal Control Applications Methods, 43(3), 842-866.
Joseph, D., Ramachandran, R., Alzabut, J., Cao, J., Niezabitowski, M., & Lim, C. P. (2022). Global exponential stability results for the host‐parasitoid model of sugarcane borer in stochastic environment with impulsive effects via non‐fragile control: An LMI approach. Optimal Control Applications Methods, 43(2), 512-531.
Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(772), 700-721.
Khan, M. A. U. (2000). The corruption prevention model. Journal of Discrete Mathematical Sciences Cryptography, 3(1-3), 173-178.
Kucharski, A. J., Russell, T. W., Diamond, C., Liu, Y., Edmunds, J., Funk, S., Eggo, R. M., Sun, F., Jit, M., & Munday, J. D. (2020). Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The lancet infectious diseases, 20(5), 553-558.
Li, R., Pei, S., Chen, B., Song, Y., Zhang, T., Yang, W., & Shaman, J. (2020). Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Science, 368(6490), 489-493.
Liu, B., Wang, Y., & Kang, B. (2014). Dynamics on a pest management SI model with control strategies of different frequencies. Nonlinear Analysis: Hybrid Systems, 12, 66-78.
Mahmoudi, M. R., Heydari, M. H., Qasem, S. N., Mosavi, A., & Band, S. S. (2021). Principal component analysis to study the relations between the spread rates of COVID-19 in high risks countries. Alexandria Engineering Journal, 60(1), 457-464.
Nugraheni, K., Trisilowati, T., & Suryanto, A. (2017). Dynamics of a fractional order eco-epidemiological model. Journal of Tropical Life Science, 7(3), 243-250.
Ojo, M. M., Gbadamosi, B., Benson, T. O., Adebimpe, O., & Georgina, A. (2021). Modeling the dynamics of Lassa fever in Nigeria. Journal of the Egyptian Mathematical Society, 29, 1-19.
Pal, A., Bhattacharyya, A., & Mondal, A. (2022). Qualitative analysis and control of predator switching on an eco-epidemiological model with prey refuge and harvesting. Results in Control Optimization, 7, 100099.
Panigoro, H. S., Suryanto, A., Kusumawinahyu, W. M., & Darti, I. (2021). Dynamics of an eco-epidemic predator–prey model involving fractional derivatives with power-law and Mittag–Leffler kernel. Symmetry, 13(5), 785.
Rahman, A., & Kuddus, M. A. (2021). Modelling the transmission dynamics of COVID-19 in six high-burden countries. BioMed Research International, 2021, 1-17.
Rajasekar, S., Pitchaimani, M., Zhu, Q., & Shi, K. (2021). Exploring the stochastic host-pathogen tuberculosis model with adaptive immune response. Mathematical Problems in Engineering, 2021, 1-23.
Rezapour, S., Rezaei, S., Khames, A., Abdelgawad, M. A., Ghoneim, M. M., & Riaz, M. B. (2022). On dynamics of an eco-epidemics system incorporating fractional operators of singular and nonsingular types. Results in Physics, 34, 105259.
Sah, P., Vilches, T. N., Moghadas, S. M., Fitzpatrick, M. C., Singer, B. H., Hotez, P. J., & Galvani, A. P. (2021). Accelerated vaccine rollout is imperative to mitigate highly transmissible COVID-19 variants. EClinicalMedicine, 35, 100865.
Saifuddin, M., Biswas, S., Samanta, S., Sarkar, S., & Chattopadhyay, J. (2016). Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator. Chaos, Solitons Fractals, 91, 270-285.
Thomas, R., Jose, S. A., Raja, R., Alzabut, J., Cao, J., & Balas, V. E. (2022). Modeling and analysis of SEIRS epidemic models using homotopy perturbation method: A special outlook to 2019-nCoV in India. International Journal of Biomathematics, 1-1.
Tilahun, G. T., Demie, S., & Eyob, A. (2020). Stochastic model of measles transmission dynamics with double dose vaccination. Infectious Disease Modelling, 5, 478-494.
Tong, Z.-W., Lv, Y.-P., Din, R. U., Mahariq, I., & Rahmat, G. (2021). Global transmission dynamic of SIR model in the time of SARS-CoV-2. Results in Physics, 25, 104253.
Van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29-48.
Yu, T., Tian, Y., Guo, H., & Song, X. (2019). Dynamical analysis of an integrated pest management predator–prey model with weak Allee effect. Journal of Biological Dynamics, 13(1), 218-244.
Zafar, Z. U. A., Ali, N., & Baleanu, D. (2021). Dynamics and numerical investigations of a fractional-order model of toxoplasmosis in the population of human and cats. Chaos, Solitons Fractals, 151, 111261.
Zafar, Z. U. A., Tunç, C., Ali, N., Zaman, G., & Thounthong, P. (2021). Dynamics of an arbitrary order model of toxoplasmosis ailment in human and cat inhabitants. Journal of Taibah University for Science, 15(1), 882-896.
Abdulrahman, S. (2014). Stability analysis of the transmission dynamics and control of corruption. Pacific Journal of Science Technology, 15(1), 99-113.
Adu, I. K., Osman, M., & Yang, C. (2017). Mathematical model of drinking epidemic. Br. J. Math. Computer Sci, 22(5).
Agrawal, A., Tenguria, A., & Modi, G. (2018). Role of epidemic model to control drinking problem. Int. J. Sci. Res. in Mathematical Statistical Sciences, 5, 4.
Aguilar-Canto, F. J., de León, U. A.-P., & Avila-Vales, E. (2022). Sensitivity theorems of a model of multiple imperfect vaccines for COVID-19. Chaos, Solitons Fractals, 156, 111844.
Alzahrani, A. K., Alshomrani, A. S., Pal, N., & Samanta, S. (2018). Study of an eco-epidemiological model with Z-type control. Chaos, Solitons Fractals, 113, 197-208.
Athithan, S., Ghosh, M., & Li, X.-Z. (2018). Mathematical modeling and optimal control of corruption dynamics. Asian-European Journal of Mathematics, 11(06), 1850090.
Balatif, O., Labzai, A., & Rachik, M. (2018). A discrete mathematical modeling and optimal control of the electoral behavior with regard to a political party. Discrete Dynamics in Nature Society, 2018, 1-14.
Brauer, F., & Castillo-Chavez, C. (2012). Mathematical models in population biology and epidemiology (Vol. 2): Springer.
Braun, M. (1993). Differential Equation and Their Application (4 ed.): Springer.
Brianzoni, S., Coppier, R., & Michetti, E. (2011). Complex dynamics in a growth model with corruption in public procurement. Discrete Dynamics in Nature Society, 2011.
Costa, M., & dos Anjos, L. (2015). Integrated pest management in a predator-prey system with Allee effects. Neotropical entomology, 44, 385-391.
Cuervo-Cazurra, A. (2016). Corruption in international business. Journal of World Business, 51(1), 35-49.
Dianavinnarasi, J., Cao, Y., Raja, R., Rajchakit, G., & Lim, C. P. (2020). Delay-dependent stability criteria of delayed positive systems with uncertain control inputs: Application in mosquito-borne morbidities control. Applied Mathematics Computational and Theoretical Chemistry, 382, 125210.
Dianavinnarasi, J., Raja, R., Alzabut, J., Cao, J., Niezabitowski, M., & Bagdasar, O. (2022). Application of Caputo–Fabrizio operator to suppress the Aedes Aegypti mosquitoes via Wolbachia: An LMI approach. Mathematics Computers in Simulation, 201, 462-485.
Dianavinnarasi, J., Raja, R., Alzabut, J., Niezabitowski, M., & Bagdasar, O. (2021). Controlling Wolbachia transmission and invasion dynamics among aedes aegypti population via impulsive control strategy. Symmetry, 13(3), 434.
Dianavinnarasi, J., Raja, R., Alzabut, J., Niezabitowski, M., Selvam, G., & Bagdasar, O. (2021). An LMI Approach-Based Mathematical Model to Control Aedes aegypti Mosquitoes Population via Biological Control. Mathematical Problems in Engineering, 2021, 1-18.
Dutta, P., Sahoo, D., Mondal, S., & Samanta, G. (2022). Dynamical complexity of a delay-induced eco-epidemic model with Beddington–DeAngelis incidence rate. Mathematics Computers in Simulation, 197, 45-90.
Ginoux, J.-M., Naeck, R., Ruhomally, Y. B., Dauhoo, M. Z., & Perc, M. (2019). Chaos in a predator–prey-based mathematical model for illicit drug consumption. Applied Mathematics Computation, 347, 502-513.
Giordano, G., Blanchini, F., Bruno, R., Colaneri, P., Di Filippo, A., Di Matteo, A., & Colaneri, M. (2020). Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nature medicine, 26(6), 855-860.
Gonzalez-Parra, G., Martínez-Rodríguez, D., & Villanueva-Micó, R. (2021). Impact of a new SARS-CoV-2 variant on the population: A mathematical modeling approach. Mathematical Computational Applications, 26(2), 25.
Helbing, D., Brockmann, D., Chadefaux, T., Donnay, K., Blanke, U., Woolley-Meza, O., Moussaid, M., Johansson, A., Krause, J., & Schutte, S. (2015). Saving human lives: What complexity science and information systems can contribute. Journal of statistical physics, 158, 735-781.
Huo, H.-F., Chen, Y.-L., & Xiang, H. (2017). Stability of a binge drinking model with delay. Journal of Biological Dynamics, 11(1), 210-225.
Huo, H.-F., Huang, S.-R., Wang, X.-Y., & Xiang, H. (2017). Optimal control of a social epidemic model with media coverage. Journal of Biological Dynamics, 11(1), 226-243.
Ivorra, B., Ferrández, M. R., Vela-Pérez, M., & Ramos, A. M. (2020). Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China. Communications in Nonlinear Science and Numerical Simulation, 88, 105303.
Jana, C., Maiti, A. P., & Maiti, D. K. (2022). Complex dynamical behavior of a ratio-dependent eco-epidemic model with Holling type-II incidence rate in the presence of two delays. Communications in Nonlinear Science Numerical Simulation, 110, 106380.
Jose, S. A., Raja, R., Alzabut, J., Rajchakit, G., Cao, J., & Balas, V. E. (2022). Mathematical modeling on transmission and optimal control strategies of corruption dynamics. Nonlinear Dynamics, 109(4), 3169-3187.
Jose, S. A., Raja, R., Dianavinnarasi, J., Baleanu, D., & Jirawattanapanit, A. (2023). Mathematical modeling of chickenpox in Phuket: Efficacy of precautionary measures and bifurcation analysis. Biomedical Signal Processing Control, 84, 104714.
Jose, S. A., Raja, R., Zhu, Q., Alzabut, J., Niezabitowski, M., & Balas, V. E. (2022a). Impact of strong determination and awareness on substance addictions: A mathematical modeling approach. Mathematical Methods in the Applied Sciences, 45(8), 4140-4160.
Jose, S. A., Raja, R., Zhu, Q., Alzabut, J., Niezabitowski, M., & Balas, V. E. (2022b). An Integrated Eco-Epidemiological Plant Pest Natural Enemy Differential Equation Model with Various Impulsive Strategies. Mathematical Problems in Engineering, 2022.
Jose, S. A., Ramachandran, R., Cao, J., Alzabut, J., Niezabitowski, M., & Balas, V. E. (2022). Stability analysis and comparative study on different eco‐epidemiological models: stage structure for prey and predator concerning impulsive control. Optimal Control Applications Methods, 43(3), 842-866.
Joseph, D., Ramachandran, R., Alzabut, J., Cao, J., Niezabitowski, M., & Lim, C. P. (2022). Global exponential stability results for the host‐parasitoid model of sugarcane borer in stochastic environment with impulsive effects via non‐fragile control: An LMI approach. Optimal Control Applications Methods, 43(2), 512-531.
Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(772), 700-721.
Khan, M. A. U. (2000). The corruption prevention model. Journal of Discrete Mathematical Sciences Cryptography, 3(1-3), 173-178.
Kucharski, A. J., Russell, T. W., Diamond, C., Liu, Y., Edmunds, J., Funk, S., Eggo, R. M., Sun, F., Jit, M., & Munday, J. D. (2020). Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The lancet infectious diseases, 20(5), 553-558.
Li, R., Pei, S., Chen, B., Song, Y., Zhang, T., Yang, W., & Shaman, J. (2020). Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Science, 368(6490), 489-493.
Liu, B., Wang, Y., & Kang, B. (2014). Dynamics on a pest management SI model with control strategies of different frequencies. Nonlinear Analysis: Hybrid Systems, 12, 66-78.
Mahmoudi, M. R., Heydari, M. H., Qasem, S. N., Mosavi, A., & Band, S. S. (2021). Principal component analysis to study the relations between the spread rates of COVID-19 in high risks countries. Alexandria Engineering Journal, 60(1), 457-464.
Nugraheni, K., Trisilowati, T., & Suryanto, A. (2017). Dynamics of a fractional order eco-epidemiological model. Journal of Tropical Life Science, 7(3), 243-250.
Ojo, M. M., Gbadamosi, B., Benson, T. O., Adebimpe, O., & Georgina, A. (2021). Modeling the dynamics of Lassa fever in Nigeria. Journal of the Egyptian Mathematical Society, 29, 1-19.
Pal, A., Bhattacharyya, A., & Mondal, A. (2022). Qualitative analysis and control of predator switching on an eco-epidemiological model with prey refuge and harvesting. Results in Control Optimization, 7, 100099.
Panigoro, H. S., Suryanto, A., Kusumawinahyu, W. M., & Darti, I. (2021). Dynamics of an eco-epidemic predator–prey model involving fractional derivatives with power-law and Mittag–Leffler kernel. Symmetry, 13(5), 785.
Rahman, A., & Kuddus, M. A. (2021). Modelling the transmission dynamics of COVID-19 in six high-burden countries. BioMed Research International, 2021, 1-17.
Rajasekar, S., Pitchaimani, M., Zhu, Q., & Shi, K. (2021). Exploring the stochastic host-pathogen tuberculosis model with adaptive immune response. Mathematical Problems in Engineering, 2021, 1-23.
Rezapour, S., Rezaei, S., Khames, A., Abdelgawad, M. A., Ghoneim, M. M., & Riaz, M. B. (2022). On dynamics of an eco-epidemics system incorporating fractional operators of singular and nonsingular types. Results in Physics, 34, 105259.
Sah, P., Vilches, T. N., Moghadas, S. M., Fitzpatrick, M. C., Singer, B. H., Hotez, P. J., & Galvani, A. P. (2021). Accelerated vaccine rollout is imperative to mitigate highly transmissible COVID-19 variants. EClinicalMedicine, 35, 100865.
Saifuddin, M., Biswas, S., Samanta, S., Sarkar, S., & Chattopadhyay, J. (2016). Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator. Chaos, Solitons Fractals, 91, 270-285.
Thomas, R., Jose, S. A., Raja, R., Alzabut, J., Cao, J., & Balas, V. E. (2022). Modeling and analysis of SEIRS epidemic models using homotopy perturbation method: A special outlook to 2019-nCoV in India. International Journal of Biomathematics, 1-1.
Tilahun, G. T., Demie, S., & Eyob, A. (2020). Stochastic model of measles transmission dynamics with double dose vaccination. Infectious Disease Modelling, 5, 478-494.
Tong, Z.-W., Lv, Y.-P., Din, R. U., Mahariq, I., & Rahmat, G. (2021). Global transmission dynamic of SIR model in the time of SARS-CoV-2. Results in Physics, 25, 104253.
Van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29-48.
Yu, T., Tian, Y., Guo, H., & Song, X. (2019). Dynamical analysis of an integrated pest management predator–prey model with weak Allee effect. Journal of Biological Dynamics, 13(1), 218-244.
Zafar, Z. U. A., Ali, N., & Baleanu, D. (2021). Dynamics and numerical investigations of a fractional-order model of toxoplasmosis in the population of human and cats. Chaos, Solitons Fractals, 151, 111261.
Zafar, Z. U. A., Tunç, C., Ali, N., Zaman, G., & Thounthong, P. (2021). Dynamics of an arbitrary order model of toxoplasmosis ailment in human and cat inhabitants. Journal of Taibah University for Science, 15(1), 882-896.