{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T20:36:40Z","timestamp":1774039000308,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,8]],"date-time":"2021-08-08T00:00:00Z","timestamp":1628380800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>We study epidemic Susceptible\u2013Infected\u2013Susceptible (SIS) models in the fractional setting. The novelty is to consider models in which the susceptible and infected populations evolve according to different fractional orders. We study a model based on the Caputo derivative, for which we establish existence results of the solutions. Furthermore, we investigate a model based on the Caputo\u2013Fabrizio operator, for which we provide existence of solutions and a study of the equilibria. Both models can be framed in the context of SIS models with time-varying total population, in which the competition between birth and death rates is macroscopically described by the fractional orders of the derivatives. Numerical simulations for both models and a direct numerical comparison are also provided.<\/jats:p>","DOI":"10.3390\/computation9080089","type":"journal-article","created":{"date-parts":[[2021,8,8]],"date-time":"2021-08-08T21:49:41Z","timestamp":1628459381000},"page":"89","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Effects of Fractional Derivatives with Different Orders in SIS Epidemic Models"],"prefix":"10.3390","volume":"9","author":[{"given":"Caterina","family":"Balzotti","sequence":"first","affiliation":[{"name":"Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, 00185 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2796-5465","authenticated-orcid":false,"given":"Mirko","family":"D\u2019Ovidio","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze di Base e Applicate per l\u2019Ingegneria, Sapienza Universit\u00e0 di Roma, 00161 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2096-6753","authenticated-orcid":false,"given":"Anna Chiara","family":"Lai","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze di Base e Applicate per l\u2019Ingegneria, Sapienza Universit\u00e0 di Roma, 00161 Rome, Italy"}]},{"given":"Paola","family":"Loreti","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze di Base e Applicate per l\u2019Ingegneria, Sapienza Universit\u00e0 di Roma, 00161 Rome, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1098\/rspa.1927.0118","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/978-3-642-61317-3_5","article-title":"Three basic epidemiological models","volume":"Volume 18","author":"Hethcote","year":"1989","journal-title":"Applied Mathematical Ecology (Trieste, 1986)"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.apm.2021.03.044","article-title":"Review of fractional epidemic models","volume":"97","author":"Chen","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s11071-006-9094-0","article-title":"Stability analysis of linear fractional differential system with multiple time delays","volume":"48","author":"Deng","year":"2007","journal-title":"Nonlinear Dyn."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Yavuz, M., and Sene, N. (2020). Stability analysis and numerical computation of the fractional predator\u2013prey model with the harvesting rate. Fractal Fract., 4.","DOI":"10.3390\/fractalfract4030035"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109870","DOI":"10.1016\/j.chaos.2020.109870","article-title":"Stability analysis of the hiv model through incommensurate fractional-order nonlinear system","volume":"137","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","first-page":"4","article-title":"Linear models of dissipation whose Q is almost frequency independent\u2014II","volume":"11","author":"Caputo","year":"2008","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_8","first-page":"113","article-title":"Notice sur la loi que la population suit dans son accroissement","volume":"10","author":"Verhulst","year":"1838","journal-title":"Corresp. Math. Phys."},{"key":"ref_9","first-page":"276","article-title":"Recherches math\u00e9matiques sur la loi d\u2019accroissement de la population","volume":"12","author":"Verhulst","year":"1845","journal-title":"J. Econ."},{"key":"ref_10","first-page":"1","article-title":"Deuxi\u00e8me m\u00e9moire sur la loi d\u2019accroissement de la population","volume":"20","author":"Verhulst","year":"1847","journal-title":"M\u00e9moires De L\u2019acad\u00e9mie Royale des Sciences des Lettres et des Beaux-Arts de Belgique"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Balzotti, C., D\u2019Ovidio, M., and Loreti, P. (2020). Fractional SIS Epidemic Models. Fractal Fract., 4.","DOI":"10.3390\/fractalfract4030044"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"195","DOI":"10.12785\/msl\/020308","article-title":"The fractional-order SIR and SIRS epidemic models with variable population size","volume":"2","year":"2013","journal-title":"Math. Sci. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.chaos.2018.10.023","article-title":"On the solution of fractional order SIS epidemic model","volume":"117","author":"Hassouna","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1016\/j.physa.2018.05.030","article-title":"Solutions of fractional logistic equations by Euler\u2019s numbers","volume":"506","author":"Loreti","year":"2018","journal-title":"Phys. A"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Brauer, F., Castillo-Chavez, C., and Feng, Z. (2019). Mathematical Models in Epidemiology, Springer. Texts in Applied Mathematics.","DOI":"10.1007\/978-1-4939-9828-9"},{"key":"ref_16","first-page":"151","article-title":"On a discrete scheme for time fractional fully nonlinear evolution equations","volume":"120","author":"Giga","year":"2020","journal-title":"Asymptot. Anal."},{"key":"ref_17","first-page":"1","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Progr. Fract. Differ. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1186\/s13662-019-2199-9","article-title":"A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying","volume":"2019","author":"Kumar","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","first-page":"142","article-title":"Analysis of an epidemic spreading model with exponential decay law","volume":"8","author":"Yavuz","year":"2020","journal-title":"Math. Sci. Appl. E-Notes"},{"key":"ref_20","unstructured":"Coddington, E.A., and Levinson, N. (1955). Theory of Ordinary Differential Equations, Tata McGraw-Hill Education."},{"key":"ref_21","first-page":"269","article-title":"Time-fractional derivatives in relaxation processes: A tutorial survey","volume":"10","author":"Mainardi","year":"2007","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4719","DOI":"10.1016\/j.aej.2020.08.034","article-title":"New Caputo-Fabrizio fractional order SEIASqEqHR model for COVID-19 epidemic transmission with genetic algorithm based control strategy","volume":"59","author":"Higazy","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"195","DOI":"10.3934\/math.2021013","article-title":"Study of mathematical model of Hepatitis B under Caputo-Fabrizo derivative","volume":"6","author":"Khan","year":"2021","journal-title":"AIMS Math."},{"key":"ref_24","first-page":"20","article-title":"A Caputo-Fabrizio fractional differential equation model for HIV\/AIDS with treatment compartment","volume":"200","author":"Moore","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_25","first-page":"975","article-title":"A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative","volume":"13","author":"Ullah","year":"2020","journal-title":"Discret. Contin. Dyn. Syst. Ser. S"},{"key":"ref_26","first-page":"87","article-title":"Properties of a new fractional derivative without singular kernel","volume":"1","author":"Losada","year":"2015","journal-title":"Progr. Fract. Differ. 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