{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T15:58:51Z","timestamp":1775577531746,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T00:00:00Z","timestamp":1665619200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Zika virus model (ZIKV) is mathematically modeled to create the perfect control strategies. The main characteristics of the model without control strategies, in particular reproduction number, are specified. Based on the basic reproduction number, if R0&lt;0, then ZIKV satisfies the disease-free equilibrium. If R0&gt;1, then ZIKV satisfies the endemic equilibrium. We use the maximum principle from Pontryagin\u2019s. This describes the critical conditions for optimal control of ZIKV. Notwithstanding, due to the prevention and treatment of mosquito populations without spraying, people infected with the disease have decreased dramatically. Be that as it may, there has been no critical decline in mosquitoes contaminated with the disease. The usage of preventive treatments and insecticide procedures to mitigate the spread of the proposed virus showed a more noticeable centrality in the decrease in contaminated people and mosquitoes. The application of preventive measures including treatment and insecticides has emerged as the most ideal way to reduce the spread of ZIKV. Best of all, to decrease the spread of ZIKV is to use avoidance, treatment and bug spraying simultaneously as control methods. Moreover, for the numerical solution of such stochastic models, we apply the spectral technique. The stochastic or random phenomenons are more realistic and make the model more informative with the additive information. Throughout this paper, the additive term is assumed as additive white noise. The Legendre polynomials and applications are implemented to transform the proposed system into a nonlinear algebraic system.<\/jats:p>","DOI":"10.3390\/sym14102137","type":"journal-article","created":{"date-parts":[[2022,10,14]],"date-time":"2022-10-14T01:44:13Z","timestamp":1665711853000},"page":"2137","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Dynamics of Stochastic Zika Virus with Treatment Class in Human Population via Spectral Method"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5022-1157","authenticated-orcid":false,"given":"Ebrahem A.","family":"Algehyne","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}]},{"given":"Farman Ullah","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, HITEC University, Taxila Cantt 47080, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7326-5534","authenticated-orcid":false,"given":"Sami Ullah","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, City University of Science and Information Technology Peshawar, Khyber Pakhtunkhwa 2500, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9438-6132","authenticated-orcid":false,"given":"Wasim","family":"Jamshed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Capital University of Science and Technology, Islamabad 44000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0229-0705","authenticated-orcid":false,"given":"El Sayed M.","family":"Tag El Din","sequence":"additional","affiliation":[{"name":"Center of Research, Faculty of Engineering, Future University, New Cairo 11835, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.chaos.2018.11.023","article-title":"Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses","volume":"118","author":"Rajasekar","year":"2019","journal-title":"J. Chaos Solitons Fractals"},{"key":"ref_2","first-page":"1","article-title":"Persistence and extinction in stochastic SIRS models with general nonlinear incidence rate","volume":"42","author":"Zhou","year":"2014","journal-title":"Electron. Differ. Eqnarrays"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.mbs.2007.05.009","article-title":"Mean-square stability of a stochastic model for bacteriophage infection with time delays","volume":"210","author":"Carletti","year":"2007","journal-title":"Math. Biosci."},{"key":"ref_4","first-page":"528","article-title":"The dynamics of a new SIR epidemic model concerning pulse vaccination strategy","volume":"197","author":"Meng","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.cnsns.2015.11.007","article-title":"Extension of the spectral element method for stability analysis of time-periodic delay-differential equations with multiple and distributed delays","volume":"35","author":"Lehotzky","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_6","first-page":"84","article-title":"A stochastic spectral finite element method for wave propagation analyses with medium uncertainties","volume":"63","author":"Zakian","year":"2018","journal-title":"Appl. Math."},{"key":"ref_7","first-page":"285","article-title":"Global stability of an SIR epidemic model with constant infectious period, Appl","volume":"199","author":"Zhang","year":"2008","journal-title":"Math. Comput."},{"key":"ref_8","unstructured":"Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F., and Trirogoff, K.N. (1962). TheMathematical Theory of Optimal Processes, Interscience Publishers."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1750096","DOI":"10.1142\/S1793524517500966","article-title":"Dynamical system of a SEIQV epidemic model with nonlinear generalized incidence rate arising in biology","volume":"7","author":"Khan","year":"2017","journal-title":"Int. J. Biomath."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1650068","DOI":"10.1142\/S1793524516500686","article-title":"Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate","volume":"9","author":"Khan","year":"2016","journal-title":"Int. J. Biomath."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.tpb.2007.12.008","article-title":"Effects of predation on hostpathogen dynamics in SIR models","volume":"73","author":"Roy","year":"2008","journal-title":"Theor. Popul. Biol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1058","DOI":"10.1016\/j.apm.2007.12.020","article-title":"Permanence and extinction for a nonautonomous SIRS epidemic model with time delay","volume":"33","author":"Zhang","year":"2009","journal-title":"Appl. Math. Model."},{"key":"ref_13","first-page":"733","article-title":"Epidemic disease in England","volume":"1","author":"Hamer","year":"1906","journal-title":"Lancet"},{"key":"ref_14","unstructured":"Ross, R.A. (1911). The Prevention of Malaria, Murray."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.nonrwa.2008.10.041","article-title":"A delay SIR epidemic model with pulse vaccination and incubation times","volume":"11","author":"Meng","year":"2010","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5370","DOI":"10.1002\/mma.4391","article-title":"Global stability of a discrete multigroup SIR model with nonlinear incidence rate","volume":"40","author":"Zhou","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_17","first-page":"11806","article-title":"Global dynamics for a new high-dimensional SIR model with distributed delay","volume":"218","author":"Zhang","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_18","first-page":"700","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1016\/S0895-7177(02)00082-1","article-title":"Qualitative analyses of SIS epidemic model with vaccination and varying total population size","volume":"35","author":"Li","year":"2002","journal-title":"Math. Comput. Model."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gordo, I., Gomes, M.G.M., Reis, D.G., and Campos, P.R.A. (2009). Genetic diversity in the SIR model of pathogen evolution. PLoS ONE, 4.","DOI":"10.1371\/journal.pone.0004876"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Lazebnik, T., and Bunimovich-Mendrazitsky, S. (2022). Generic approach for mathematical model of multi-strain pandemics. PLoS ONE, 17.","DOI":"10.1371\/journal.pone.0260683"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Lazebnik, T., Bunimovich-Mendrazitsky, S., and Shami, L. (2021). Pandemic management by a spatio-temporal mathematical model. Int. J. Nonlinear Sci. Numer.","DOI":"10.1515\/ijnsns-2021-0063"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Kiaee, S.N., Khodabin, M., Ezzati, R., and Lopes, A.M. (2022). A New Approach to Approximate Solutions of Single Time-Delayed Stochastic Integral Equations via Orthogonal Functions. Symmetry, 14.","DOI":"10.3390\/sym14102085"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Eidinejad, Z., Saadati, R., O\u2019Regan, D., and Alshammari, F.S. (2022). Minimum Superstability of Stochastic Ternary Antiderivations in Symmetric Matrix-Valued FB-Algebras and Symmetric Matrix-Valued FC-?-Algebras. Symmetry, 14.","DOI":"10.3390\/sym14102064"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Tan, J., Xiang, P., Zhao, H., Yu, J., Ye, B., and Yang, D. (2022). Stochastic Analysis of Train Running Safety on Bridge with Earthquake-Induced Irregularity under Aftershock. Symmetry, 14.","DOI":"10.3390\/sym14101998"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ishtiaq, A., and Khan, S.U. (2022). Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method. Symmetry, 14.","DOI":"10.3390\/sym14091838"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1080\/10255842.2021.1970143","article-title":"Transmission dynamic of stochastic hepatitis C model by spectral collocation method","volume":"25","author":"Gul","year":"2022","journal-title":"Comput. Methods Biomech. Biomed. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"8230","DOI":"10.1002\/mma.8183","article-title":"On dynamics of stochastic avian influenza model with asymptomatic carrier using spectral method","volume":"45","author":"Ali","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"035301","DOI":"10.1063\/1.5016680","article-title":"Application of Legendre spectral-collocation method to delay differential and stochastic delay differential equation","volume":"8","author":"Ullah","year":"2018","journal-title":"AIP Adv."},{"key":"ref_30","first-page":"161","article-title":"A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis","volume":"1","author":"Ullah","year":"2019","journal-title":"J. Adv. Differ. Equ."},{"key":"ref_31","unstructured":"Ullah, K.S., and Ishtiaq, A. (2019). Numerical Analysis of Stochastic SIR Model by Legendre Spectral Collocation Method, SAGE Publications Sage. Advances in Mechanical Engineering."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"110008","DOI":"10.1016\/j.chaos.2020.110008","article-title":"Analysis of stochastic delayed SIRS model with exponential birth and saturated incidence rate","volume":"138","author":"Ishtiaq","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s40314-019-0897-2","article-title":"Convergence and error analysis of a spectral collo- cation method for solving system of nonlinear Fredholm integral equations of second kind","volume":"38","author":"Ullah","year":"2019","journal-title":"Comput. Appl. Math."},{"key":"ref_34","first-page":"1","article-title":"Applications of Legendre spectral collocation method for solving system of time delay differential equations","volume":"12","author":"Ullah","year":"2020","journal-title":"Adv.-Mechan-Ical Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1186\/s13662-018-1759-8","article-title":"Extinction and persistence of a stochastic SIRS epidemic model with saturated incidence rate and transfer from infectious to susceptible","volume":"2018","author":"Song","year":"2018","journal-title":"Adv. Differ. Equations"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2137\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:53:24Z","timestamp":1760144004000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2137"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,13]]},"references-count":35,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102137"],"URL":"https:\/\/doi.org\/10.3390\/sym14102137","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,13]]}}}