{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T20:42:26Z","timestamp":1757623346903,"version":"3.44.0"},"reference-count":125,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s40314-025-03330-8","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T07:02:46Z","timestamp":1753167766000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stochastic and spatio-temporal models in understanding the extinction, persistence and control of a within-host viral dynamics"],"prefix":"10.1007","volume":"44","author":[{"given":"Bishal","family":"Chhetri","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0006-0025-3288","authenticated-orcid":false,"given":"B. V.","family":"Rathish Kumar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"3330_CR1","doi-asserted-by":"crossref","unstructured":"Abhijeet, Mohamed MNG, Sharma A, Chakravorty S (2024) Convexity in optimal control problems, In: 2024 IEEE 63rd conference on decision and control (CDC), pp 261\u2013266. https:\/\/doi.org\/10.1109\/CDC56724.2024.10886652","DOI":"10.1109\/CDC56724.2024.10886652"},{"issue":"1","key":"3330_CR2","doi-asserted-by":"crossref","first-page":"17306","DOI":"10.1038\/s41598-020-74386-1","volume":"10","author":"M Aguiar","year":"2020","unstructured":"Aguiar M, Ortuondo EM, Bidaurrazaga Van-Dierdonck J, Mar J, Stollenwerk N (2020) Modelling COVID 19 in the Basque country from introduction to control measure response. Sci Rep 10(1):17306","journal-title":"Sci Rep"},{"key":"3330_CR3","first-page":"446","volume":"37","author":"R Ahmad","year":"1988","unstructured":"Ahmad R (1988) Introduction to stochastic differential equations. R Stat Soc 37:446","journal-title":"R Stat Soc"},{"issue":"4","key":"3330_CR4","doi-asserted-by":"crossref","first-page":"1590","DOI":"10.1111\/bcp.15089","volume":"88","author":"S Akinbolade","year":"2022","unstructured":"Akinbolade S, Coughlan D, Fairbairn R, McConkey G, Powell H, Ogunbayo D, Craig D (2022) Combination therapies for COVID-19: an overview of the clinical trials landscape. Br J Clin Pharmacol 88(4):1590\u20131597","journal-title":"Br J Clin Pharmacol"},{"issue":"06","key":"3330_CR5","doi-asserted-by":"crossref","first-page":"941","DOI":"10.4236\/jamp.2020.86073","volume":"8","author":"MS Alam","year":"2020","unstructured":"Alam MS, Kamrujjaman M, Islam MS (2020) Parameter sensitivity and qualitative analysis of dynamics of ovarian tumor growth model with treatment strategy. J Appl Math Phys 8(06):941\u2013955","journal-title":"J Appl Math Phys"},{"key":"3330_CR6","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/978-3-540-78911-6_3","volume-title":"Mathematical epidemiology","author":"LJ Allen","year":"2008","unstructured":"Allen LJ (2008) An introduction to stochastic epidemic models. In: Brauer F, Van den Driessche P, Allen LJ (eds) Mathematical epidemiology. Springer, Berlin, pp 81\u2013130"},{"key":"3330_CR7","doi-asserted-by":"crossref","DOI":"10.1201\/b12537","volume-title":"An introduction to stochastic processes with applications to biology","author":"LJ Allen","year":"2010","unstructured":"Allen LJ (2010) An introduction to stochastic processes with applications to biology. CRC Press, Cambridge"},{"key":"3330_CR8","volume":"95","author":"AES Almocera","year":"2021","unstructured":"Almocera AES, Quiroz G, Hernandez-Vargas EA (2021) Stability analysis in COVID-19 within-host model with immune response. Commun Nonlinear Sci Numer Simul 95:105584","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"1","key":"3330_CR9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-03162-2","volume":"2021","author":"RT Alqahtani","year":"2021","unstructured":"Alqahtani RT (2021) Mathematical model of sir epidemic system (COVID-19) with fractional derivative: stability and numerical analysis. Adv Difference Equ 2021(1):1\u201316","journal-title":"Adv Difference Equ"},{"key":"3330_CR10","doi-asserted-by":"crossref","unstructured":"Aronna MS, Guglielmi R, Moschen LM (2020) A model for COVID-19 with isolation, quarantine and testing as control measures, arXiv preprint arXiv:2005.07661","DOI":"10.1101\/2020.05.29.20116897"},{"key":"3330_CR11","volume-title":"Introduction to stochastic control theory","author":"KJ \u00c5str\u00f6","year":"2012","unstructured":"\u00c5str\u00f6 KJ (2012) Introduction to stochastic control theory. Courier Corporation, Honolulu"},{"issue":"7","key":"3330_CR12","doi-asserted-by":"crossref","first-page":"1564","DOI":"10.1111\/all.14364","volume":"75","author":"AK Azkur","year":"2020","unstructured":"Azkur AK, Akdis M, Azkur D, Sokolowska M, van de Veen W, Br\u00fcggen M-C, Mahony LO, Gao Y, Nadeau K, Akdis CA (2020) Immune response to SARS-CoV-2 and mechanisms of immunopathological changes in COVID-19. Allergy 75(7):1564\u20131581","journal-title":"Allergy"},{"key":"3330_CR13","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2021.110788","volume":"145","author":"A Babaei","year":"2021","unstructured":"Babaei A, Jafari H, Banihashemi S, Ahmadi M (2021) Mathematical analysis of a stochastic model for spread of coronavirus. Chaos, Solitons Fractals 145:110788","journal-title":"Chaos, Solitons Fractals"},{"issue":"2","key":"3330_CR14","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.jmaa.2003.12.004","volume":"292","author":"A Bahar","year":"2004","unstructured":"Bahar A, Mao X (2004) Stochastic delay Lotka\u2013Volterra model. J Math Anal Appl 292(2):364\u2013380","journal-title":"J Math Anal Appl"},{"key":"3330_CR15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-018-1466-5","volume":"2018","author":"M Bayram","year":"2018","unstructured":"Bayram M, Partal T, Orucova Buyukoz G (2018) Numerical methods for simulation of stochastic differential equations. Adv Difference Equ 2018:1\u201310","journal-title":"Adv Difference Equ"},{"key":"3330_CR16","volume-title":"Perturbation methods in optimal control","author":"A Bensoussan","year":"1988","unstructured":"Bensoussan A, Tomson C (1988) Perturbation methods in optimal control. Gauthier-Villars, Paris"},{"issue":"1","key":"3330_CR17","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s11071-020-05958-z","volume":"102","author":"SK Biswas","year":"2020","unstructured":"Biswas SK, Ghosh JK, Sarkar S, Ghosh U (2020) COVID-19 pandemic in India: a mathematical model study. Nonlinear Dyn 102(1):537\u2013553","journal-title":"Nonlinear Dyn"},{"key":"3330_CR18","doi-asserted-by":"crossref","DOI":"10.1201\/9780429428357","volume-title":"Introduction to probability","author":"JK Blitzstein","year":"2019","unstructured":"Blitzstein JK, Hwang J (2019) Introduction to probability. Chapman and Hall\/CRC, Boca Raton"},{"key":"3330_CR19","first-page":"229","volume":"62","author":"SM Blower","year":"1994","unstructured":"Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int Stat Rev\/Revue Internationale de Statistique 62:229\u2013243","journal-title":"Int Stat Rev\/Revue Internationale de Statistique"},{"issue":"4","key":"3330_CR20","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1038\/s41591-021-01290-0","volume":"27","author":"R Burioni","year":"2021","unstructured":"Burioni R, Topol EJ (2021) Assessing the human immune response to SARS-CoV-2 variants. Nat Med 27(4):571\u2013572","journal-title":"Nat Med"},{"key":"3330_CR21","first-page":"221","volume":"305","author":"Y Cai","year":"2017","unstructured":"Cai Y, Kang Y, Wang W (2017) A stochastic SIRS epidemic model with nonlinear incidence rate. Appl Math Comput 305:221\u2013240","journal-title":"Appl Math Comput"},{"key":"3330_CR22","doi-asserted-by":"crossref","first-page":"1192","DOI":"10.1016\/j.eng.2020.03.007","volume":"6","author":"Q Cai","year":"2020","unstructured":"Cai Q, Yang M, Liu D, Chen J, Shu D, Xia J, Liao X, Gu Y, Cai Q, Yang Y et al (2020) Experimental treatment with favipiravir for COVID-19: an open-label control study. Engineering 6:1192\u20131198","journal-title":"Engineering"},{"issue":"1","key":"3330_CR23","first-page":"153","volume":"12","author":"K Chang","year":"2022","unstructured":"Chang K, Zhang Q, Yuan H (2022) Stationary distribution and control strategy of a stochastic dengue model with spatial diffusion. J Appl Anal Comput 12(1):153\u2013178","journal-title":"J Appl Anal Comput"},{"issue":"1","key":"3330_CR24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40249-019-0617-6","volume":"9","author":"T-M Chen","year":"2020","unstructured":"Chen T-M, Rui J, Wang Q-P, Zhao Z-Y, Cui J-A, Yin L (2020) A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infect Dis Poverty 9(1):1\u20138","journal-title":"Infect Dis Poverty"},{"key":"3330_CR25","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.chaos.2019.07.054","volume":"129","author":"Y Cheng","year":"2019","unstructured":"Cheng Y, Li M, Zhang F (2019) A dynamics stochastic model with HIV infection of CD4+ T-cells driven by L\u00e9vy noise. Chaos, Solitons Fractals 129:62\u201370","journal-title":"Chaos, Solitons Fractals"},{"key":"3330_CR26","first-page":"245","volume":"8","author":"F Chen","year":"2024","unstructured":"Chen F, Li X (2024) Exponential stability and optimal control of a stochastic brucellosis model with spatial diffusion and nonlocal transmission. Int J Biomath 8:245","journal-title":"Int J Biomath"},{"issue":"2","key":"3330_CR27","doi-asserted-by":"crossref","first-page":"2491","DOI":"10.1016\/j.aej.2020.12.011","volume":"60","author":"B Chhetri","year":"2021","unstructured":"Chhetri B, Bhagat VM, Vamsi D, Ananth V, Mandale R, Muthusamy S, Sanjeevi CB et al (2021) Within-host mathematical modeling on crucial inflammatory mediators and drug interventions in COVID-19 identifies combination therapy to be most effective and optimal. Alex Eng J 60(2):2491\u20132512","journal-title":"Alex Eng J"},{"issue":"2","key":"3330_CR28","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s10441-022-09440-8","volume":"70","author":"B Chhetri","year":"2022","unstructured":"Chhetri B, Bhagat VM, Vamsi D, Ananth V, Prakash B, Muthusamy S, Deshmukh P, Sanjeevi CB (2022) Optimal drug regimen and combined drug therapy and its efficacy in the treatment of COVID-19: a within-host modeling study. Acta Biotheor 70(2):16","journal-title":"Acta Biotheor"},{"issue":"3","key":"3330_CR29","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1136\/thorax.2003.012658","volume":"59","author":"C Chu","year":"2004","unstructured":"Chu C, Cheng V, Hung I, Wong M, Chan K, Chan K, Kao R, Poon L, Wong C, Guan Y et al (2004) Role of lopinavir\/ritonavir in the treatment of SARS: initial virological and clinical findings. Thorax 59(3):252\u2013256","journal-title":"Thorax"},{"issue":"2","key":"3330_CR30","doi-asserted-by":"crossref","first-page":"1084","DOI":"10.1016\/j.jmaa.2007.11.005","volume":"341","author":"N Dalal","year":"2008","unstructured":"Dalal N, Greenhalgh D, Mao X (2008) A stochastic model for internal HIV dynamics. J Math Anal Appl 341(2):1084\u20131101","journal-title":"J Math Anal Appl"},{"issue":"1","key":"3330_CR31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-021-86873-0","volume":"11","author":"M Dashtbali","year":"2021","unstructured":"Dashtbali M, Mirzaie M (2021) A compartmental model that predicts the effect of social distancing and vaccination on controlling COVID-19. Sci Rep 11(1):1\u201311","journal-title":"Sci Rep"},{"key":"3330_CR32","doi-asserted-by":"crossref","unstructured":"Davoudi-Monfared E, Rahmani H, Khalili H, Hajiabdolbaghi M, Salehi M, Abbasian L, Kazemzadeh H, Yekaninejad MS (2020) Efficacy and safety of interferon beta-1a in treatment of severe COVID-19: a randomized clinical trial, medRxiv","DOI":"10.1101\/2020.05.28.20116467"},{"key":"3330_CR33","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/B978-0-12-812102-3.00004-X","volume-title":"The autoimmune diseases","author":"PJ Delves","year":"2020","unstructured":"Delves PJ (2020) Innate and adaptive systems of immunity. In: Mackay IR, Rose NR (eds) The autoimmune diseases. Elsevier, Amsterdam, pp 45\u201361"},{"key":"3330_CR34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12597-020-00491-4","volume":"58","author":"AK Dhaiban","year":"2021","unstructured":"Dhaiban AK, Jabbar BK (2021) An optimal control model of COVID-19 pandemic: a comparative study of five countries. Opsearch 58:1\u201320","journal-title":"Opsearch"},{"issue":"12","key":"3330_CR35","doi-asserted-by":"crossref","first-page":"123101","DOI":"10.1063\/5.0063050","volume":"31","author":"A Din","year":"2021","unstructured":"Din A (2021) The stochastic bifurcation analysis and stochastic delayed optimal control for epidemic model with general incidence function. Chaos Interdiscipl J Nonlinear Sci 31(12):123101","journal-title":"Chaos Interdiscipl J Nonlinear Sci"},{"key":"3330_CR36","volume":"141","author":"A Din","year":"2020","unstructured":"Din A, Li Y, Khan T, Zaman G (2020) Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China. Chaos, Solitons Fractals 141:110286","journal-title":"Chaos, Solitons Fractals"},{"key":"3330_CR37","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/s41478-020-00276-4","volume":"29","author":"M El Fatini","year":"2021","unstructured":"El Fatini M, Sekkak I, Taki R, El Guendouz T (2021) A control treatment for a stochastic epidemic model with relapse and Crowly\u2013Martin incidence. J Anal 29:713\u2013729","journal-title":"J Anal"},{"key":"3330_CR38","doi-asserted-by":"crossref","DOI":"10.1016\/j.virusres.2020.198204","volume":"295","author":"A Elaiw","year":"2021","unstructured":"Elaiw A, Alshamrani NH (2021) Analysis of a within-host HIV\/HTLV-I co-infection model with immunity. Virus Res 295:198204","journal-title":"Virus Res"},{"key":"3330_CR39","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1007\/s40435-016-0283-5","volume":"6","author":"A El-Alami Laaroussi","year":"2018","unstructured":"El-Alami Laaroussi A, Rachik M, Elhia M (2018) An optimal control problem for a spatiotemporal sir model. Int J Dyn Control 6:384\u2013397","journal-title":"Int J Dyn Control"},{"issue":"4","key":"3330_CR40","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s12064-022-00379-5","volume":"141","author":"I Elbaz","year":"2022","unstructured":"Elbaz I, Sohaly M, El-Metwally H (2022) Modeling the stochastic within-host dynamics SARS-CoV-2 infection with discrete delay. Theory Biosci 141(4):365\u2013374","journal-title":"Theory Biosci"},{"issue":"1","key":"3330_CR41","doi-asserted-by":"crossref","first-page":"11675","DOI":"10.1038\/s41598-023-38705-6","volume":"13","author":"IM Elbaz","year":"2023","unstructured":"Elbaz IM, El-Metwally H, Sohaly M (2023) Viral kinetics, stability and sensitivity analysis of the within-host COVID-19 model. Sci Rep 13(1):11675","journal-title":"Sci Rep"},{"issue":"21\u201322","key":"3330_CR42","doi-asserted-by":"crossref","first-page":"9131","DOI":"10.1016\/j.apm.2016.05.007","volume":"40","author":"Y Emvudu","year":"2016","unstructured":"Emvudu Y, Bongor D, Ko\u00efna R (2016) Mathematical analysis of HIV\/AIDS stochastic dynamic models. Appl Math Model 40(21\u201322):9131\u20139151","journal-title":"Appl Math Model"},{"key":"3330_CR43","doi-asserted-by":"crossref","unstructured":"Fabbri G, Gozzi F, Swiech A (2017) Stochastic optimal control in infinite dimension. Springer, Probability and Stochastic Modelling","DOI":"10.1007\/978-3-319-53067-3"},{"key":"3330_CR44","volume-title":"Deterministic and stochastic optimal control","author":"WH Fleming","year":"2012","unstructured":"Fleming WH, Rishel RW (2012) Deterministic and stochastic optimal control, vol 1. Springer Science & Business Media, Berlin"},{"issue":"1","key":"3330_CR45","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1046\/j.1523-1739.1994.08010124.x","volume":"8","author":"P Foley","year":"1994","unstructured":"Foley P (1994) Predicting extinction times from environmental stochasticity and carrying capacity. Conserv Biol 8(1):124\u2013137","journal-title":"Conserv Biol"},{"key":"3330_CR46","first-page":"75","volume-title":"Stochastic differential equations","author":"A Friedman","year":"1975","unstructured":"Friedman A (1975) Stochastic differential equations and applications. In: Arnold L (ed) Stochastic differential equations. Springer, New York, pp 75\u2013148"},{"issue":"2","key":"3330_CR47","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.jtbi.2004.10.004","volume":"233","author":"GA Funk","year":"2005","unstructured":"Funk GA, Jansen VA, Bonhoeffer S, Killingback T (2005) Spatial models of virus-immune dynamics. J Theor Biol 233(2):221\u2013236","journal-title":"J Theor Biol"},{"issue":"6","key":"3330_CR48","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1007\/s42979-021-00919-8","volume":"2","author":"I Ghosh","year":"2021","unstructured":"Ghosh I (2021) Within host dynamics of SARS-CoV-2 in humans: modeling immune responses and antiviral treatments. SN Comput Sci 2(6):482","journal-title":"SN Comput Sci"},{"issue":"3","key":"3330_CR49","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1137\/10081856X","volume":"71","author":"A Gray","year":"2011","unstructured":"Gray A, Greenhalgh D, Hu L, Mao X, Pan J (2011) A stochastic differential equation sis epidemic model. SIAM J Appl Math 71(3):876\u2013902","journal-title":"SIAM J Appl Math"},{"key":"3330_CR50","first-page":"1","volume":"140","author":"P Groisman","year":"2007","unstructured":"Groisman P, Rossi JD (2007) Explosion time in stochastic differential equations with small diffusion. Electron J Differ Equ (EJDE) 140:1\u20139","journal-title":"Electron J Differ Equ (EJDE)"},{"key":"3330_CR51","doi-asserted-by":"crossref","unstructured":"Hernandez-Vargas EA, Velasco-Hernandez JX (2020) In-host modelling of COVID-19 kinetics in humans, medrxiv 2020\u201303","DOI":"10.1101\/2020.03.26.20044487"},{"issue":"3","key":"3330_CR52","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1137\/S0036144500378302","volume":"43","author":"DJ Higham","year":"2001","unstructured":"Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 43(3):525\u2013546","journal-title":"SIAM Rev"},{"key":"3330_CR53","first-page":"1","volume":"2022","author":"S Hussain","year":"2022","unstructured":"Hussain S, Madi EN, Khan H, Gulzar H, Etemad S, Rezapour S, Kaabar MK (2022) On the stochastic modeling of COVID-19 under the environmental white noise. J Func Spaces 2022:1\u20139","journal-title":"J Func Spaces"},{"key":"3330_CR54","doi-asserted-by":"crossref","unstructured":"Ishikawa M (2012) Stochastic optimal control of an sir epidemic model with vaccination, In: Proceedings of the ISCIE international symposium on stochastic systems theory and its applications, Vol. 2012, the ISCIE symposium on stochastic systems theory and its applications, pp 57\u201362","DOI":"10.5687\/sss.2012.57"},{"issue":"4","key":"3330_CR55","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1002\/oca.710","volume":"23","author":"HR Joshi","year":"2002","unstructured":"Joshi HR (2002) Optimal control of an HIV immunology model. Opt Control Appl Methods 23(4):199\u2013213","journal-title":"Opt Control Appl Methods"},{"issue":"23","key":"3330_CR56","first-page":"1145","volume":"9","author":"MA Khan","year":"2015","unstructured":"Khan MA, Ali Z, Dennis L, Khan I, Islam S, Ullah M, Gul T (2015) Stability analysis of an SVIR epidemic model with non-linear saturated incidence rate. Appl Math Sci 9(23):1145\u20131158","journal-title":"Appl Math Sci"},{"key":"3330_CR57","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.chaos.2018.01.036","volume":"108","author":"T Khan","year":"2018","unstructured":"Khan T, Khan A, Zaman G (2018) The extinction and persistence of the stochastic hepatitis B epidemic model. Chaos, Solitons Fractals 108:123\u2013128","journal-title":"Chaos, Solitons Fractals"},{"issue":"1","key":"3330_CR58","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1080\/10273669708833004","volume":"1","author":"DE Kirschner","year":"1997","unstructured":"Kirschner DE, Webb GF (1997) A mathematical model of combined drug therapy of HIV infection. Comput Math Methods Med 1(1):25\u201334","journal-title":"Comput Math Methods Med"},{"issue":"12","key":"3330_CR59","doi-asserted-by":"crossref","first-page":"5702","DOI":"10.1002\/jmv.28037","volume":"94","author":"Y Kojima","year":"2022","unstructured":"Kojima Y, Nakakubo S, Kamada K, Yamashita Y, Takei N, Nakamura J, Matsumoto M, Horii H, Sato K, Shima H et al (2022) Combination therapy with remdesivir and immunomodulators improves respiratory status in COVID-19: a retrospective study. J Med Virol 94(12):5702\u20135712","journal-title":"J Med Virol"},{"issue":"4","key":"3330_CR60","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1016\/j.spl.2012.12.021","volume":"83","author":"A Lahrouz","year":"2013","unstructured":"Lahrouz A, Omari L (2013) Extinction and stationary distribution of a stochastic sirs epidemic model with non-linear incidence. Stat Prob Lett 83(4):960\u2013968","journal-title":"Stat Prob Lett"},{"issue":"5","key":"3330_CR61","doi-asserted-by":"crossref","first-page":"2667","DOI":"10.3390\/ijerph18052667","volume":"18","author":"A Leontitsis","year":"2021","unstructured":"Leontitsis A, Senok A, Alsheikh-Ali A, Al Nasser Y, Loney T, Alshamsi A (2021) Seahir: A specialized compartmental model for covid-19. Int J Environ Res Public Health 18(5):2667","journal-title":"Int J Environ Res Public Health"},{"issue":"2","key":"3330_CR62","doi-asserted-by":"crossref","first-page":"523","DOI":"10.3934\/dcds.2009.24.523","volume":"24","author":"X Li","year":"2009","unstructured":"Li X, Mao X (2009) Population dynamical behavior of non-autonomous Lotka\u2013Volterra competitive system with random perturbation. Discrete Continuous Dyn Syst-Ser A 24(2):523\u2013593","journal-title":"Discrete Continuous Dyn Syst-Ser A"},{"key":"3330_CR63","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1007\/s11538-015-0101-9","volume":"77","author":"D Li","year":"2015","unstructured":"Li D, Cui J, Liu M, Liu S (2015) The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate. Bull Math Biol 77:1705\u20131743","journal-title":"Bull Math Biol"},{"issue":"2","key":"3330_CR64","doi-asserted-by":"crossref","first-page":"73","DOI":"10.5582\/ddt.2020.01015","volume":"14","author":"Z Li","year":"2020","unstructured":"Li Z, Wang X, Cao D, Sun R, Li C, Li G (2020) Rapid review for the anti-coronavirus effect of remdesivir. Drug Discover Therap 14(2):73\u201376","journal-title":"Drug Discover Therap"},{"issue":"5","key":"3330_CR65","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1093\/imamat\/hxv002","volume":"80","author":"M Liu","year":"2015","unstructured":"Liu M (2015) Global asymptotic stability of stochastic Lotka\u2013Volterra systems with infinite delays. IMA J Appl Math 80(5):1431\u20131453","journal-title":"IMA J Appl Math"},{"key":"3330_CR66","volume":"26","author":"P Liu","year":"2021","unstructured":"Liu P, Din A, Huang L, Yusuf A (2021) Stochastic optimal control analysis for the hepatitis B epidemic model. Results Phys 26:104372","journal-title":"Results Phys"},{"key":"3330_CR67","doi-asserted-by":"crossref","unstructured":"Li C, Xu J, Liu J, Zhou Y (2020) The within-host viral kinetics of sars-cov-2, bioRxiv","DOI":"10.1101\/2020.02.29.965418"},{"key":"3330_CR68","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-58647-2","volume-title":"Stochastic partial differential equations","author":"SV Lototsky","year":"2017","unstructured":"Lototsky SV, Rozovsky BL et al (2017) Stochastic partial differential equations, vol 11. Springer, Berlin"},{"issue":"18","key":"3330_CR69","doi-asserted-by":"crossref","first-page":"3677","DOI":"10.1016\/j.physa.2009.05.036","volume":"388","author":"Q Lu","year":"2009","unstructured":"Lu Q (2009) Stability of sirs system with random perturbations. XXPhys A 388(18):3677\u20133686","journal-title":"XXPhys A"},{"issue":"5","key":"3330_CR70","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1051\/mmnp\/201712502","volume":"12","author":"M Mahrouf","year":"2017","unstructured":"Mahrouf M, Hattaf K, Yousfi N (2017) Dynamics of a stochastic viral infection model with immune response. Math Model Nat Phenomena 12(5):15\u201332","journal-title":"Math Model Nat Phenomena"},{"key":"3330_CR71","volume-title":"Stochastic differential equations and applications","author":"X Mao","year":"2007","unstructured":"Mao X (2007) Stochastic differential equations and applications. Elsevier, Hoboken"},{"issue":"1","key":"3330_CR72","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0304-4149(01)00126-0","volume":"97","author":"X Mao","year":"2002","unstructured":"Mao X, Marion G, Renshaw E (2002) Environmental Brownian noise suppresses explosions in population dynamics. Stoch Process Appl 97(1):95\u2013110","journal-title":"Stoch Process Appl"},{"issue":"1","key":"3330_CR73","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.jtbi.2008.04.011","volume":"254","author":"S Marino","year":"2008","unstructured":"Marino S, Hogue IB, Ray CJ, Kirschner DE (2008) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 254(1):178\u2013196","journal-title":"J Theor Biol"},{"issue":"1","key":"3330_CR74","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1080\/00401706.2000.10485979","volume":"42","author":"MD McKay","year":"2000","unstructured":"McKay MD, Beckman RJ, Conover WJ (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55\u201361","journal-title":"Technometrics"},{"issue":"4","key":"3330_CR75","doi-asserted-by":"crossref","first-page":"2227","DOI":"10.1002\/oca.2976","volume":"44","author":"M Mehdaoui","year":"2023","unstructured":"Mehdaoui M, Lamrani Alaoui A, Tilioua M (2023) Analysis of an optimal control problem for a spatio-temporal SIR model with nonlinear density dependent diffusion terms. Opt Control Appl Methods 44(4):2227\u20132256","journal-title":"Opt Control Appl Methods"},{"key":"3330_CR76","doi-asserted-by":"crossref","first-page":"2004","DOI":"10.1007\/s11538-010-9521-8","volume":"72","author":"RL Miller Neilan","year":"2010","unstructured":"Miller Neilan RL, Schaefer E, Gaff H, Fister KR, Lenhart S (2010) Modeling optimal intervention strategies for cholera. Bull Math Biol 72:2004\u20132018","journal-title":"Bull Math Biol"},{"issue":"5","key":"3330_CR77","doi-asserted-by":"crossref","first-page":"e639","DOI":"10.1016\/S2214-109X(20)30114-5","volume":"8","author":"O Mitj\u00e0","year":"2020","unstructured":"Mitj\u00e0 O, Clotet B (2020) Use of antiviral drugs to reduce covid-19 transmission. Lancet Glob Health 8(5):e639\u2013e640","journal-title":"Lancet Glob Health"},{"key":"3330_CR78","doi-asserted-by":"crossref","first-page":"8276317","DOI":"10.1155\/2018\/8276317","volume":"2018","author":"B Mobisa","year":"2018","unstructured":"Mobisa B, Lawi G, Nthiiri J (2018) Modelling in vivo HIV dynamics under combined antiretroviral treatment. J Appl Math 2018:8276317","journal-title":"J Appl Math"},{"key":"3330_CR79","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-03162-2","volume":"2021","author":"BJ Nath","year":"2021","unstructured":"Nath BJ, Dehingia K, Mishra VN, Chu Y-M, Sarmah HK (2021) Mathematical analysis of a within-host model of SARS-CoV-2. Adv Differ Equ 2021:1\u201311","journal-title":"Adv Differ Equ"},{"issue":"1","key":"3330_CR80","doi-asserted-by":"crossref","first-page":"20220148","DOI":"10.1515\/cmb-2022-0148","volume":"11","author":"D Nayak","year":"2023","unstructured":"Nayak D, Chhetri B, Vamsi Dasu KK, Muthusamy S, Bhagat VM (2023) A comprehensive and detailed within-host modeling study involving crucial biomarkers and optimal drug regimen for type i lepra reaction: A deterministic approach. Comput Math Biophys 11(1):20220148","journal-title":"Comput Math Biophys"},{"key":"3330_CR81","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2020.109846","volume":"135","author":"F Nda\u00efrou","year":"2020","unstructured":"Nda\u00efrou F, Area I, Nieto JJ, Torres DF (2020) Mathematical modeling of covid-19 transmission dynamics with a case study of Wuhan. Chaos, Solitons Fractals 135:109846","journal-title":"Chaos, Solitons Fractals"},{"key":"3330_CR82","first-page":"6185","volume":"20","author":"MZ Ndii","year":"2017","unstructured":"Ndii MZ, Supriatna AK (2017) Stochastic mathematical models in epidemiology. Information 20:6185\u20136196","journal-title":"Information"},{"issue":"3","key":"3330_CR83","first-page":"8","volume":"13","author":"NN Nguyen","year":"2019","unstructured":"Nguyen NN, Yin G (2019) Stochastic partial differential equation sis epidemic models: modeling and analysis. Commun Stoch Anal 13(3):8","journal-title":"Commun Stoch Anal"},{"issue":"2","key":"3330_CR84","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1017\/jpr.2020.15","volume":"57","author":"DH Nguyen","year":"2020","unstructured":"Nguyen DH, Nguyen NN, Yin G (2020) Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model. J Appl Probab 57(2):613\u2013636","journal-title":"J Appl Probab"},{"issue":"1","key":"3330_CR85","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmaa.2024.128361","volume":"538","author":"DT Nguyen","year":"2024","unstructured":"Nguyen DT, Du NH, Nguyen SL (2024) Asymptotic behavior for a stochastic behavioral change sir model. J Math Anal Appl 538(1):128361","journal-title":"J Math Anal Appl"},{"issue":"10","key":"3330_CR86","doi-asserted-by":"crossref","first-page":"e08207","DOI":"10.1016\/j.heliyon.2021.e08207","volume":"7","author":"M Odaka","year":"2021","unstructured":"Odaka M, Inoue K (2021) Modeling viral dynamics in SARS-CoV-2 infection based on differential equations and numerical analysis. Heliyon 7(10):e08207","journal-title":"Heliyon"},{"key":"3330_CR87","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-14394-6","volume-title":"Stochastic differential equations","author":"B \u00d8ksendal","year":"2003","unstructured":"\u00d8ksendal B, \u00d8ksendal B (2003) Stochastic differential equations. Springer, Berlin"},{"issue":"7","key":"3330_CR88","doi-asserted-by":"crossref","first-page":"1585","DOI":"10.3390\/cells10071585","volume":"10","author":"A Paolini","year":"2021","unstructured":"Paolini A, Borella R, De Biasi S, Neroni A, Mattioli M, Lo Tartaro D, Simonini C, Franceschini L, Cicco G, Piparo AM et al (2021) Cell death in coronavirus infections: uncovering its role during covid-19. Cells 10(7):1585","journal-title":"Cells"},{"issue":"4","key":"3330_CR89","doi-asserted-by":"crossref","first-page":"966","DOI":"10.1137\/0328054","volume":"28","author":"S Peng","year":"1990","unstructured":"Peng S (1990) A general stochastic maximum principle for optimal control problems. SIAM J Control Optim 28(4):966\u2013979","journal-title":"SIAM J Control Optim"},{"key":"3330_CR90","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1741-7007-11-96","volume":"11","author":"AS Perelson","year":"2013","unstructured":"Perelson AS, Ribeiro RM (2013) Modeling the within-host dynamics of HIV infection. BMC Biol 11:1\u201310","journal-title":"BMC Biol"},{"issue":"1","key":"3330_CR91","doi-asserted-by":"crossref","first-page":"20220144","DOI":"10.1515\/cmb-2022-0144","volume":"11","author":"DB Prakash","year":"2023","unstructured":"Prakash DB, Vamsi DKK (2023) Stochastic optimal and time-optimal control studies for additional food provided prey-predator systems involving holling type iii functional response. Comput Math Biophy 11(1):20220144","journal-title":"Comput Math Biophy"},{"key":"3330_CR92","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1007\/s40265-020-01367-z","volume":"80","author":"JG Rizk","year":"2020","unstructured":"Rizk JG, Kalantar-Zadeh K, Mehra MR, Lavie CJ, Rizk Y, Forthal DN (2020) Pharmaco-immunomodulatory therapy in covid-19. Drugs 80:1267\u20131292","journal-title":"Drugs"},{"key":"3330_CR93","volume-title":"A primer on Pontryagin\u2019s principle in optimal control","author":"IM Ross","year":"2009","unstructured":"Ross IM (2009) A primer on Pontryagin\u2019s principle in optimal control. Collegiate Publ, Pentagon City"},{"key":"3330_CR94","first-page":"20140886","volume":"12","author":"B Rotem","year":"2014","unstructured":"Rotem B, Koelle K (2014) minimal within-host dengue models highlight the specific roles of the immune response in primary and secondary dengue infections. Jr Soc Interface 12:20140886","journal-title":"Jr Soc Interface"},{"issue":"6","key":"3330_CR95","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.3390\/v13061141","volume":"13","author":"M Sadria","year":"2021","unstructured":"Sadria M, Layton AT (2021) Modeling within-host SARS-CoV-2 infection dynamics and potential treatments. Viruses 13(6):1141","journal-title":"Viruses"},{"key":"3330_CR96","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2020.110173","volume":"140","author":"P Samui","year":"2020","unstructured":"Samui P, Mondal J, Khajanchi S (2020) A mathematical model for COVID-19 transmission dynamics with a case study of India. Chaos, Solitons Fractals 140:110173","journal-title":"Chaos, Solitons Fractals"},{"key":"3330_CR97","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2020.110049","volume":"139","author":"K Sarkar","year":"2020","unstructured":"Sarkar K, Khajanchi S, Nieto JJ (2020) Modeling and forecasting the covid-19 pandemic in India. Chaos, Solitons Fractals 139:110049","journal-title":"Chaos, Solitons Fractals"},{"issue":"2","key":"3330_CR98","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1109\/TASE.2021.3111338","volume":"19","author":"P Scarabaggio","year":"2021","unstructured":"Scarabaggio P, Carli R, Cavone G, Epicoco N, Dotoli M (2021) Nonpharmaceutical stochastic optimal control strategies to mitigate the covid-19 spread. IEEE Trans Autom Sci Eng 19(2):560\u2013575","journal-title":"IEEE Trans Autom Sci Eng"},{"issue":"8","key":"3330_CR99","doi-asserted-by":"crossref","first-page":"3275","DOI":"10.3934\/dcdsb.2018244","volume":"23","author":"S Sengupta","year":"2018","unstructured":"Sengupta S, Das P, Mukherjee D (2018) Stochastic non-autonomous Holling type-III prey-predator model with predator\u2019s intra-specific competition. Discrete Contin Dyn Syst-Ser B 23(8):3275\u20133296","journal-title":"Discrete Contin Dyn Syst-Ser B"},{"issue":"9","key":"3330_CR100","doi-asserted-by":"crossref","first-page":"3243","DOI":"10.1073\/pnas.0806852106","volume":"106","author":"J Shaman","year":"2009","unstructured":"Shaman J, Kohn M (2009) Absolute humidity modulates influenza survival, transmission, and seasonality. Proc Natl Acad Sci 106(9):3243\u20133248","journal-title":"Proc Natl Acad Sci"},{"issue":"8","key":"3330_CR101","doi-asserted-by":"crossref","first-page":"2787","DOI":"10.1128\/AAC.48.8.2787-2792.2004","volume":"48","author":"SL Spruance","year":"2004","unstructured":"Spruance SL, Reid JE, Grace M, Samore M (2004) Hazard ratio in clinical trials. Antimicrob Agents Chemother 48(8):2787\u20132792","journal-title":"Antimicrob Agents Chemother"},{"issue":"1","key":"3330_CR102","first-page":"5780719","volume":"2022","author":"AK Srivastav","year":"2022","unstructured":"Srivastav AK, Stollenwerk N, Aguiar M (2022) Deterministic and stochastic dynamics of covid-19: The case study of Italy and Spain. Comput Math Methods 2022(1):5780719","journal-title":"Comput Math Methods"},{"issue":"1","key":"3330_CR103","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.camwa.2018.09.032","volume":"77","author":"H Sun","year":"2019","unstructured":"Sun H, Wang J (2019) Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay. Comput Math Appl 77(1):284\u2013301","journal-title":"Comput Math Appl"},{"issue":"02","key":"3330_CR104","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1142\/S0218339018500110","volume":"26","author":"S Sun","year":"2018","unstructured":"Sun S, Zhang X (2018) Asymptotic behavior of a stochastic delayed chemostat model with nutrient storage. J Biol Syst 26(02):225\u2013246","journal-title":"J Biol Syst"},{"issue":"2","key":"3330_CR105","doi-asserted-by":"crossref","first-page":"998","DOI":"10.1016\/j.jmaa.2018.08.033","volume":"468","author":"W Sun","year":"2018","unstructured":"Sun W, Xue L, Yan X (2018) Stability of a dengue epidemic model with independent stochastic perturbations. J Math Anal Appl 468(2):998\u20131017","journal-title":"J Math Anal Appl"},{"issue":"1","key":"3330_CR106","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1186\/s13662-021-03396-8","volume":"2021","author":"A Tesfay","year":"2021","unstructured":"Tesfay A, Saeed T, Zeb A, Tesfay D, Khalaf A, Brannan J (2021) Dynamics of a stochastic covid-19 epidemic model with jump-diffusion. Adv Difference Equ 2021(1):228","journal-title":"Adv Difference Equ"},{"key":"3330_CR107","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-021-03597-1","volume":"2021","author":"AW Tesfaye","year":"2021","unstructured":"Tesfaye AW, Satana TS (2021) Stochastic model of the transmission dynamics of covid-19 pandemic. Adv Difference Equ 2021:1\u201321","journal-title":"Adv Difference Equ"},{"issue":"sup2","key":"3330_CR108","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1080\/17513758.2016.1269957","volume":"11","author":"X Wang","year":"2017","unstructured":"Wang X, Wang J (2017) Modeling the within-host dynamics of cholera: bacterial-viral interaction. J Biol Dyn 11(sup2):484\u2013501","journal-title":"J Biol Dyn"},{"key":"3330_CR109","doi-asserted-by":"crossref","DOI":"10.1016\/j.mbs.2020.108438","volume":"328","author":"S Wang","year":"2020","unstructured":"Wang S, Pan Y, Wang Q, Miao H, Brown AN, Rong L (2020) Modeling the viral dynamics of SARS-CoV-2 infection. Math Biosci 328:108438","journal-title":"Math Biosci"},{"issue":"1","key":"3330_CR110","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1186\/s13662-023-03791-3","volume":"2023","author":"PJ Witbooi","year":"2023","unstructured":"Witbooi PJ, Maku Vyambwera S, van Schalkwyk GJ, Muller GE (2023) Stability and control in a stochastic model of malaria population dynamics. Adv Contin Discrete Models 2023(1):45","journal-title":"Adv Contin Discrete Models"},{"issue":"86","key":"3330_CR111","doi-asserted-by":"crossref","first-page":"20121018","DOI":"10.1098\/rsif.2012.1018","volume":"10","author":"J Wu","year":"2013","unstructured":"Wu J, Dhingra R, Gambhir M, Remais JV (2013) Sensitivity analysis of infectious disease models: methods, advances and their application. J R Soc Interface 10(86):20121018","journal-title":"J R Soc Interface"},{"issue":"3","key":"3330_CR112","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.jtbi.2009.01.001","volume":"257","author":"R Xu","year":"2009","unstructured":"Xu R, Ma Z (2009) An HBV model with diffusion and time delay. J Theor Biol 257(3):499\u2013509","journal-title":"J Theor Biol"},{"issue":"1","key":"3330_CR113","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.jmaa.2011.11.072","volume":"388","author":"Q Yang","year":"2012","unstructured":"Yang Q, Jiang D, Shi N, Ji C (2012) The ergodicity and extinction of stochastically perturbed sir and SEIR epidemic models with saturated incidence. J Math Anal Appl 388(1):248\u2013271","journal-title":"J Math Anal Appl"},{"key":"3330_CR114","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s12985-020-01472-1","volume":"18","author":"P Yin","year":"2021","unstructured":"Yin P, Meng J, Chen J, Gao J, Wang D, Liu S, Guo Q, Zhu M, Zhang G, Liu Y et al (2021) Antiviral drugs arbidol and interferon alpha-1b contribute to reducing the severity of COVID-19 patients: a retrospective cohort study. Virol J 18:1\u201311","journal-title":"Virol J"},{"key":"3330_CR115","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1466-3","volume-title":"Stochastic controls: Hamiltonian systems and HJB equations","author":"J Yong","year":"1999","unstructured":"Yong J, Zhou XY (1999) Stochastic controls: Hamiltonian systems and HJB equations, vol 43. Springer Science & Business Media, Berlin"},{"key":"3330_CR116","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s12190-018-01226-x","volume":"60","author":"P Yosyingyong","year":"2019","unstructured":"Yosyingyong P, Viriyapong R (2019) Global stability and optimal control for a hepatitis b virus infection model with immune response and drug therapy. J Appl Math Comput 60:537\u2013565","journal-title":"J Appl Math Comput"},{"key":"3330_CR117","doi-asserted-by":"crossref","first-page":"3452402","DOI":"10.1155\/2020\/3452402","volume":"2020","author":"A Zeb","year":"2020","unstructured":"Zeb A, Alzahrani E, Erturk VS, Zaman G (2020) Mathematical model for coronavirus disease 2019 (COVID-19) containing isolation class. Biomed Res Int 2020:3452402","journal-title":"Biomed Res Int"},{"key":"3330_CR118","volume":"175","author":"X Zhang","year":"2023","unstructured":"Zhang X (2023) Ultimate boundedness of a stochastic chemostat model with periodic nutrient input and discrete delay. Chaos, Solitons Fractals 175:113956","journal-title":"Chaos, Solitons Fractals"},{"key":"3330_CR119","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1016\/j.apm.2019.10.013","volume":"78","author":"X Zhang","year":"2020","unstructured":"Zhang X, Yuan R (2020) Sufficient and necessary conditions for stochastic near-optimal controls: a stochastic chemostat model with non-zero cost inhibiting. Appl Math Model 78:601\u2013626","journal-title":"Appl Math Model"},{"key":"3330_CR120","volume":"533","author":"C Zhang","year":"2019","unstructured":"Zhang C, Gao J, Sun H, Wang J (2019) Dynamics of a reaction\u2013diffusion SVIR model in a spatial heterogeneous environment. XXPhys A 533:122049","journal-title":"XXPhys A"},{"key":"3330_CR121","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-019-2438-0","volume":"2020","author":"Z Zhang","year":"2020","unstructured":"Zhang Z, Zeb A, Hussain S, Alzahrani E (2020) Dynamics of COVID-19 mathematical model with stochastic perturbation. Adv Differ Equ 2020:1\u201312","journal-title":"Adv Differ Equ"},{"key":"3330_CR122","first-page":"718","volume":"243","author":"Y Zhao","year":"2014","unstructured":"Zhao Y, Jiang D (2014) The threshold of a stochastic sis epidemic model with vaccination. Appl Math Comput 243:718\u2013727","journal-title":"Appl Math Comput"},{"issue":"4","key":"3330_CR123","doi-asserted-by":"crossref","first-page":"1954","DOI":"10.1137\/17M1144106","volume":"78","author":"G Zhao","year":"2018","unstructured":"Zhao G, Ruan S (2018) Spatial and temporal dynamics of a nonlocal viral infection model. SIAM J Appl Math 78(4):1954\u20131980","journal-title":"SIAM J Appl Math"},{"issue":"20","key":"3330_CR124","doi-asserted-by":"crossref","first-page":"4916","DOI":"10.1016\/j.physa.2013.06.009","volume":"392","author":"Y Zhao","year":"2013","unstructured":"Zhao Y, Jiang D, Regan DO (2013) The extinction and persistence of the stochastic sis epidemic model with vaccination. XXPhys A 392(20):4916\u20134927","journal-title":"XXPhys A"},{"issue":"1","key":"3330_CR125","doi-asserted-by":"crossref","DOI":"10.1155\/2012\/854723","volume":"2012","author":"M Zorom","year":"2012","unstructured":"Zorom M, Zongo P, Barbier B, Some B (2012) Optimal control of a spatio-temporal model for malaria: synergy treatment and prevention. J Appl Math 2012(1):854723","journal-title":"J Appl Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03330-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03330-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03330-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T05:06:44Z","timestamp":1757394404000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03330-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,22]]},"references-count":125,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["3330"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03330-8","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,7,22]]},"assertion":[{"value":"13 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 July 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 July 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"The authors declare that no AI tools were used in the creation of this article.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Use of AI tools declaration"}}],"article-number":"372"}}