{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T12:47:51Z","timestamp":1788266871362,"version":"build-2803163510"},"reference-count":78,"publisher":"Public Library of Science (PLoS)","issue":"12","license":[{"start":{"date-parts":[[2024,12,10]],"date-time":"2024-12-10T00:00:00Z","timestamp":1733788800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS ONE"],"abstract":"<jats:p>The co-epidemics of HIV\/AIDS and Tuberculosis (TB) outbreak is one of a serious disease in Ethiopia that demands integrative approaches to combat its transmission. In contrast, epidemiological co-infection models often considered a single latent case and recovered individuals with TB. To bridge this gap, we presented a new optimal HIV-TB co-infection model that considers both high risk and low risk latent TB cases with taking into account preventive efforts of both HIV and TB diseases, case finding for TB and HIV\/AIDS treatment. This study aimed to develop optimal HIV\/AIDS-TB co-infection mathematical model to explore the best cost-effective measure to mitigate the disease burden. The model is analysed analytically by firstly segregating TB and HIV only sub models followed by the full TB-HIV co-infection model. The Disease Free Equilibrium (DFE) and Endemic Equilibrium (EE) points are found and the basic reproduction number <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> is obtained using the next generation matrix method (NGM). Based on the threshold value <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub>, the stabilities of equilibria for each sub-model are analysed. The DFE point is locally asymptotically stable when <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> &lt; 1 and unstable when <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> &gt; 1. The EE point is also asymptotically stable when <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> &gt; 1 and does not exist otherwise. At <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> = 1, the existence of backward bifurcation phenomena is discussed. To curtail the cost and disease fatality, an optimal control model is formulated via time based controlling efforts. The optimal mathematical model is analysed both analytically and numerically. The numerical results are presented for two or more control measures at a time. In addition, the Incremental Cost-Effectiveness Ratio(ICER) has identified the best strategy which is crucial in limited resource. Hence, the model outcomes illustrated that applying HIV\/AIDS prevention efforts and TB case finding concurrently is the most cost-effective strategy to offer substantial relief from the burden of the pandemic in the community. All results found in this study have significant public health lessons. We anticipated that the results will notify evidence based approaches to control the disease. Thus, this study will aids in the fight against HIV\/AIDS, TB, and their co-infection policy-makers and other concerned organizations.<\/jats:p>","DOI":"10.1371\/journal.pone.0312539","type":"journal-article","created":{"date-parts":[[2024,12,10]],"date-time":"2024-12-10T18:31:08Z","timestamp":1733855468000},"page":"e0312539","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":12,"title":["Co-infection mathematical model for HIV\/AIDS and tuberculosis with optimal control in Ethiopia"],"prefix":"10.1371","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7692-4697","authenticated-orcid":true,"given":"Tigabu Kasie","family":"Ayele","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emile Franc","family":"Doungmo Goufo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stella","family":"Mugisha","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2024,12,10]]},"reference":[{"issue":"8","key":"pone.0312539.ref001","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1056\/NEJMoa0905848","article-title":"Timing of initiation of antiretroviral drugs during tuberculosis therapy","volume":"362","author":"Salim S Abdool Karim","year":"2010","journal-title":"New England Journal of Medicine"},{"issue":"2","key":"pone.0312539.ref002","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1128\/CMR.00042-10","article-title":"Hiv and tuberculosis: a deadly human syndemic","volume":"24","author":"Candice K Kwan","year":"2011","journal-title":"Clinical microbiology reviews"},{"issue":"2","key":"pone.0312539.ref003","doi-asserted-by":"crossref","first-page":"e1002464","DOI":"10.1371\/journal.ppat.1002464","article-title":"Tuberculosis and hiv co-infection","volume":"8","author":"Pawlowski Andrzej","year":"2012","journal-title":"PLoS pathogens"},{"issue":"10","key":"pone.0312539.ref004","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1097\/00002030-199907090-00002","article-title":"Does tuberculosis accelerate the progression of hiv disease? evidence from basic science and epidemiology","volume":"13","author":"Del Amo Julia","year":"1999","journal-title":"Aids"},{"issue":"1","key":"pone.0312539.ref005","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3923\/crt.2009.15.20","article-title":"The mathematical models of the dynamical behaviour of tuberculosis disease in the upper east region of the northern part of Ghana: A case study of bawku","volume":"1","author":"IA Adetunde","year":"2008","journal-title":"Current Research in Tuberculosis"},{"issue":"10153","key":"pone.0312539.ref006","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1016\/S0140-6736(18)32361-4","article-title":"Advancing global tuberculosis control after the unga-hlm","volume":"392","author":"Ben Marais","year":"2018","journal-title":"The Lancet"},{"key":"pone.0312539.ref007","unstructured":"World health organization (who), tuberculosis\/hiv fact sheet no 104, archived from the original on 23 August 2012, retrieved 11th February 2016."},{"issue":"9","key":"pone.0312539.ref008","first-page":"158","article-title":"Mathematical modelling of pulmonary and extra-pulmonary tuberculosis","volume":"4","author":"Nita H Shah","year":"2013","journal-title":"International Journal of Mathematics Trends and Technology"},{"key":"pone.0312539.ref009","unstructured":"Center for disease control and prevention (cdc), basic tb facts; archived from the original on 6th February, retrieved 17th May 2016."},{"key":"pone.0312539.ref010","unstructured":"World health organization, global tuberculosis report, 2017."},{"key":"pone.0312539.ref011","first-page":"66","article-title":"Hiv-tb co-infection; a simple mathematical model","volume":"7","author":"Kaur Navjot","year":"2015","journal-title":"J. Adv. Res. Dyn. Control Syst"},{"key":"pone.0312539.ref012","article-title":"Dynamic of a tb-hiv coinfection epidemic model with latent age","author":"Xiaoyan Wang","year":"2013","journal-title":"Journal of Applied Mathematics"},{"key":"pone.0312539.ref013","unstructured":"Gbolahan Bolarin. A mathematical model of complacency in hiv\/aids scenario: Sex-structure approach, 2012."},{"key":"pone.0312539.ref014","first-page":"8261208","article-title":"An optimal treatment control of tb-hiv coinfection","volume":"2016","author":"Fatmawati Hengki","year":"2016","journal-title":"J. Math. Sci"},{"key":"pone.0312539.ref015","unstructured":"World health organization (who), tuberculosis (tb): Tb\/hiv. Geneva, Switzerland, 2018."},{"key":"pone.0312539.ref016","unstructured":"World health organization (who), tuberculosis (tb): Tb\/hiv. Geneva, Switzerland, 2017."},{"issue":"1","key":"pone.0312539.ref017","doi-asserted-by":"crossref","first-page":"145","DOI":"10.3934\/mbe.2008.5.145","article-title":"Mathematical analysis of the transmission dynamics of hiv\/tb co-infection in the presence of treatment","volume":"5","author":"Sharomi Oluwaseun","year":"2008","journal-title":"Mathematical Biosciences & Engineering"},{"key":"pone.0312539.ref018","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.cam.2016.02.051","article-title":"Hiv-tb co-infection treatment: Modeling and optimal control theory perspectives","volume":"307","author":"Abhishek Mallela","year":"2016","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"pone.0312539.ref019","unstructured":"Global information and education on hiv and aids: https:\/\/www.avert.org\/global-hiv-and-aids-statistics."},{"key":"pone.0312539.ref020","unstructured":"HIV Global. Statistics. Unaids, 2018."},{"key":"pone.0312539.ref021","unstructured":"World health organization (who), tuberculosis (tb): Tb\/hiv. Geneva, Switzerland, 2019."},{"key":"pone.0312539.ref022","unstructured":"Global WHO. Global tuberculosis report. Glob. Tubercl. Rep, 2020."},{"issue":"4","key":"pone.0312539.ref023","doi-asserted-by":"crossref","first-page":"815","DOI":"10.3934\/mbe.2009.6.815","article-title":"Modeling tb and hiv co-infections","volume":"6","author":"Lih-Ing W Roeger","year":"2009","journal-title":"Mathematical Biosciences & Engineering"},{"key":"pone.0312539.ref024","doi-asserted-by":"crossref","unstructured":"V Lusiana, PS Putra, N Nuraini, and E Soewono. Mathematical modeling of transmission co-infection tuberculosis in hiv community. In AIP Conference Proceedings, volume 1825, page 020012. AIP Publishing LLC, 2017.","DOI":"10.1063\/1.4978981"},{"issue":"9","key":"pone.0312539.ref025","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1056\/NEJMoa1507198","article-title":"A trial of early antiretrovirals and isoniazid preventive therapy in Africa","volume":"373","author":"Temprano ANRS 12136 Study Group","year":"2015","journal-title":"New England Journal of Medicine"},{"issue":"3","key":"pone.0312539.ref026","doi-asserted-by":"crossref","first-page":"186","DOI":"10.4314\/gmj.v54i3.10","article-title":"Management of tb\/hiv co-infection: the state of the evidence","volume":"54","author":"Kwasi Torpey","year":"2020","journal-title":"Ghana Medical Journal"},{"key":"pone.0312539.ref027","unstructured":"Ethiopia-national strategic plan tuberculosis and leprosy control (2013-2020\/21): https:\/\/www.afro.who.int\/sites\/default\/les\/2019-04\/."},{"key":"pone.0312539.ref028","unstructured":"World health organization (who), tuberculosis (tb): Tb\/hiv. Geneva, Switzerland, 2020."},{"issue":"Supplement3","key":"pone.0312539.ref029","doi-asserted-by":"crossref","first-page":"S201","DOI":"10.1086\/651492","article-title":"Hiv infection associated tuberculosis: the epidemiology and the response","volume":"50","author":"Getahun Haileyesus","year":"2010","journal-title":"Clinical Infectious Diseases"},{"issue":"16","key":"pone.0312539.ref030","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1056\/NEJMoa1013911","article-title":"Earlier versus later start of antiretroviral therapy in hiv-infected adults with tuberculosis","volume":"365","author":"Francois-Xavier Blanc","year":"2011","journal-title":"New England Journal of Medicine"},{"issue":"2","key":"pone.0312539.ref031","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s10867-014-9342-3","article-title":"The role of screening and treatment in the transmission dynamics of hiv\/aids and tuberculosis co-infection: a mathematical study","volume":"40","author":"Kaur Navjot","year":"2014","journal-title":"Journal of biological physics"},{"issue":"1","key":"pone.0312539.ref032","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s12190-022-01734-x","article-title":"Mathematical model for control of tuberculosis epidemiology","volume":"69","author":"Mayowa M Ojo","year":"2023","journal-title":"Journal of Applied Mathematics and Computing"},{"issue":"4","key":"pone.0312539.ref033","first-page":"144","article-title":"Analysis and dynamics of tuberculosis outbreak: A mathematical modelling approach","volume":"22","author":"Festus Abiodun Oguntolu","year":"2022","journal-title":"Advances in Systems Science and Applications"},{"issue":"5","key":"pone.0312539.ref034","doi-asserted-by":"crossref","first-page":"1","DOI":"10.9734\/JAMCS\/2018\/41850","article-title":"Mathematical modeling of tb-hiv co infection, case study of tigania west sub county, Kenya","volume":"27","author":"Grace Gakii Muthuri","year":"2018","journal-title":"Journal of Advances in Mathematics and Computer science"},{"issue":"5","key":"pone.0312539.ref035","doi-asserted-by":"crossref","first-page":"48","DOI":"10.3390\/pr6050048","article-title":"Optimal control strategy for tb-hiv\/aids co-infection model in the presence of behaviour modification","volume":"6","author":"Temesgen Debas Awoke","year":"2018","journal-title":"Processes"},{"key":"pone.0312539.ref036","doi-asserted-by":"crossref","first-page":"104772","DOI":"10.1016\/j.rinp.2021.104772","article-title":"On analysis generalization of tb-hiv dynamics by a two-scale reduction process","volume":"30","author":"Emile Franc Doungmo Goufo","year":"2021","journal-title":"Results in Physics"},{"issue":"3","key":"pone.0312539.ref037","doi-asserted-by":"crossref","first-page":"2979","DOI":"10.1016\/j.aej.2021.01.041","article-title":"An epidemic prediction from analysis of a combined hiv-covid-19 co-infection model via abc-fractional operator","volume":"60","author":"Ahmed Idris","year":"2021","journal-title":"Alexandria Engineering Journal"},{"key":"pone.0312539.ref038","doi-asserted-by":"crossref","first-page":"110030","DOI":"10.1016\/j.chaos.2020.110030","article-title":"Hiv and shifting epicenters for covid-19, an alert for some countries","volume":"139","author":"Emile Franc Doungmo Goufo","year":"2020","journal-title":"Chaos, Solitons & Fractals"},{"issue":"1","key":"pone.0312539.ref039","first-page":"1","article-title":"Modelling intracellular delay and therapy interruptions within ghanaian hiv population","volume":"2020","author":"Ko F Owusu","year":"2020","journal-title":"Advances in Difference Equations"},{"key":"pone.0312539.ref040","doi-asserted-by":"crossref","first-page":"100316","DOI":"10.1016\/j.imu.2020.100316","article-title":"Hiv-tb co-infection treatment control using multi-objective optimized sliding mode","volume":"19","author":"S Hadipour Lakmesari","year":"2020","journal-title":"Informatics in Medicine Unlocked"},{"issue":"1","key":"pone.0312539.ref041","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1080\/25765299.2020.1840771","article-title":"Mathematical analysis and numerical simulation of co-infection of tb-hiv","volume":"27","author":"Ahmad Aqeel","year":"2020","journal-title":"Arab Journal of Basic and Applied Sciences"},{"issue":"4","key":"pone.0312539.ref042","doi-asserted-by":"crossref","first-page":"4701","DOI":"10.1007\/s11071-021-06518-9","article-title":"A fractional order hiv-tb co-infection model in the presence of exogenous reinfection and recurrent tb","volume":"104","author":"Rajiv Aggarwal","year":"2021","journal-title":"Nonlinear Dynamics"},{"key":"pone.0312539.ref043","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/j.matcom.2023.01.014","article-title":"A mathematical model for the co-dynamics of covid-19 and tuberculosis","volume":"207","author":"Mayowa M Ojo","year":"2023","journal-title":"Mathematics and Computers in Simulation"},{"issue":"1","key":"pone.0312539.ref044","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1080\/25765299.2022.2085445","article-title":"Optimal control analysis for the coinfection of covid-19 and tb","volume":"29","author":"Kassahun Getnet Mekonen","year":"2022","journal-title":"Arab Journal of Basic and Applied Sciences"},{"key":"pone.0312539.ref045","doi-asserted-by":"crossref","first-page":"100978","DOI":"10.1016\/j.imu.2022.100978","article-title":"Hiv and covid-19 co-infection: A mathematical model and optimal control","author":"N Ringa","year":"2022","journal-title":"Informatics in Medicine Unlocked"},{"issue":"1","key":"pone.0312539.ref046","first-page":"120131","article-title":"A dynamic model for tuberculosis transmission and optimal treatment strategies in south Korea","volume":"279","author":"Whang Sungim","year":"2011","journal-title":"Journal of Theoretical Biology"},{"issue":"1","key":"pone.0312539.ref047","first-page":"1","article-title":"Latent tuberculosis infection and associated risk indicators in pastoral communities in southern Ethiopia: A community based cross-sectional study","volume":"18","author":"Takele Teklu","year":"2018","journal-title":"Public health"},{"issue":"7","key":"pone.0312539.ref048","doi-asserted-by":"crossref","first-page":"e012636","DOI":"10.1136\/bmjopen-2016-012636","article-title":"Prevalence and risk factors of latent tuberculosis infection in Africa: a systematic review and meta-analysis protocol","volume":"7","author":"Tariro J Basera","year":"2017","journal-title":"BMJ open"},{"key":"pone.0312539.ref049","unstructured":"Global tb full report, world health organization, Geneva, Switzerland, 2020."},{"issue":"9","key":"pone.0312539.ref050","doi-asserted-by":"crossref","first-page":"4116","DOI":"10.1016\/j.jfranklin.2022.04.005","article-title":"Optimal control of tb transmission based on an age structured hiv-tb co-infection model","volume":"359","author":"Zhong-Kai Guo","year":"2022","journal-title":"Journal of the Franklin Institute"},{"issue":"1","key":"pone.0312539.ref051","doi-asserted-by":"crossref","first-page":"5556734","DOI":"10.1155\/2024\/5556734","article-title":"Mathematical modeling of covid-19 disease dynamics with contact tracing: A case study in Ethiopia","volume":"2024","author":"Shimelis Bekele Zerefe","year":"2024","journal-title":"Journal of Applied Mathematics"},{"key":"pone.0312539.ref052","unstructured":"Nadhirah Abdul Halim. Tuberculosis Model: A Mathematical Analysis\/Nadhirah bt Abdul Halim. PhD thesis, University of Malaya, 2013."},{"key":"pone.0312539.ref053","unstructured":"WHO, report on global hiv\/aids, 2020."},{"key":"pone.0312539.ref054","unstructured":"Unicef. For every child, end aids: seventh stocktaking report, 2016."},{"issue":"9635","key":"pone.0312539.ref055","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/S0140-6736(08)61115-0","article-title":"Relation between hiv viral load and infectiousness: A model-based analysis","volume":"372","author":"David P Wilson","year":"2008","journal-title":"The Lancet"},{"issue":"1-2","key":"pone.0312539.ref056","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Pauline Van den Driessche","year":"2002","journal-title":"Mathematical biosciences"},{"key":"pone.0312539.ref057","doi-asserted-by":"crossref","unstructured":"Carlos Castillo-Chavez, Sally Blower, Pauline Van den Driessche, Denise Kirschner, and Abdul-Aziz Yakubu. Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory, volume 126 Springer Science & Business Media, 2002.","DOI":"10.1007\/978-1-4613-0065-6"},{"issue":"5","key":"pone.0312539.ref058","first-page":"841","article-title":"Stability analysis for differential infectivity epidemic models","volume":"4","author":"Ma Zhien","year":"2003","journal-title":"Non-linear Analysis: Real World Applications"},{"key":"pone.0312539.ref059","unstructured":"Dandan Hou. Global dynamics of a staged progression model for infectious diseases, 2018."},{"key":"pone.0312539.ref060","unstructured":"JP LaSalle. The stability of dynamical systems, society for industrial and applied mathematics, Philadelphia, 1976."},{"key":"pone.0312539.ref061","article-title":"Modeling the effects of vaccination and treatment on tuberculosis transmission dynamics","author":"Ashenafi Kelemu Mengistu","year":"2019","journal-title":"Journal of Applied Mathematics"},{"issue":"2","key":"pone.0312539.ref062","doi-asserted-by":"crossref","first-page":"361","DOI":"10.3934\/mbe.2004.1.361","article-title":"Dynamical models of tuberculosis and their applications","volume":"1","author":"Castillo-Chavez Carlos","year":"2004","journal-title":"Mathematical Biosciences & Engineering"},{"key":"pone.0312539.ref063","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.amc.2014.12.096","article-title":"Transmission dynamics and optimal control of measles epidemics","volume":"256","author":"Pang Liuyong","year":"2015","journal-title":"Applied Mathematics and Computation"},{"key":"pone.0312539.ref064","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.matcom.2013.11.005","article-title":"Qualitative analysis and optimal control of an epidemic model with vaccination and treatment","volume":"100","author":"Buonomo Bruno","year":"2014","journal-title":"Mathematics and Computers in Simulation"},{"key":"pone.0312539.ref065","doi-asserted-by":"crossref","unstructured":"Maia Martcheva. An introduction to mathematical epidemiology, volume 61. Springer, 2015.","DOI":"10.1007\/978-1-4899-7612-3"},{"issue":"1","key":"pone.0312539.ref066","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.biosystems.2010.12.010","article-title":"Impact of chemotherapy on optimal control of malaria disease with infected immigrants","volume":"104","author":"Oluwole Daniel Makinde","year":"2011","journal-title":"BioSystems"},{"issue":"6","key":"pone.0312539.ref067","doi-asserted-by":"crossref","first-page":"3802","DOI":"10.1016\/j.apm.2012.08.004","article-title":"Impact of optimal control on the treatment of hiv\/aids and screening of unaware infective","volume":"37","author":"Kazeem Oare Okosun","year":"2013","journal-title":"Applied mathematical modelling"},{"key":"pone.0312539.ref068","volume-title":"Deterministic and stochastic optimal control, volume 1","author":"Wendell H Fleming","year":"2012"},{"key":"pone.0312539.ref069","unstructured":"Earl A Coddington and Norman Levinson. Theory of ordinary differential equations. Tata McGraw-Hill Education, 1955."},{"key":"pone.0312539.ref070","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-77647-5","volume-title":"Optimal control of non-linear processes","author":"Dieter Grass","year":"2008"},{"key":"pone.0312539.ref071","unstructured":"Earl A Coddington. An introduction to ordinary differential equations. Courier Corporation, 2012."},{"key":"pone.0312539.ref072","unstructured":"Boltyanskii V. G. Gamkrelidze R. V. Pontryagin, L. S. and E. Mishchenko. The mathematical theory of optimal processes, 1986."},{"key":"pone.0312539.ref073","doi-asserted-by":"crossref","first-page":"104263","DOI":"10.1016\/j.rinp.2021.104263","article-title":"Mathematical modeling of hiv\/aids with optimal control: A case study in Ethiopia","volume":"26","author":"Tigabu Kasia Ayele","year":"2021","journal-title":"Results in Physics"},{"key":"pone.0312539.ref074","unstructured":"Cristiana J Silva and Delfim FM Torres. A tb-hiv\/aids co-infection model and optimal control treatment. ArXiv preprint arXiv: 1501.03322, 2015."},{"key":"pone.0312539.ref075","doi-asserted-by":"crossref","first-page":"2050312120921055","DOI":"10.1177\/2050312120921055","article-title":"A 5-year experience characterizing the demographic and clinical profile and directly observed treatment short-course treatment outcome in national tuberculosis center of duhok province, iraqi kurdistan","volume":"8","author":"Muayad Aghali Merza","year":"2020","journal-title":"SAGE Open Medicine"},{"issue":"1","key":"pone.0312539.ref076","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.mbs.2010.10.007","article-title":"Epidemiological models with prevalence dependent endogenous self-protection measure","volume":"229","author":"Semu Mitiku Kassa","year":"2011","journal-title":"Mathematical biosciences"},{"key":"pone.0312539.ref077","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0218339024400084","article-title":"Mathematical modeling of tuberculosis with drug resistance in the presence of optimal control: A case study in Ethiopia","author":"Tigabu Kasie Ayele","year":"2024","journal-title":"Journal of Biological Systems"},{"issue":"1","key":"pone.0312539.ref078","first-page":"831506","article-title":"Optimal control of hiv\/aids in the workplace in the presence of careless individuals","volume":"2014","author":"Baba Seidu","year":"2014","journal-title":"Computational and mathematical methods in medicine"}],"container-title":["PLOS ONE"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0312539","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,10]],"date-time":"2024-12-10T18:31:47Z","timestamp":1733855507000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0312539"}},"subtitle":[],"editor":[{"given":"Joshua Kiddy K.","family":"Asamoah","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2024,12,10]]},"references-count":78,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12,10]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0312539","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,10]]}}}