{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T16:31:37Z","timestamp":1774542697063,"version":"3.50.1"},"reference-count":40,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T00:00:00Z","timestamp":1704326400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:p>In this study, we present a nonlinear deterministic mathematical model for co-infection of pneumonia and COVID-19 transmission dynamics. To understand the dynamics of the co-infection of COVID-19 and pneumonia sickness, we developed and examined a compartmental based ordinary differential equation type mathematical model. Firstly, we showed the limited region and non-negativity of the solution, which demonstrate that the model is biologically relevant and mathematically well-posed. Secondly, the Jacobian matrix and the Lyapunov function are used to illustrate the local and global stability of the equilibrium locations. If the related reproduction numbers <jats:inline-formula><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">R<\/mml:mi><\/mml:mrow><\/mml:mrow><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:math><\/jats:inline-formula>, <jats:inline-formula><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:msubsup><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">R<\/mml:mi><\/mml:mrow><\/mml:mrow><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:math><\/jats:inline-formula>, and <jats:inline-formula><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">R<\/mml:mi><\/mml:mrow><\/mml:mrow><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:inline-formula> are smaller than unity, then pneumonia, COVID-19, and their co-infection have disease-free equilibrium points that are both locally and globally asymptotically stable otherwise the endemic equilibrium points are stable. Sensitivity analysis is used to determine how each parameter affects the spread or control of the illnesses. Moreover, we applied the optimal control theory to describe the optimal control model that incorporates four controls, namely, prevention of pneumonia, prevention of COVID-19, treatment of infected pneumonia and treatment of infected COVID-19. Then the Pontryagin's maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the numerical simulation of optimality system reveals that the combination of treatment and prevention is the most optimal to minimize the diseases.<\/jats:p>","DOI":"10.3389\/fams.2023.1286914","type":"journal-article","created":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T05:03:01Z","timestamp":1704344581000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":10,"title":["Pneumonia and COVID-19 co-infection modeling with optimal control analysis"],"prefix":"10.3389","volume":"9","author":[{"given":"Beza Zeleke","family":"Aga","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Temesgen Duressa","family":"Keno","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Debela Etefa","family":"Terfasa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hailay Weldegiorgis","family":"Berhe","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1965","published-online":{"date-parts":[[2024,1,4]]},"reference":[{"key":"B1","volume-title":"Pneumonia Fact Sheet: Media Centre","year":"2013"},{"key":"B2","year":"2020","journal-title":"Childhood Pneumonia: Everything you need to Know"},{"key":"B3","volume-title":"Technical Basis for the WHO Recommendations on the Management of Pneumonia in Children at First Level Health Facilities","year":"2008"},{"key":"B4","volume-title":"Coronavirus Disease 2019 (COVID-19): Situation Report, 68","year":"2020"},{"key":"B5","volume-title":"COVID-19 Weekly Epidemiological Update, Edition 70","year":"2021"},{"key":"B6","year":"","journal-title":"WHO Coronavirus Dashboard"},{"key":"B7","volume-title":"Mathematical Epidemiology of Infectious Diseases","author":"Diekmann","year":"2000"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/s13360-021-02046-y","article-title":"Mathematical modeling of COVID-19 in India and its states with optimal control","author":"Bandekar","year":"2021","journal-title":"Model Earth Syst Environ"},{"key":"B9","doi-asserted-by":"publisher","first-page":"812","DOI":"10.1080\/07362994.2021.1962343","article-title":"Modeling and analysis of COVID-19 in India with treatment function through different phases of lockdown and unlock","volume":"40","author":"Bandekar","year":"2021","journal-title":"Stoch Anal Appl"},{"key":"B10","doi-asserted-by":"publisher","first-page":"590","DOI":"10.1080\/17513758.2020.1788182","article-title":"Optimal control strategies for the transmission risk of COVID-19","volume":"14","author":"Lemecha Obsu","year":"2020","journal-title":"J Biol Dyn"},{"key":"B11","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1016\/j.idm.2020.02.002","article-title":"Real-time forecasts of the COVID-19 epidemic in china from February 5th to February 24th, 2020","volume":"5","author":"Roosa","year":"2020","journal-title":"Infect Dis Model"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1101\/2020.01.31.20019901","article-title":"Early dynamics of transmission and control of COVID-19: a mathematical modelling study","author":"Kucharski","year":"2020","journal-title":"Lancet Infect Dis"},{"key":"B13","doi-asserted-by":"publisher","first-page":"100134","DOI":"10.1016\/j.rinam.2020.100134","article-title":"Modeling the effect of contaminated objects for the transmission dynamics of COVID-19 pandemic with self protection behavior changes","volume":"9","author":"Mekonen","year":"2021","journal-title":"Results Appl Math"},{"key":"B14","doi-asserted-by":"publisher","first-page":"100123","DOI":"10.1016\/j.rinam.2020.100123","article-title":"Model the transmission dynamics of COVID-19 propagation with public health intervention","volume":"7","author":"Mamo","year":"2020","journal-title":"Results Appl Math"},{"key":"B15","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1017\/S0950268800056648","article-title":"Acquisition and invasiveness of different serotypes of streptococcus pneumoniae in young children","volume":"111","author":"Smith","year":"1993","journal-title":"Epidemiol Infect"},{"key":"B16","doi-asserted-by":"publisher","first-page":"336","DOI":"10.3201\/eid0503.990304","article-title":"Bacterial vaccines and serotype replacement: lessons from haemophilus influenzae and prospects for streptococcus pneumoniae","volume":"5","author":"Lipsitch","year":"1999","journal-title":"Emerg Infect Dis"},{"key":"B17","doi-asserted-by":"publisher","first-page":"2206","DOI":"10.1128\/AAC.48.6.2206-2213.2004","article-title":"Short and long term effects of pneumococcal conjugate vaccination of children on penicillin resistance","volume":"48","author":"Temime","year":"2004","journal-title":"Antimicrob Agents Chemother"},{"key":"B18","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1017\/S0950268804001980","article-title":"Estimating the transmission parameter of pneumococcal carriage in households","volume":"132","author":"Melegaro","year":"2004","journal-title":"Epidemiol Infect"},{"key":"B19","doi-asserted-by":"publisher","first-page":"413","DOI":"10.12988\/ijma.2013.13037","article-title":"Modeling co-infection of paediatric malaria and pneumonia","volume":"7","author":"Lawi","year":"2013","journal-title":"Int J Math Anal"},{"key":"B20","doi-asserted-by":"publisher","first-page":"954","DOI":"10.1053\/rmed.2000.0865","article-title":"Risk factors for community-acquire pneumonia diagnosed upon hospital admission","volume":"94","author":"Farr","year":"2000","journal-title":"Respir Med"},{"key":"B21","author":"Pessoa","year":"2010","journal-title":"Modelling the Dynamics of Streptococcus pneumoniae Transmission in Children"},{"key":"B22","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.prrv.2010.09.011","article-title":"Pneumonia management in the developing world","volume":"12","author":"Singh","year":"2011","journal-title":"Paediatr Respir Rev"},{"key":"B23","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1016\/j.apm.2021.06.016","article-title":"Malaria and COVID-19 co-dynamics: a mathematical model and optimal control","volume":"99","author":"Tchoumi","year":"2021","journal-title":"Appl Math Model"},{"key":"B24","doi-asserted-by":"publisher","first-page":"111008","DOI":"10.1016\/j.chaos.2021.111008","article-title":"Modeling, analysis and prediction of new variants of COVID-19 and dengue co-infection on complex network","volume":"150","author":"Rehman","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"B25","doi-asserted-by":"publisher","first-page":"128607","DOI":"10.1016\/j.physa.2023.128607","article-title":"Modeling SARS-CoV-2 and HBV co-dynamics with optimal control","volume":"615","author":"Omame","year":"2023","journal-title":"Physica A"},{"key":"B26","doi-asserted-by":"publisher","first-page":"2449710","DOI":"10.1155\/2022\/2449710","article-title":"Mathematical modeling and analysis of TB and COVID-19 coinfection","volume":"2022","author":"Mekonen","year":"2022","journal-title":"J Appl Math"},{"key":"B27","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/j.ijtb.2020.05.006","article-title":"COVID-19 and tuberculosis: a mathematical model based forecasting in Delhi, India","volume":"67","author":"Marimuthu","year":"2020","journal-title":"Indian J Tuberc"},{"key":"B28","doi-asserted-by":"publisher","first-page":"111486","DOI":"10.1016\/j.chaos.2021.111486","article-title":"A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana-Baleanu derivative","volume":"153","author":"Omame","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"B29","first-page":"1","article-title":"A mathematical model for the transmissionand control of malaria and typhoid co-infection using sirs approach","volume":"2","author":"Adeboye","year":"2015","journal-title":"Niger Res J Math"},{"key":"B30","doi-asserted-by":"publisher","first-page":"541","DOI":"10.12988\/ijma.2015.412403","article-title":"A model on the impact of treating typhoid with anti-malarial: dynamics ofmalaria concurrent and co-infection with typhoid","volume":"9","author":"Akinyi","year":"2015","journal-title":"Appl Math Sci"},{"key":"B31","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1016\/j.amc.2017.07.063","article-title":"Co-dynamics of pneumonia and typhoid fever diseases with cost effective optimal control analysis","volume":"316","author":"Tilahun","year":"2018","journal-title":"Appl Math Comput"},{"key":"B32","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/s0025-5564(02)00108-6","article-title":"Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Van den Driessche","year":"2002","journal-title":"Math Biosci"},{"key":"B33","doi-asserted-by":"publisher","first-page":"1272","DOI":"10.1007\/s11538-008-9299-0","article-title":"Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model","volume":"70","author":"Chitnis","year":"2008","journal-title":"Bull Math Biol"},{"key":"B34","doi-asserted-by":"publisher","first-page":"134","DOI":"10.5614\/j.math.fund.sci.2021.53.1.10","article-title":"Optimal control and cost-effectiveness analysis of SIRS malaria disease model with temperature variability factor","volume":"53","author":"Temesgen","year":"2021","journal-title":"J Math Fund Sci"},{"key":"B35","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1142\/S0218339021500170","article-title":"Impact of temperature variability on SIRS Malaria Model","volume":"29","author":"Duressa","year":"2021","journal-title":"J Biol Syst"},{"key":"B36","author":"Fleming","year":"2012","journal-title":"Deterministic and Stochastic Optimal Control"},{"key":"B37","volume-title":"Theory of Ordinary Differential Equations","author":"Levinson","year":"1955"},{"key":"B38","volume-title":"The Mathematical Theory of Optimal Processes","author":"Pontryagin","year":"1986"},{"key":"B39","doi-asserted-by":"crossref","DOI":"10.1201\/9781420011418","volume-title":"Optimal Control Applied to Biological Models","author":"Lenhart","year":"2007"},{"key":"B40","unstructured":"2020"}],"container-title":["Frontiers in Applied Mathematics and Statistics"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1286914\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T05:03:10Z","timestamp":1704344590000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1286914\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,4]]},"references-count":40,"alternative-id":["10.3389\/fams.2023.1286914"],"URL":"https:\/\/doi.org\/10.3389\/fams.2023.1286914","relation":{},"ISSN":["2297-4687"],"issn-type":[{"value":"2297-4687","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,4]]},"article-number":"1286914"}}