{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T21:30:25Z","timestamp":1777498225027,"version":"3.51.4"},"reference-count":26,"publisher":"Wiley","license":[{"start":{"date-parts":[[2021,9,16]],"date-time":"2021-09-16T00:00:00Z","timestamp":1631750400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2021,9,16]]},"abstract":"<jats:p>In the paper, we have considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV\/AIDS and pneumonia coinfection in a human population at different infection stages. Our model revealed that the disease-free equilibrium points of the HIV\/AIDS and pneumonia submodels are both locally and globally asymptotically stable whenever the associated basic reproduction numbers (<jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi mathvariant=\"script\">R<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>H<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                     <\/a:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <d:math xmlns:d=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <d:msub>\n                           <d:mrow>\n                              <d:mi mathvariant=\"script\">R<\/d:mi>\n                           <\/d:mrow>\n                           <d:mrow>\n                              <d:mi>P<\/d:mi>\n                           <\/d:mrow>\n                        <\/d:msub>\n                     <\/d:math>\n                  <\/jats:inline-formula>) are less than unity. Both the submodel endemic equilibrium points are locally and globally asymptotically stable whenever the associated basic reproduction numbers (<jats:inline-formula>\n                     <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <g:msub>\n                           <g:mrow>\n                              <g:mi mathvariant=\"script\">R<\/g:mi>\n                           <\/g:mrow>\n                           <g:mrow>\n                              <g:mi>P<\/g:mi>\n                           <\/g:mrow>\n                        <\/g:msub>\n                     <\/g:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <j:math xmlns:j=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <j:msub>\n                           <j:mrow>\n                              <j:mi mathvariant=\"script\">R<\/j:mi>\n                           <\/j:mrow>\n                           <j:mrow>\n                              <j:mi>H<\/j:mi>\n                           <\/j:mrow>\n                        <\/j:msub>\n                     <\/j:math>\n                  <\/jats:inline-formula>) are greater than unity. The full HIV\/AIDS-pneumonia coinfection model has both locally and globally asymptotically stable disease-free equilibrium points whenever the basic reproduction number of the coinfection model<jats:inline-formula>\n                     <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <m:mfenced open=\"(\" close=\")\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mrow>\n                                    <m:mi mathvariant=\"script\">R<\/m:mi>\n                                 <\/m:mrow>\n                                 <m:mrow>\n                                    <m:mi>H<\/m:mi>\n                                    <m:mi>P<\/m:mi>\n                                 <\/m:mrow>\n                              <\/m:msub>\n                           <\/m:mrow>\n                        <\/m:mfenced>\n                     <\/m:math>\n                  <\/jats:inline-formula>is less than unity. Using standard values of parameters collected from different kinds of literature, we found that the numerical values of the basic reproduction numbers of the HIV\/AIDS-only submodel and pneumonia-only submodel are 17 and 7, respectively, and the basic reproduction number of the HIV\/AIDS-pneumonia coinfection model is <jats:inline-formula>\n                     <r:math xmlns:r=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <r:mi mathvariant=\"normal\">max<\/r:mi>\n                        <r:mfenced open=\"{\" close=\"}\">\n                           <r:mrow>\n                              <r:mn>7<\/r:mn>\n                              <r:mo>,<\/r:mo>\n                              <r:mn>17<\/r:mn>\n                           <\/r:mrow>\n                        <\/r:mfenced>\n                        <r:mo>=<\/r:mo>\n                        <r:mn>17<\/r:mn>\n                     <\/r:math>\n                  <\/jats:inline-formula>. Applying sensitive analysis, we identified the most influential parameters to change the behavior of the solution of the considered coinfection dynamical system are the HIV\/AIDS and pneumonia transmission rates <jats:inline-formula>\n                     <w:math xmlns:w=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <w:msub>\n                           <w:mrow>\n                              <w:mi>\u03b2<\/w:mi>\n                           <\/w:mrow>\n                           <w:mrow>\n                              <w:mn>1<\/w:mn>\n                           <\/w:mrow>\n                        <\/w:msub>\n                     <\/w:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <y:math xmlns:y=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <y:msub>\n                           <y:mrow>\n                              <y:mi>\u03b2<\/y:mi>\n                           <\/y:mrow>\n                           <y:mrow>\n                              <y:mn>2<\/y:mn>\n                           <\/y:mrow>\n                        <\/y:msub>\n                     <\/y:math>\n                  <\/jats:inline-formula>, respectively. The coinfection model was numerically simulated to investigate the stability of the coinfection endemic equilibrium point, the impacts of transmission rates, and treatment strategies for HIV\/AIDS-only, pneumonia-only, and HIV\/AIDS-pneumonia coinfected individuals. Finally, we observed that numerical simulations indicate that treatment against infection at every stage lowers the rate of infection or disease prevalence.<\/jats:p>","DOI":"10.1155\/2021\/5444605","type":"journal-article","created":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T18:50:12Z","timestamp":1631904612000},"page":"1-21","source":"Crossref","is-referenced-by-count":40,"title":["HIV\/AIDS-Pneumonia Coinfection Model with Treatment at Each Infection Stage: Mathematical Analysis and Numerical Simulation"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9073-6075","authenticated-orcid":true,"given":"Shewafera Wondimagegnhu","family":"Teklu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Debre Berhan University, Debre Berhan, Ethiopia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Temesgen Tibebu","family":"Mekonnen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Debre Berhan University, Debre Berhan, Ethiopia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","article-title":"Texts in applied mathematics","volume-title":"An Introduction to Mathematical Epidemiology","author":"M. 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