article · Journal of Applied Mathematics
A nonlinear compartmental mathematical model evaluates how treatment interventions affect the spread of HIV/AIDS and pneumonia coinfection across different stages of infection. Mathematical analysis confirms that disease-free states remain stable when the basic reproduction numbers for the separate diseases and the coinfection model are below one, whereas endemic states stabilise when these values exceed one. Using standard values gathered from existing literature, baseline reproduction numbers were calculated as seventeen for HIV/AIDS and seven for pneumonia, yielding an overall coinfection reproduction number of seventeen. Sensitivity analysis indicates that the transmission rates for both diseases are the primary drivers influencing system dynamics. Numerical simulations demonstrate that administering treatments at every infection stage effectively reduces overall disease prevalence and transmission across single-infection and coinfected groups.
HIV/AIDS and pneumonia frequently co-occur, presenting severe public health challenges. By mathematically simulating how these two conditions interact and identifying transmission rates as critical drivers, this research highlights the value of multi-stage medical care. It demonstrates that treating individuals at every phase of infection can substantially lower overall disease prevalence in single-infection and coinfected populations.
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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 ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msub> <a:mrow> <a:mi mathvariant="script">R</a:mi> </a:mrow> <a:mrow> <a:mi>H</a:mi> </a:mrow> </a:msub> </a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M2"> <d:msub> <d:mrow> <d:mi mathvariant="script">R</d:mi> </d:mrow> <d:mrow> <d:mi>P</d:mi> </d:mrow> </d:msub> </d:math> ) are less than unity. Both the submodel endemic equilibrium points are locally and globally asymptotically stable whenever the associated basic reproduction numbers ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M3"> <g:msub> <g:mrow> <g:mi mathvariant="script">R</g:mi> </g:mrow> <g:mrow> <g:mi>P</g:mi> </g:mrow> </g:msub> </g:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"> <j:msub> <j:mrow> <j:mi mathvariant="script">R</j:mi> </j:mrow> <j:mrow> <j:mi>H</j:mi> </j:mrow> </j:msub> </j:math> ) 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M5"> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="script">R</m:mi> </m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mi>P</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:math> 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 <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M6"> <r:mi mathvariant="normal">max</r:mi> <r:mfenced open="{" close="}"> <r:mrow> <r:mn>7</r:mn> <r:mo>,</r:mo> <r:mn>17</r:mn> </r:mrow> </r:mfenced> <r:mo>=</r:mo> <r:mn>17</r:mn> </r:math> . 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 <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M7"> <w:msub> <w:mrow> <w:mi>β</w:mi> </w:mrow> <w:mrow> <w:mn>1</w:mn> </w:mrow> </w:msub> </w:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" id="M8"> <y:msub> <y:mrow> <y:mi>β</y:mi> </y:mrow> <y:mrow> <y:mn>2</y:mn> </y:mrow> </y:msub> </y:math> , 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.
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DOI: 10.1155/2021/5444605
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