HIV/AIDS-Pneumonia Coinfection Model with Treatment at Each Infection Stage: Mathematical Analysis and Numerical Simulation
Shewafera Wondimagegnhu Teklu, Temesgen Tibebu Mekonnen
Abstract
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.