article · Computational and Mathematical Methods in Medicine
A compartmental mathematical model captures the spread and management of simultaneous HIV/AIDS and pneumonia infections within a population, accounting for pneumonia vaccination and disease treatment across different infection stages. Mathematical analysis reveals six distinct equilibrium states, covering disease-free and endemic scenarios for single infections as well as co-infections. While reducing the reproduction number below one guarantees the eradication of HIV/AIDS alone, this threshold is insufficient for pneumonia alone or combined co-infections. Both pneumonia and co-infection models exhibit backward bifurcations, resulting in multiple coexisting endemic states even when the reproduction number is below one. Sensitivity evaluations identify the transmission rates of both pathogens as primary drivers of system dynamics, with pneumonia transmission enabling backward bifurcations. Applying published literature data yielded a baseline co-infection reproduction number of 9.69.
Managing overlapping epidemics is challenging because standard disease control benchmarks may fail when infections interact. By showing that reducing transmission metrics below standard thresholds is not sufficient to eradicate pneumonia or co-infections, this work provides critical theoretical guidance for healthcare planners designing vaccination and treatment programmes for dual epidemics.
This work represents early-stage theoretical and mathematical research. While the mathematical framework could eventually inform public health planning tools or epidemiological decision-support software, the abstract indicates no immediate commercial application or industry translation pathway.
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In this paper, we proposed and analyzed a realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections at each infection stage in a population. The model exhibits six equilibriums: HIV/AIDS only disease-free, pneumonia only disease-free, HIV/AIDS-pneumonia coepidemic disease-free, HIV/AIDS only endemic, pneumonia only endemic, and HIV/AIDS-pneumonia coepidemic endemic equilibriums. The HIV/AIDS only submodel has a globally asymptotically stable disease-free equilibrium if <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:mn>1</a:mn> </a:mrow> </a:msub> <a:mo><</a:mo> <a:mn>1</a:mn> <a:mo>.</a:mo> </a:math> Using center manifold theory, we have verified that both the pneumonia only submodel and the HIV/AIDS-pneumonia coepidemic model undergo backward bifurcations whenever <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:mn>2</d:mn> </d:mrow> </d:msub> <d:mo><</d:mo> <d:mn>1</d:mn> <d:mtext> </d:mtext> </d:math> and <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:mn>3</g:mn> </g:mrow> </g:msub> <g:mo>=</g:mo> <g:mi mathvariant="normal">max</g:mi> <g:mfenced open="{" close="}"> <g:mrow> <g:msub> <g:mrow> <g:mi mathvariant="script">R</g:mi> </g:mrow> <g:mrow> <g:mn>1</g:mn> </g:mrow> </g:msub> <g:mo>,</g:mo> <g:msub> <g:mrow> <g:mi mathvariant="script">R</g:mi> </g:mrow> <g:mrow> <g:mn>2</g:mn> </g:mrow> </g:msub> </g:mrow> </g:mfenced> <g:mo><</g:mo> <g:mn>1</g:mn> </g:math> , respectively. Thus, for pneumonia infection and HIV/AIDS-pneumonia coinfection, the requirement of the basic reproduction numbers to be less than one, even though necessary, may not be sufficient to completely eliminate the disease. Our sensitivity analysis results demonstrate that the pneumonia disease transmission rate <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M4"> <o:mtext> </o:mtext> <o:msub> <o:mrow> <o:mi>β</o:mi> </o:mrow> <o:mrow> <o:mn>2</o:mn> </o:mrow> </o:msub> </o:math> and the HIV/AIDS transmission rate <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M5"> <q:mtext> </q:mtext> <q:msub> <q:mrow> <q:mi>β</q:mi> </q:mrow> <q:mrow> <q:mn>1</q:mn> </q:mrow> </q:msub> </q:math> play an important role to change the qualitative dynamics of HIV/AIDS and pneumonia coinfection. The pneumonia infection transmission rate <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M6"> <s:msub> <s:mrow> <s:mi>β</s:mi> </s:mrow> <s:mrow> <s:mn>2</s:mn> </s:mrow> </s:msub> </s:math> gives rises to the possibility of backward bifurcation for HIV/AIDS and pneumonia coinfection if <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M7"> <u:msub> <u:mrow> <u:mi mathvariant="script">R</u:mi> </u:mrow> <u:mrow> <u:mn>3</u:mn> </u:mrow> </u:msub> <u:mo>=</u:mo> <u:mi mathvariant="normal">max</u:mi> <u:mfenced open="{" close="}"> <u:mrow> <u:msub> <u:mrow> <u:mi mathvariant="script">R</u:mi> </u:mrow> <u:mrow> <u:mn>1</u:mn> </u:mrow> </u:msub> <u:mo>,</u:mo> <u:msub> <u:mrow> <u:mi mathvariant="script">R</u:mi> </u:mrow> <u:mrow> <u:mn>2</u:mn> </u:mrow> </u:msub> </u:mrow> </u:mfenced> <u:mo><</u:mo> <u:mn>1</u:mn> </u:math> , and hence, the existence of multiple endemic equilibria some of which are stable and others are unstable. Using standard data from different literatures, our results show that the complete HIV/AIDS and pneumonia coinfection model reproduction number is <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M8"> <cb:msub> <cb:mrow> <cb:mi mathvariant="script">R</cb:mi> </cb:mrow> <cb:mrow> <cb:mn>3</cb:mn> </cb:mrow> </cb:msub> <cb:mo>=</cb:mo> <cb:mi mathvariant="normal">max</cb:mi> <cb:mfenced open="{" close="}"> <cb:mrow> <cb:msub> <cb:mrow> <cb:mi mathvariant="script">R</cb:mi> </cb:mrow> <cb:mrow> <cb:mn>1</cb:mn> </cb:mrow> </cb:msub> <cb:mo>,</cb:mo> <cb:msub> <cb:mrow> <cb:mi mathvariant="script">R</cb:mi> </cb:mrow> <cb:mrow> <cb:mn>2</cb:mn> </cb:mrow> </cb:msub> </cb:mrow> </cb:mfenced> <cb:mo>=</cb:mo> <cb:mi mathvariant="normal">max</cb:mi> <cb:mfenced open="{" close="}"> <cb:mrow> <cb:mn>1.386</cb:mn> <cb:mo>,</cb:mo> <cb:mn>9.69</cb:mn> <cb:mtext> </cb:mtext> </cb:mrow> </cb:mfenced> <cb:mo>=</cb:mo> <cb:mn>9.69</cb:mn> <cb:mtext> </cb:mtext> </cb:math> at <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" id="M9">
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DOI: 10.1155/2022/3105734
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