Mathematical Modelling Of Hiv/Aids Co-Infection With Tuberculosis And Pneumonia Incorporating Protection

ABSTRACT

The synergistic relationship between HIV/AIDS and respiratory infections, such as tu-

berculosis(TB) and pneumonia not only results in high mortality rates but is also a source

of economic burden borne by many nations in the sub-saharan Africa. The search for a

cure or vaccine for HIV/AIDS has yielded no conclusive results so far. Treatment fail-

ure and lack of adherence to treatment schedule which results in the evolution of drug

resistant strains of diseases are challenges to grapple with in the management of diseases

such as HIV/AIDS and TB. Due to global economic recession, provision and access to

subsidised medication may not be sustainable in the long run. Existing HIV/AIDS - TB

models do not consider protection, which may be less costly as an intervention measure.

Notably, the interaction between HIV/AIDS and pneumonia which contribute to a signif-

icant number of mortality cases in HIV/AIDS, has not been mathematically explored. In

this work, two deterministic models based on systems of ordinary dierential equations,

one on the co-infection of HIV/AIDS with TB and the second on the co-infection between

HIV/AIDS and pneumonia are formulated and analyzed to investigate protection as a

control strategy. Using the next generation matrix approach the reproduction numbers

for the models are determined and the respective disease free equilibrium points are shown

not to be globally asymptotically stable. This implies that reoccurrence of the disease is

possible especially when the conditions favoring such reoccurrence are prevailing. Four

cases of maximum protection are considered. In all cases, the endemic states are shown

to exist provided that the reproduction number is greater than unity. By use of Routh-

Hurwitz criterion and suitable Lyapunov functions, the endemic states are shown to be

locally and globally asymptotically stable respectively. This implies that with maximum

protection against one infection, the other disease can be controlled with intervention

measures possibly resulting in minimal deaths. This is illustrated by the numerical sim-

ulations which shows that protection as a strategy reduces the disease prevalence in all

the cases considered. Thus, from the ndings, emphasis should be placed on advocacy

for protection against infection as a strategy for reducing disease prevalence.