Mathematical modelling of the impact of testing, treatment and control of HIV transmission in Kenya

Figures & data

Figure 1. Graphical representation of HIV transmission. Susceptible individuals flow into the undiagnosed (I) via infection rate. Unaware individuals recieve diagnosis and flow into ${\mathit{I}}_{\mathbf{1}}$ and ${\mathit{I}}_{\mathbf{2}}$ based on the CD4 cell counts. The tested individuals are then linked to ART treatment in classes ${\mathit{I}}_{\mathbf{1}}^{\mathit{T}}$ and ${\mathit{I}}_{\mathbf{2}}^{\mathit{T}}$.

Table 1. Parameter values and ranges per year obtained from data fitting

Par	Min	Max	Value	Source
$\mathrm{\Pi }$	0.0308N	0.0409N	0.0345N	WB (2017)
${\beta }_1$	0.0	2.0	1.9652	Okosun et al. (2013); Tripathi, Naresh, and Sharma (2007)
${\beta }_2$	0.0	2.0	2.000	Okosun et al. (2013); Tripathi et al. (2007)
${\beta }_3$	0.0	2.0	1.4153	Okosun et al. (2013); Tripathi et al. (2007)
${\beta }_4$	0.0	1.5	0.0160	Okosun et al. (2013)
${\beta }_5$	0.0	1.5	0.0376	Okosun et al. (2013)
${\alpha }_1$	0.1	1.5	1.4143	Okosun et al. (2013)
${\alpha }_2$	0.1	1.5	1.4363	Okosun et al. (2013)
${\eta }_1$	0.1	1.5	1.4732	Okosun et al. (2013)
${\gamma }_1$	0.0	1.2	0.5186	Assumed
${\gamma }_2$	0.0	1.5	1.4360	Assumed
${\eta }_2$	0.01	2.0	1.9960	Assumed
${\eta }_3$	0.0	1.0	0.3785	Assumed
$\mu$	0.014	0.02	0.0158	WB (2017)
${\delta }_1$	0.083	0.125	0.0845	Okosun et al. (2013)
${\delta }_2$	0.083	0.125	0.0835	Okosun et al. (2013)
ϕ	0.001	0.9	0.0511	Assumed
ϕ	0.001	0.9	0.7916	Assumed
ϕ	0.01	0.9	0.7990	Assumed

Figure 2. (a) shows the trend of new HIV infections while (b) shows the projected new HIV infections in Kenya by 2030.

Table 2. Parameter values and ranges used in numerical simulations as well as determination of sensitivity indices of ${\mathcal{R}}_0$. The parameter rates are given per annum

Par	Index	Par	Index	Par	Index	Par	Index
${\beta }_1$	+0.4331	${\alpha }_1$	−0.0360	${\eta }_2$	−0.0787	ϕ	−0.0539
${\beta }_2$	+0.3864	${\alpha }_2$	−0.1117	${\eta }_3$	+0.0772	ϕ	−0.0817
${\beta }_3$	+0.2433	${\eta }_1$	−0.0606	$\mu$	−0.1933	ϕ	−0.1542
${\beta }_4$	+0.2319	${\gamma }_1$	−0.0486	${\delta }_1$	−0.0180
${\beta }_5$	+0.1053	${\gamma }_2$	−0.1426	${\delta }_2$	−0.1547

Figure 3. Simulation result showing the effect of efforts made to protect susceptible individuals from infections on the disease prevalence.

Figure 4. Simulation result showing the testing and counselling on the disease prevalence.

Figure 5. Simulation results showing the advantage of combining personal protection, testing and counselling and ART treatment on the disease prevalence.

WB. (2017). World bank data. Birth rate, crude (per 1,000 people). Retrieved October, 2017 from https://data.worldbank.org/indicator/SP.DYN.CBRT.IN

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Okosun, K., Makinde, O., & Takaidza, I. (2013). Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives. Applied Mathematical Modelling, 37(6), 3802–3820. doi:10.1016/j.apm.2012.08.004

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Tripathi, A., Naresh, R., & Sharma, D. (2007). Modeling the effect of screening of unaware infectives on the spread of HIV infection. Applied Mathematics and Computation, 184(2), 1053–1068. doi:10.1016/j.amc.2006.07.007

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