Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis

Figures & data

Figure 1. Schema of the compartmental model.

Table 1. Description of state variables and parameters of model (1(1) $\begin{array}{rl}\dot{S}& =\mathrm{\Lambda }-\beta SI-({\mu }_S+\epsilon )S,\\ \dot{V}& =\epsilon S-{\mu }_{V}V-\theta \beta VI,\\ \dot{E}& =\beta SI+\theta \beta VI-({\mu }_E+\omega )E,\\ \dot{L}& =(1-p)\omega E-({\mu }_L+\nu )L,\\ \dot{I}& =p\omega E+\nu L-{\mu }_{I}I,\\ N& =S+V+E+L+I\end{array}$(1) ).

Variables	Description
S	Susceptible human individuals
V	Vaccinated human individuals
E	Earlier latent
L	Later latent
I	TB-infected human individuals
Parameters	Description
Λ	Recruitment of susceptible
ϵ	Vaccination coverage rate
ω	Progression rate to latent earlier stage
β	Effective contact rate
$1-\theta$	Vaccine efficacy
${\mu }_S$, ${\mu }_V$	Natural death rate of human individuals
${\mu }_E$, ${\mu }_L$,	Natural death rate latent individuals
${\mu }_I$	Death rate for I including natural and disease-caused death
p	Fraction of individuals who progress directly in TB active stage
1−p	Progression fraction to the late latent stage
ν	Progression rate to active disease

Figure 2. Sensitivity analysis between ${\mathcal{R}}_0$ and each parameter.

Table 2. Values and ranges of the parameters of model (1(1) $\begin{array}{rl}\dot{S}& =\mathrm{\Lambda }-\beta SI-({\mu }_S+\epsilon )S,\\ \dot{V}& =\epsilon S-{\mu }_{V}V-\theta \beta VI,\\ \dot{E}& =\beta SI+\theta \beta VI-({\mu }_E+\omega )E,\\ \dot{L}& =(1-p)\omega E-({\mu }_L+\nu )L,\\ \dot{I}& =p\omega E+\nu L-{\mu }_{I}I,\\ N& =S+V+E+L+I\end{array}$(1) ).

Parameters	Values	Range	Source
Λ	10	$[1;20]$	[15,16]
β	Variable	$[1e-5;1e-1]$	Assumed
θ	0.50	$[0.2;0.90]$	[6]
ω	0.0645	$[1e-2;1e-1]$	[8]
ν	0.00375	$[2e-3;1e-2]$	[3]
p	1.14e−3	$[6.70e-5;1.97e-3]$	[13]
${\mu }_S$, ${\mu }_V$, ${\mu }_E$, ${\mu }_L$	0.01874	$[1e-2;1e-1]$	[13]
${\mu }_I$	0.1577	$[1e-2;5e-1]$	[13]
ϵ	Variable	$[0.1,0.98]$	Assumed

Figure 3. Simulation results showing the global stability of the DFE.

Figure 4. Simulation results showing the global stability of endemic equilibrium.

Figure 5. Impact of effective contact rate on the spread of TB.

Figure 6. Impact of vaccination on the spread of TB.

Figure 7. Impact of contact rate and vaccination coverage rate on the spread of TB.

Figure 8. M/F age TB distribution.

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