Analysis of a model of gambiense sleeping sickness in humans and cattle

Figures & data

Figure 1. Flow diagram of HAT transmission dynamics.

Figure 2. Numbers of humans, cattle and flies in each compartment with baseline parameter values as in Table 1 with $I_F=1$, $N_H=300$, $N_C=50$ and $L^{\ast }=4545$, giving approximate equilibrium values $S_C^{\ast }=5,E_C^{\ast }=5,I_C^{\ast }=10,R_C^{\ast }=30,S_H^{\ast }=165,E_H^{\ast }=12,I_H^{\ast }=31,R_H^{\ast }=92,S_F^{\ast }=4929,E_F^{\ast }=31,I_F^{\ast }=40$. For these parameter values, the reproduction numbers are ${\mathcal{R}}_0=3.0298,{\mathcal{R}}_{0H}=1.9051,{\mathcal{R}}_{0C}=4.2505$.

Table 1. Description of model parameters, indicating baselines, ranges and references. The first six parameters describing larval and adult fly populations are fixed at their baseline values.

Parameter	Description	Baseline value	Value range with time $\mathrm{unit}=1$ day
${\sigma }_L$	Rate of maturation from pupal to adult fly	$\frac{1}{30}$	$\frac{1}{60}$ to $\frac{1}{22}$ [18]
$d_F$	Fly death rate	$\frac{1}{33}$	$\frac{1}{30}$ to $\frac{1}{62}$ [3]
W	Proportion of female flies	$\frac{6}{10}$	Estimated guess, female flies are more abundant than males
$d_L$	Pupa death rate	$\frac{1}{100}$	Estimated guess, in natural conditions very few pupae die
$K_L$	Pupa carrying capacity	300,000	Estimated guess
$b_L$	Larva birth rate	$\frac{0.6}{9}$	An adult female is expected to produce one larva every 9 days. [18, 20]
$b_H$	Human population $\mathrm{birthrate}=\mathrm{deathrate}$	$\frac{1}{50\times 365}$	10 to 40 births per 1000 per year [14]
$b_C$	Cattle population $\mathrm{birthrate}=\mathrm{deathrate}$	$\frac{1}{15\times 365}$	Estimated guess
$q_F$	Incubation rate of the flies	$\frac{1}{25}$	$\frac{1}{25}$ to $\frac{1}{30}$ [[3], [21, Table 2]]
$q_H,q_C$	Incubation rate of humans, cattle	$\frac{1}{12}$	$\frac{1}{10}$ to $\frac{1}{14}$, incubation period is between 10 and 14 days [3]
a	Fly biting rate	$\frac{1}{4}$	$\frac{1}{10}$ to $\frac{1}{3}$, a fly is expected to have 1 bite every 3–10 days [21]
p	Proportion of tsetse fly bites on cattle	0.7	[21, Table 2]
b,u	Probability that an infectious fly infects a human, cattle	0.62	[21, Table 2]
c,v	Probability that a fly becomes infected after biting an infectious human, cattle	0.01	Estimated guess [21, Table 2]
${\gamma }_H,{\gamma }_C$	Human, cattle rate of progression from stage I to stage II	$\frac{1}{30},\frac{1}{25}$	[21, Table 2]
${\kappa }_H,{\kappa }_C$	Human, cattle rate of progression from stage II to recovery and loss of temporary immunity for untreated humans, cattle; and humans, cattle rate of progression from start of treatment to recovery and loss of temporary immunity for treated humans, cattle	$\frac{1}{90},\frac{1}{75}$	[21, Table 2]

Table 2. Elasticity indices of ${\mathcal{R}}_0$ relative to different model parameters.

Parameter	Formula ${\displaystyle \frac{\mathrm{\partial }{\mathcal{R}}_0}{\mathrm{\partial }\nu }\frac{\nu }{{\mathcal{R}}_0}}$	Values of the parameter	Elasticity index
a	1	0.25	1
p	$-p(1-p){\displaystyle \frac{{\mathcal{R}}_{0H}^2}{{\mathcal{R}}_0^2}+p^2\frac{{\mathcal{R}}_{0C}^2}{{\mathcal{R}}_0^2}}$	0.7	0.8814
u	${\displaystyle \frac{1}{2}{p}^2\frac{{\mathcal{R}}_{0C}^2}{{\mathcal{R}}_0^2}}$	0.62	0.4822
v	${\displaystyle \frac{1}{2}{p}^2\frac{{\mathcal{R}}_{0C}^2}{{\mathcal{R}}_0^2}}$	0.01	0.4822
${\gamma }_C$	$-{\displaystyle \frac{1}{2}{p}^2\frac{{\mathcal{R}}_{0C}^2}{{\mathcal{R}}_0^2}\frac{{\gamma }_C}{{\gamma }_C+b_C}}$	0.04	−0.4800
$q_F$	${\displaystyle \frac{1}{2}\frac{d_F}{d_F+q_F}}$	0.04	0.2155
b	${\displaystyle \frac{1}{2}(1-p{)}^2\frac{{\mathcal{R}}_{0H}^2}{{\mathcal{R}}_0^2}}$	0.62	0.0178
c	${\displaystyle \frac{1}{2}(1-p{)}^2\frac{{\mathcal{R}}_{0H}^2}{{\mathcal{R}}_0^2}}$	0.01	0.0178
${\gamma }_H$	$-{\displaystyle \frac{1}{2}(1-p{)}^2\frac{{\mathcal{R}}_{0H}^2}{{\mathcal{R}}_0^2}\frac{{\gamma }_H}{{\gamma }_H+b_H}}$	0.0333	−0.0178
$b_C$	$-{\displaystyle \frac{1}{2}{p}^2\frac{{\gamma }_C+q_C+2b_C}{({\gamma }_C+b_C)(q_C+b_C)}{b}_C\frac{{\mathcal{R}}_{0C}^2}{{\mathcal{R}}_0^2}}$	0.00018	−0.0032
$q_C$	${\displaystyle \frac{1}{2}{p}^2\frac{{\mathcal{R}}_{0C}^2}{{\mathcal{R}}_0^2}\frac{b_C}{q_C+b_C}}$	0.0011	0.0010
$b_H$	$-{\displaystyle \frac{1}{2}(1-p{)}^2\frac{{\gamma }_H+q_H+2b_H}{({\gamma }_H+b_H)(q_H+b_H)}{b}_H\frac{{\mathcal{R}}_{0H}^2}{{\mathcal{R}}_0^2}}$	0.00005	−0.00004
$q_H$	${\displaystyle \frac{1}{2}(1-p{)}^2\frac{{\mathcal{R}}_{0H}^2}{{\mathcal{R}}_0^2}\frac{b_H}{q_H+b_H}}$	0.00001	0.00001

Figure 3. Numbers of humans, cattle and flies in each compartment with baseline parameter values as in Table 1 except $a=\frac{0.25}{2}$ with $I_F\left(0\right)=1$, $N_H=300$, $N_C=50$, $L^{\ast }=4545$, giving approximate equilibrium values $S_C^{\ast }=21,E_C^{\ast }=3,I_C^{\ast }=7,R_C^{\ast }=19,S_H^{\ast }=269,E_H^{\ast }=3,I_H^{\ast }=7,R_H^{\ast }=21,S_F^{\ast }=4979,E_F^{\ast }=9,I_F^{\ast }=12$. For these parameter values, the reproduction numbers are ${\mathcal{R}}_0=1.5149,{\mathcal{R}}_{0H}=0.9526,{\mathcal{R}}_{0C}=2.1253$.

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