Global stability analysis and modelling onchocerciasis transmission dynamics with control measures

Figures & data

Figure 1. Human onchocerciasis reported cases in Ghana from 2008–2015.

Table 1. Human onchocerciasis reported cases in Ghana from 2008 to 2015.

Year	Cases	Cummulative cases
2008	2225	2225
2009	2111	4336
2010	1728	6064
2011	1263	7327
2012	724	8051
2013	462	8513
2014	609	9122
2015	380	9502

Table 2. Reported number of individuals treated for human onchocerciasis in Ghana from 2005 to 2019.

Year	Individuals treated
2019	6526501
2018	6441223
2017	4403121
2016	4331666
2015	3416583
2014	3372058
2013	3372058
2012	3466716
2011	1776626
2010	1491516
2009	544959
2008	1784353
2007	No data
2006	1069137
2005	675066

Figure 2. Transfer diagram for the onchocerciasis transmission dynamics.

Table 3. Description of state variables of the model.

Variable	Description
$S_h$	Susceptible humans
$S_{\mathit{eh}}$	Susceptible educated humans
$I_h$	Infected humans
$T_h$	Treated humans
$A_v$	Immature black flies
$S_v$	Susceptible black flies
$I_v$	Infected black flies
$N_h$	Total human population
$N_v$	Total black fly population
${\lambda }_h$	Force of infection for the human population
${\lambda }_v$	Force of infection for the black fly (vector)

Table 4. Parameter description. susceptible human

Parameter	Description
${\pi }_h$	Human recruitment rate
${\mu }_h$	Natural death rate of human
${\beta }_h$	Prob. of transmission of infections from an infected human to a susceptible black fly
$\mathit{p}$	Proportion of infected immigrants
$\mathit{\theta }$	Educational campaign compliance rate
$\mathit{\sigma }$	Efficacy of educational campaign
$\mathit{\gamma }$	Progression rate of infected humans to treated humans
$\mathit{\phi }$	Progression rate from treated human to susceptible educated humans
$\mathit{r}$	Modification parameter accounting for the reduction of transmission of infections from a treated human to a susceptible black fly
${\pi }_v$	Black fly egg deposition rate
$\mathit{K}$	Carrying capacity of aquatic vector
$\mathit{b}$	Black fly biting rate
${\beta }_v$	Probability of transmission of infections from an infected black fly to a susceptible human
$\mathit{\psi }$	Maturity rate of immature black fly
$\mathit{\epsilon }$	Efficacy of larviciding
${\mu }_v$	Natural death rate of mature black fly
${\mu }_a$	Natural death rate of immature black fly
${\mu }_{\mathrm{\ell }}$	Death rate of immature black fly due to larviciding
${\mu }_t$	Death rate of mature black fly due to trapping

Table 5. Number of possible positive roots of $\mathit{f}({\lambda }_h^{\ast \ast })$.

Case	$a_0$	$a_1$	$a_2$	$a_3$	Sign change	+ve roots
i	+	+	+	$-$	1	0, 1
ii	+	+	$-$	$-$	1	0, 1
iii	+	$-$	+	$-$	3	1, 3
iv	+	$-$	$-$	$-$	1	0, 1

Figure 3. Backward bifurcation.

Table 6. The values of the sensitivity indices.

Parameter	Sensitivity Index
b	$+1.00$
${\beta }_h$	$+0.50$
${\beta }_v$	$+0.50$
${\pi }_h$	$-0.50$
${\mu }_h$	$+0.49997$
${\pi }_v$	$+$3.6 x10^–3
$\mathit{k}$	$+0.50$
$\mathit{\theta }$	$-$1.84877x10^–5
$\mathit{\sigma }$	$-0.2143$
$\mathit{\gamma }$	$-0.471$
$\mathit{\phi }$	$-0.02852$
$\mathit{\epsilon }$	$-0.33495$
r	$+0.02853$
$\mathit{\psi }$	$+0.50243$
${\mu }_a$	$-$4.04995x10^–4
${\mu }_{\mathrm{\ell }}$	$-2$.02498x10^–3
${\mu }_t$	$-$0.028798
${\mu }_v$	$-$0.9748

Table 7. The model parameter values.

Parameter	Value/Range	Source
${\pi }_h$	0.031	[8]
${\mu }_h$	1/23178	[8]
${\beta }_h$	0.0114	[3]
$\mathit{\theta }$	0.5	[20]
$\mathit{\sigma }$	0.3	[20]
$\mathit{\gamma }$	0.65	[27]
$\mathit{\phi }$	0.0891	[9,12]
$\mathit{r}$	0.0083	[27]
${\pi }_v$	600	[27]
$\mathit{K}$	6000	[20,27]
$\mathit{b}$	0.0855	[8]
${\beta }_v$	0.0111	[3]
$\mathit{\psi }$	1.926	[20]
$\mathit{\epsilon }$	0.4	[20]
${\mu }_v$	1.354	[27]
$\mathit{\mu }\mathit{a}$	2	[20]
${\mu }_{\mathrm{\ell }}$	0.4	[20]
$\mathit{\mu }\mathit{t}$	0–0.9	[8]

Figure 4. Susceptible human class with controls.

Figure 5. Susceptible educated human class with controls.

Figure 6. Infected human class with controls.

Figure 7. Treated human class with controls.

Figure 8. Aquatic vector class with controls.

Figure 9. Susceptible vector class with controls.

Figure 10. Infected vector class with controls.

Figure 11. Infected human class without infected immigrants.

Figure 12. Infected vector class without infected immigrants.

Tumwiine J, Muhumuza R. Modelling the impact of trapping blackfly vectors on the transmission of onchocerciasis. Int J Math Model Comput. 2020;10(4 (Fall):311–332.

 Google Scholar

Otieno Omondi E, Orwa TO, Nyabadza F. Titus Okello Orwa, and Farai Nyabadza. Application of optimal control to the onchocerciasis transmission model with treatment. Math Biosci. 2018;297:43–57. doi:10.1016/j.mbs.2017.11.009

 PubMed Web of Science ®Google Scholar

Hassan A, Shaban N. Onchocerciasis dynamics: modelling the effects of treatment, education and vector control. J Biol Dyn. 2020;14(1):245–268. PMID: 32266871. doi: 10.1080/17513758.2020.1745306

 PubMed Web of Science ®Google Scholar

Otieno Omondi E, Nyabadza F, Bonyah E, et al. Modeling the infection dynamics of onchocerciasis and its treatment. J Biol Syst. 2017;25(2):247–277. doi:10.1142/S0218339017500139

 Web of Science ®Google Scholar

Sesan Idowu A, Michael Ogunmiloro O. Transmission dynamics of onchocerciasis with two classes of infection and saturated treatment function. Int J Model Simul Sci Comput. 2020;11(5):2050047. doi:10.1142/S1793962320500476

 Web of Science ®Google Scholar

Michael Ogunmiloro O, Sesan Idowu A. Bifurcation, sensitivity, and optimal control analysis of onchocerciasis disease transmission model with two groups of infectives and saturated treatment function. Math Methods Appl Sci. 2024;47(5):3387–3411. doi: 10.1002/mma.8317

 Web of Science ®Google Scholar

Data availability statement

The data used in the analysis of this manuscript is taken from published articles and are cited in this paper. These published articles are also cited at relevant places within the text as references.