Modelling and optimal control of pneumonia disease with cost-effective strategies

Figures & data

Figure 1. Reported cases of Pneumonia disease in Ethiopia from 2009 to 2015.

Figure 2. Flow diagram of the model.

Table 1. Sensitivity indices table.

Parameter symbol	Sensitivity indices
k	+ve
ε	+ve
τ	+ve
φ	+ve
χ	+ve
p	−ve
$\vartheta$	−ve
μ	−ve
α	−ve
ρ	−ve
β	−ve
η	−ve
q	−ve

Table 2. Parameter values for the Pneumonia model.

Parameter symbol	Value	Source
k	0.5	Estimated
ε	0.002	Emile.J,2012
τ	0.89–0.99	Jacob.O,2012
φ	0.0025	Emile.J,2012
χ	0.001–0.01096 per day	Jacob.O,2012
p	0.2	Emile.J,2012
$\vartheta$	0.008	Assumed
μ	0.01	Estimated
α	0.057	Estimated
ρ	0.05	Jacob.O,2012
β	0.0115	Jacob.O,2012
η	0.2	Jacob.O,2012
q	0.5–1	Jacob.O,2012
ϒ	1.2	Assumed
δ	0.1	Emile. J, 2012
Ψ	0.2	Assumed

Figure 3. Simulations of optimal control with prevention only.

Figure 4. Simulations optimal control with treatment only.

Figure 5. Simulations of optimal control with screening only.

Figure 6. Simulations optimal control with prevention and treatment interventions.

Figure 7. Simulations of optimal control with prevention and screening intervention.

Figure 8. Simulations of optimal control with treatment and screening intervention.

Figure 9. Simulations of optimal control with Prevention, Treatment and screening interventions.

Figure 10. Backward bifurcation of the force of infection at equilibrium against the effective reproduction number $R_{\mathrm{eff}}$.

Table 3. Number of infectious averted and total cost of each strategies.

Strategies	Description	Total infectious averted	Total cost (USD)
A	Vaccination and screening	101,417	5906.1
B	Treatment and screening	116,099	5472.7
C	Prevention and treatment	117,142	5292
D	Prevention, treatment and screening	119,465	6948.8

Figure 11. Cost Function of the intervention strategies for the period of 10 months.