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# Stability analysis on an economic epidemiological model with vaccination

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• In this paper, an economic epidemiological model with vaccination is studied. The stability of the endemic steady-state is analyzed and some bifurcation properties of the system are investigated. It is established that the system exhibits saddle-point and period-doubling bifurcations when adult susceptible individuals are vaccinated. Furthermore, it is shown that susceptible individuals also have the tendency of opting for more number of contacts even if the vaccine is inefficacious and thus causes the disease endemic to increase in the long run. Results from sensitivity analysis with specific disease parameters are also presented. Finally, it is shown that the qualitative behaviour of the system is affected by contact levels.

Mathematics Subject Classification: Primary: 92D30, 92B05, 90C3; Secondary: 37N40.

 Citation: • • Figure 1.  Graph of utility function. $\phi=1 \text{ and } h=2$

Figure 2.  Graph of utility function. $\delta=0.05 \text{ and } \phi=1$

Figure 3.  Graph of infection prevalence verses number of contacts. $\delta=\mu=0.05, \nu=0.8, m = 0.6, n = 0.5$

Figure 4.  Graph of $pn$ vs $L$

Figure 6.  Sensitivity analysis of number of contacts

Figure 5.  Simulation of the proportion susceptible, infected, vaccinated babies and number of contacts

Figure 7.  The parameter values for the plot of the graphs are given in Table 3. n = 0, µ = 0:05 and ν = 0:1

Figure 8.  The parameter values for plot of graphs are given in Table 3. n = 0:6; µ = 0:05 and ν = 0:2

Figure 9.  The parameter values for plot of graphs are given in Table 3. n = 0:6; µ = 0:05; ν = 0:1

Figure 10.  The parameter values for plot of graphs are given in Table 3. n = 0:6; µ = 0:05; ν = 0:4

Figure 11.  Period-doubling bifurcation diagram for a varying number of contacts. The bifurcation parameter is λ.

Figure 12.  Period-doubling bifurcation diagram for a fixed number of contacts (c = 8). The bifurcation parameter is λ.

Figure 13.  Period-doubling bifurcation diagram for a varying number of contacts. The bifurcation parameter is n.

Table 1.  Parameter values

 Parameters Values Sources $m$ 92.9 %  $n$ 0.0 Assumed $\sigma$ 40 % Assumed $\lambda$ 0.09091 per day  $\delta$ 0.05 Assumed $\beta$ 0.96  $\phi$ 1 Assumed $\mu$ 0.02755 per year  $\nu$ 10 % Assumed

Table 2.  Corresponding endemic steady state values for Table 1

 $s^*$ $i^*$ $v^*$ $p$ 0.533 0.266 0.200 0.214

Table 3.  Fixed parameter values

 Parameters m $\sigma$ $\lambda$ $\delta$ $\beta$ $\phi$ Values 0.8 0.6 0.6 0.05 0.96 3

Table 4.  Parameter values satisfying proposition 3

 Cases Parameters $|\lambda|$ $L<1$ $\nu=0.2$ $|\lambda_{1}|=0.556$ $\mu=0.05$ $|\lambda_{2}|=0.714$ $n=0.6$ $12$ $\nu=0.8$ $|\lambda_{1}|=1.426$ $\mu=0.6$ $|\lambda_{2}|=0.183$ $n=0.7$

Table 5.  Corresponding endemic steady state values

 $s^*$ $i^*$ $v^*$ $c^*$ $p$ 0.19 0.194 0.616 8.80 0.663

Table 6.  Corresponding endemic steady state values

 $s^*$ $i^*$ $v^*$ $c^*$ $p$ 0.135 0.059 0.807 9.261 0.282

Table 7.  Parameter values for bifurcation analysis

 Parameter $\mu$ $\nu$ n m $\sigma$ $\delta$ $\beta$ $\phi$ Value 0.05 0.5 0.6 0.5 0.6 0.05 0.96 3
• Figures(13)

Tables(7)

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