# American Institute of Mathematical Sciences

August  2017, 14(4): 975-999. doi: 10.3934/mbe.2017051

## Stability analysis on an economic epidemiological model with vaccination

 1 Department of Statistical and Actuarial Sciences, University of Western Ontario, London, N6A 5B7, Canada 2 Department of Mathematics, Trent University, Peterborough, K9L 0G2, Canada

* Corresponding author

Received  October 2015 Accepted  January 2017 Published  February 2017

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.

Citation: Wisdom S. Avusuglo, Kenzu Abdella, Wenying Feng. Stability analysis on an economic epidemiological model with vaccination. Mathematical Biosciences & Engineering, 2017, 14 (4) : 975-999. doi: 10.3934/mbe.2017051
##### References:

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##### References:
Graph of utility function. $\phi=1 \text{ and } h=2$
Graph of utility function. $\delta=0.05 \text{ and } \phi=1$
Graph of infection prevalence verses number of contacts. $\delta=\mu=0.05, \nu=0.8, m = 0.6, n = 0.5$
Graph of $pn$ vs $L$
Sensitivity analysis of number of contacts
Simulation of the proportion susceptible, infected, vaccinated babies and number of contacts
The parameter values for the plot of the graphs are given in Table 3. n = 0, µ = 0:05 and ν = 0:1
The parameter values for plot of graphs are given in Table 3. n = 0:6; µ = 0:05 and ν = 0:2
The parameter values for plot of graphs are given in Table 3. n = 0:6; µ = 0:05; ν = 0:1
The parameter values for plot of graphs are given in Table 3. n = 0:6; µ = 0:05; ν = 0:4
Period-doubling bifurcation diagram for a varying number of contacts. The bifurcation parameter is λ.
Period-doubling bifurcation diagram for a fixed number of contacts (c = 8). The bifurcation parameter is λ.
Period-doubling bifurcation diagram for a varying number of contacts. The bifurcation parameter is n.
Parameter values
 Parameters Values Sources $m$ 92.9 % [27] $n$ 0.0 Assumed $\sigma$ 40 % Assumed $\lambda$ 0.09091 per day [24] $\delta$ 0.05 Assumed $\beta$ 0.96 [4] $\phi$ 1 Assumed $\mu$ 0.02755 per year [24] $\nu$ 10 % Assumed
 Parameters Values Sources $m$ 92.9 % [27] $n$ 0.0 Assumed $\sigma$ 40 % Assumed $\lambda$ 0.09091 per day [24] $\delta$ 0.05 Assumed $\beta$ 0.96 [4] $\phi$ 1 Assumed $\mu$ 0.02755 per year [24] $\nu$ 10 % Assumed
Corresponding endemic steady state values for Table 1
 $s^*$ $i^*$ $v^*$ $p$ 0.533 0.266 0.200 0.214
 $s^*$ $i^*$ $v^*$ $p$ 0.533 0.266 0.200 0.214
Fixed parameter values
 Parameters m $\sigma$ $\lambda$ $\delta$ $\beta$ $\phi$ Values 0.8 0.6 0.6 0.05 0.96 3
 Parameters m $\sigma$ $\lambda$ $\delta$ $\beta$ $\phi$ Values 0.8 0.6 0.6 0.05 0.96 3
Parameter values satisfying proposition 3
 Cases Parameters $|\lambda|$ $L<1$ $\nu=0.2$ $|\lambda_{1}|=0.556$ $\mu=0.05$ $|\lambda_{2}|=0.714$ $n=0.6$ $12$ $\nu=0.8$ $|\lambda_{1}|=1.426$ $\mu=0.6$ $|\lambda_{2}|=0.183$ $n=0.7$
 Cases Parameters $|\lambda|$ $L<1$ $\nu=0.2$ $|\lambda_{1}|=0.556$ $\mu=0.05$ $|\lambda_{2}|=0.714$ $n=0.6$ $12$ $\nu=0.8$ $|\lambda_{1}|=1.426$ $\mu=0.6$ $|\lambda_{2}|=0.183$ $n=0.7$
 $s^*$ $i^*$ $v^*$ $c^*$ $p$ 0.19 0.194 0.616 8.80 0.663
 $s^*$ $i^*$ $v^*$ $c^*$ $p$ 0.19 0.194 0.616 8.80 0.663
 $s^*$ $i^*$ $v^*$ $c^*$ $p$ 0.135 0.059 0.807 9.261 0.282
 $s^*$ $i^*$ $v^*$ $c^*$ $p$ 0.135 0.059 0.807 9.261 0.282
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
 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
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