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:
[1]

M. Andrews and C. T. Bauch, The impacts of simultaneous disease intervention decisions on epidemic outcomes, Journal of Theoretical Biology, 395 (2016), 1-10.  doi: 10.1016/j.jtbi.2016.01.027.

[2]

M. Andrews and C. T. Bauch, Disease interventions can interfere with one another through disease-behaviour interactions PLOS Computational Biology 11(2015), e1004291. doi: 10.1371/journal.pcbi.1004291.

[3]

D. Aadland, D. Finnof and X. D. K. Huang, Syphilis Cycles University Library of Munich, Germany in its series MPRA Paper with number 8722. http://ideas.repec.org/p/pra/mprapa/8722.html, 2007.

[4]

D. AadlandD. Finnoff and X. D. K. Huang, Syphilis cycles, The B.E, Journal of Economic Analysis and Policy, De Gruyter, 14 (2013), 297-348. 

[5]

D. Aadland, D. Finnoff and K. X. D. Huang, The Equilibrium Dynamics of Economic Epidemiology (2011) https://www.researchgate.net/publication/50310816.

[6]

D. Aadland, D. Finnof and X. D. K. Huang, The Dynamic of Economics Epidemiology Equilibria Association of Environmental and Resource Economists 2nd Annual Summer Conference, Asheville, NC, June 2012.

[7]

D. Aadland, D. Finnof and X. D. K. Huang, The Equilibrium Dynamics of Economic Epidemiology Vanderbilt University Department of Economics Working Paper Series 13-00003, http://ideas.repec.org/p/van/wpaper/vuecon-sub-13-00003.html, March 2013.

[8]

A. AhituvV. Hotz and T. Philipson, Is aids self-limiting? evidence on the prevalence elasticity of the demand for condoms, Journal of Human Resources, 31 (1996), 869-898. 

[9]

J. ArinoK. L. CookeP. Van Den Driessche and J. Velasco-Hern{á}ndez, An epidemiology model that includes a leaky vaccine with a general waning function, Discrete and Continuous Dynamical Systems Series B, 4 (2004), 479-495.  doi: 10.3934/dcdsb.2004.4.479.

[10]

M. C. Auld, Choices, beliefs, and infectious disease dynamics, Journal of Health Economics, 22 (2003), 361-377.  doi: 10.1016/S0167-6296(02)00103-0.

[11]

L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci, 124 (2003), 83-105.  doi: 10.1016/0025-5564(94)90025-6.

[12]

J. L. Aron and I. B. Schwartz, Seasonality and period-doubling bifurcations in an epidemic model, Journal of Theoretical Biology, 110 (1984), 665-679.  doi: 10.1016/S0022-5193(84)80150-2.

[13]

W. S. AvusugloK. Abdella and W. Feng, Stability analysis on an economic epidemiology model on syphilis, Communications in Applied Analysis, 18 (2014), 59-78. 

[14]

M. AguiarB. Kooi and N. Stollenwerk, Epidemiology of dengue fever: A model with temporary cross-immunity and possible secondary infection shows bifurcations and chaotic behaviour in wide parameter regions, Math. Model. Nat. Phenom., 3 (2008), 48-70.  doi: 10.1051/mmnp:2008070.

[15]

A. M. Bate and F. M. Hilker, Complex dynamics in an eco-epidemiological model, Bulletin of Mathematical Biology, 75 (2013), 2059-2078.  doi: 10.1007/s11538-013-9880-z.

[16]

C. T. Bauch and A. P. Galvani, Social factors in epidemiology, Science, 342 (2013), 47-49.  doi: 10.1126/science.1244492.

[17]

C. T. Bauch and R. McElreath, Disease dynamics and costly punishment can foster socially imposed monogamy Nature Communications 7 (2016), 11219. doi: 10.1038/ncomms11219.

[18]

S. M. Blower and A. R. McLean, Prophylactic vaccines, risk behaviour change, and the probability of eradicating HIV in San Francisco, Science, 265 (1994), 1451-1454. 

[19]

F. Brauer, Models for the spread of universally fatal diseases, Journal of Mathematical Biology, 28 (1990), 451-462.  doi: 10.1007/BF00178328.

[20]

F. Brauer, Epidemic models in populations of varying size, Mathematical Approaches to Problems in Resource Management and Epidemiology, 81 (1989), 109-123.  doi: 10.1007/978-3-642-46693-9_9.

[21]

R. O. BarattaM. C. GinterM. A. PriceJ. W. WalkerR. G. SkinnerE. C. Prather and J. K. David, Measles (rubeola) in previously immunized children, Pediatrics, 46 (1970), 397-402. 

[22]

M. P. Do Carmo and M. P. Do Carmo, Differential Forms and Applications Translated from the 1971 Portuguese original, Universitext, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57951-6.

[23]

E. P. FenichelC. Castillo-ChavezM. G. CeddiaG. ChowellP. A. G. ParraG. J. HicklingG. HollowayR. HoranB. MorinC. PerringsM. SpringbornL. Velazquez and C. Villalobos, Adaptive human behavior in epidemiological models, PNAS, 108 (2011), 6306-6311.  doi: 10.1073/pnas.1011250108.

[24]

M. O. FredJ. K. SigeyJ. A. OkelloJ. M. Okwoyo and G. J. Kang'ethe, Mathematical Modeling on the Control of Measles by Vaccination: Case Study of KISII County, Kenya, The SIJ Transactions on Computer Science Engineering and its Applications (CSEA), The Standard International Journals (The SIJ), 2 (2014), 61-69. 

[25]

M. Grossman, On the concept of health capital and the demand for health, Journal of Political Economy, 80 (1972), 223-255.  doi: 10.1086/259880.

[26]

Z. HuW. Ma and S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Mathematical Biosciences, 238 (2012), 12-20.  doi: 10.1016/j.mbs.2012.03.010.

[27]

Kenya National Bureau of Statistics. 2013. Kisii County Multiple Indicator Cluster Survey 2011 Final Report. Nairobi, Kenya: Kenya National Bureau of Statistics, pp. 33.

[28]

L. Marcos and R. Jesus, Multiparametric bifurcations for a model in epidemiology, J. Mathematical Biology, 35 (1996), 21-36.  doi: 10.1007/s002850050040.

[29]

M. Mark, Mathematical Modelling (4th Edition), ISBN 978-0-12-386912-8, ScienceDirect, 2012.

[30]

R. M. May, Nonlinear phenomena in ecology and epidemiology, Annals of the New York Academy of Sciences, 357 (1980), 267-281. 

[31]

R. E. Mickens, Analysis of a discrete-time model for periodic diseases with pulse vaccination, Journal of Difference Equations and Applications, 9 (2003), 541-551.  doi: 10.1080/1023619031000078306.

[32]

Z. MukandavireA. B. GumelW. Garira and J. M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection, Mathematical Biosciences and Engineering, 6 (2009), 333-362.  doi: 10.3934/mbe.2009.6.333.

[33]

A. M. Niger and A. B. Gumel, Mathematical analysis of the role of repeated exposure on malaria transmission dynamics, Differential Equations and Dynamical Systems, 16 (2008), 251-287.  doi: 10.1007/s12591-008-0015-1.

[34]

T. Oraby and C. T. Bauch, The influence of social norms on dynamics of paediatric vaccinating behaviour, Proc. R. Soc. B. 281 (2014), 20133172. doi: 10.1098/rspb.2013.3172.

[35]

T. Philipson and R. A. Posner, Private Choices and Public Health: An Economic Interpretation of the AIDS Epidemic Harvard University, Cambridge, MA, 1993.

[36]

C. PerringsC. Castillo-ChavezG. ChowellP. DaszakE. P. FenichelD. FinnoffR. D. HoranA. M. KilpatrickA. P. KinzigN. V. KuminoffS. LevinB. MorinK. F. Smith and M. Springborn, Merging economics and epidemiology to improve the prediction and management of infectious disease, EcoHealth, 11 (2014), 464-475.  doi: 10.1007/s10393-014-0963-6.

[37]

T. Philipson, Economic epidemiology and infectious diseases, Handbook of Health Economics, 1 (2000), 1761-1799.  doi: 10.3386/w7037.

[38]

S. A. Plotkin, W. A. Orenstein and P. A. Offit, Vaccines 5th ed. (2008), Pennsylvania: Elsevier Inc.

[39]

V. F. Reyna, How people make decisions that involve risk, American Psychological Society, 13 (2004), 60-66.  doi: 10.1111/j.0963-7214.2004.00275.x.

[40]

L. W. Rauh and R. Schmidt, Measles immunization with killed virus vaccine. Serum antibody titers and experience with exposure to measles epidemic, Bulletin of the World Health Organization, 78 (2000), 226-231. 

[41]

E. ShimG. B. Chapman and A. P. Galvani, Medical decision making, Decision Making with Regard to Antiviral Intervention during an Influenza Pandemic, 30 (2010), E64-E81.  doi: 10.1177/0272989X10374112.

[42]

S. TullyM. Cojocaru and C. T. Bauch, Sexual behaviour, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways, Scientific Reports, 5 (2015), 15411. 

[43]

Z. WangM. A. AndrewsZ.-X. WuL. Wang and C. T. Bauch, Coupled disease-behavior dynamics on complex networks: A review, Physics of Life Reviews, 15 (2015), 1-29.  doi: 10.1016/j.plrev.2015.07.006.

[44]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.  doi: 10.1016/j.mbs.2005.12.022.

[45]

Z. Yicang and L. Hanwu, Stability of periodic solutions for an SIS model with pulse vaccination, Mathematical and Computer Modelling, 38 (2003), 299-308.  doi: 10.1016/S0895-7177(03)90088-4.

[46]

M. T. Caserta, ed. (September 2013), http://www.merckmanuals.com/professional/pediatrics/miscellaneous-viral-infections-in-infants-and-children/measles, Merck Manual Professional. Merck Sharp and Dohme Corp. Retrieved 15 January 2017.

[47]

Government of Canada, http://healthycanadians.gc.ca/publications/healthy-living-vie-saine/4-canadian-immunization-guide-canadien-immunisation/index-eng.php?page=12 (accessed January 12,2017).

[48]

http://www.saskatchewan.ca/residents/health/diseases-and-conditions/measles, Government of Saskatchewan, Retrieved 15 January 2017.

[49]

http://www.who.int/mediacentre/factsheets/fs286/en/, November 2016, Retrieved 15 January 2017.

[50]

http://www.immune.org.nz/duration-protection-efficacy-and-effectiveness

show all references

References:
[1]

M. Andrews and C. T. Bauch, The impacts of simultaneous disease intervention decisions on epidemic outcomes, Journal of Theoretical Biology, 395 (2016), 1-10.  doi: 10.1016/j.jtbi.2016.01.027.

[2]

M. Andrews and C. T. Bauch, Disease interventions can interfere with one another through disease-behaviour interactions PLOS Computational Biology 11(2015), e1004291. doi: 10.1371/journal.pcbi.1004291.

[3]

D. Aadland, D. Finnof and X. D. K. Huang, Syphilis Cycles University Library of Munich, Germany in its series MPRA Paper with number 8722. http://ideas.repec.org/p/pra/mprapa/8722.html, 2007.

[4]

D. AadlandD. Finnoff and X. D. K. Huang, Syphilis cycles, The B.E, Journal of Economic Analysis and Policy, De Gruyter, 14 (2013), 297-348. 

[5]

D. Aadland, D. Finnoff and K. X. D. Huang, The Equilibrium Dynamics of Economic Epidemiology (2011) https://www.researchgate.net/publication/50310816.

[6]

D. Aadland, D. Finnof and X. D. K. Huang, The Dynamic of Economics Epidemiology Equilibria Association of Environmental and Resource Economists 2nd Annual Summer Conference, Asheville, NC, June 2012.

[7]

D. Aadland, D. Finnof and X. D. K. Huang, The Equilibrium Dynamics of Economic Epidemiology Vanderbilt University Department of Economics Working Paper Series 13-00003, http://ideas.repec.org/p/van/wpaper/vuecon-sub-13-00003.html, March 2013.

[8]

A. AhituvV. Hotz and T. Philipson, Is aids self-limiting? evidence on the prevalence elasticity of the demand for condoms, Journal of Human Resources, 31 (1996), 869-898. 

[9]

J. ArinoK. L. CookeP. Van Den Driessche and J. Velasco-Hern{á}ndez, An epidemiology model that includes a leaky vaccine with a general waning function, Discrete and Continuous Dynamical Systems Series B, 4 (2004), 479-495.  doi: 10.3934/dcdsb.2004.4.479.

[10]

M. C. Auld, Choices, beliefs, and infectious disease dynamics, Journal of Health Economics, 22 (2003), 361-377.  doi: 10.1016/S0167-6296(02)00103-0.

[11]

L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci, 124 (2003), 83-105.  doi: 10.1016/0025-5564(94)90025-6.

[12]

J. L. Aron and I. B. Schwartz, Seasonality and period-doubling bifurcations in an epidemic model, Journal of Theoretical Biology, 110 (1984), 665-679.  doi: 10.1016/S0022-5193(84)80150-2.

[13]

W. S. AvusugloK. Abdella and W. Feng, Stability analysis on an economic epidemiology model on syphilis, Communications in Applied Analysis, 18 (2014), 59-78. 

[14]

M. AguiarB. Kooi and N. Stollenwerk, Epidemiology of dengue fever: A model with temporary cross-immunity and possible secondary infection shows bifurcations and chaotic behaviour in wide parameter regions, Math. Model. Nat. Phenom., 3 (2008), 48-70.  doi: 10.1051/mmnp:2008070.

[15]

A. M. Bate and F. M. Hilker, Complex dynamics in an eco-epidemiological model, Bulletin of Mathematical Biology, 75 (2013), 2059-2078.  doi: 10.1007/s11538-013-9880-z.

[16]

C. T. Bauch and A. P. Galvani, Social factors in epidemiology, Science, 342 (2013), 47-49.  doi: 10.1126/science.1244492.

[17]

C. T. Bauch and R. McElreath, Disease dynamics and costly punishment can foster socially imposed monogamy Nature Communications 7 (2016), 11219. doi: 10.1038/ncomms11219.

[18]

S. M. Blower and A. R. McLean, Prophylactic vaccines, risk behaviour change, and the probability of eradicating HIV in San Francisco, Science, 265 (1994), 1451-1454. 

[19]

F. Brauer, Models for the spread of universally fatal diseases, Journal of Mathematical Biology, 28 (1990), 451-462.  doi: 10.1007/BF00178328.

[20]

F. Brauer, Epidemic models in populations of varying size, Mathematical Approaches to Problems in Resource Management and Epidemiology, 81 (1989), 109-123.  doi: 10.1007/978-3-642-46693-9_9.

[21]

R. O. BarattaM. C. GinterM. A. PriceJ. W. WalkerR. G. SkinnerE. C. Prather and J. K. David, Measles (rubeola) in previously immunized children, Pediatrics, 46 (1970), 397-402. 

[22]

M. P. Do Carmo and M. P. Do Carmo, Differential Forms and Applications Translated from the 1971 Portuguese original, Universitext, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57951-6.

[23]

E. P. FenichelC. Castillo-ChavezM. G. CeddiaG. ChowellP. A. G. ParraG. J. HicklingG. HollowayR. HoranB. MorinC. PerringsM. SpringbornL. Velazquez and C. Villalobos, Adaptive human behavior in epidemiological models, PNAS, 108 (2011), 6306-6311.  doi: 10.1073/pnas.1011250108.

[24]

M. O. FredJ. K. SigeyJ. A. OkelloJ. M. Okwoyo and G. J. Kang'ethe, Mathematical Modeling on the Control of Measles by Vaccination: Case Study of KISII County, Kenya, The SIJ Transactions on Computer Science Engineering and its Applications (CSEA), The Standard International Journals (The SIJ), 2 (2014), 61-69. 

[25]

M. Grossman, On the concept of health capital and the demand for health, Journal of Political Economy, 80 (1972), 223-255.  doi: 10.1086/259880.

[26]

Z. HuW. Ma and S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Mathematical Biosciences, 238 (2012), 12-20.  doi: 10.1016/j.mbs.2012.03.010.

[27]

Kenya National Bureau of Statistics. 2013. Kisii County Multiple Indicator Cluster Survey 2011 Final Report. Nairobi, Kenya: Kenya National Bureau of Statistics, pp. 33.

[28]

L. Marcos and R. Jesus, Multiparametric bifurcations for a model in epidemiology, J. Mathematical Biology, 35 (1996), 21-36.  doi: 10.1007/s002850050040.

[29]

M. Mark, Mathematical Modelling (4th Edition), ISBN 978-0-12-386912-8, ScienceDirect, 2012.

[30]

R. M. May, Nonlinear phenomena in ecology and epidemiology, Annals of the New York Academy of Sciences, 357 (1980), 267-281. 

[31]

R. E. Mickens, Analysis of a discrete-time model for periodic diseases with pulse vaccination, Journal of Difference Equations and Applications, 9 (2003), 541-551.  doi: 10.1080/1023619031000078306.

[32]

Z. MukandavireA. B. GumelW. Garira and J. M. Tchuenche, Mathematical analysis of a model for HIV-malaria co-infection, Mathematical Biosciences and Engineering, 6 (2009), 333-362.  doi: 10.3934/mbe.2009.6.333.

[33]

A. M. Niger and A. B. Gumel, Mathematical analysis of the role of repeated exposure on malaria transmission dynamics, Differential Equations and Dynamical Systems, 16 (2008), 251-287.  doi: 10.1007/s12591-008-0015-1.

[34]

T. Oraby and C. T. Bauch, The influence of social norms on dynamics of paediatric vaccinating behaviour, Proc. R. Soc. B. 281 (2014), 20133172. doi: 10.1098/rspb.2013.3172.

[35]

T. Philipson and R. A. Posner, Private Choices and Public Health: An Economic Interpretation of the AIDS Epidemic Harvard University, Cambridge, MA, 1993.

[36]

C. PerringsC. Castillo-ChavezG. ChowellP. DaszakE. P. FenichelD. FinnoffR. D. HoranA. M. KilpatrickA. P. KinzigN. V. KuminoffS. LevinB. MorinK. F. Smith and M. Springborn, Merging economics and epidemiology to improve the prediction and management of infectious disease, EcoHealth, 11 (2014), 464-475.  doi: 10.1007/s10393-014-0963-6.

[37]

T. Philipson, Economic epidemiology and infectious diseases, Handbook of Health Economics, 1 (2000), 1761-1799.  doi: 10.3386/w7037.

[38]

S. A. Plotkin, W. A. Orenstein and P. A. Offit, Vaccines 5th ed. (2008), Pennsylvania: Elsevier Inc.

[39]

V. F. Reyna, How people make decisions that involve risk, American Psychological Society, 13 (2004), 60-66.  doi: 10.1111/j.0963-7214.2004.00275.x.

[40]

L. W. Rauh and R. Schmidt, Measles immunization with killed virus vaccine. Serum antibody titers and experience with exposure to measles epidemic, Bulletin of the World Health Organization, 78 (2000), 226-231. 

[41]

E. ShimG. B. Chapman and A. P. Galvani, Medical decision making, Decision Making with Regard to Antiviral Intervention during an Influenza Pandemic, 30 (2010), E64-E81.  doi: 10.1177/0272989X10374112.

[42]

S. TullyM. Cojocaru and C. T. Bauch, Sexual behaviour, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways, Scientific Reports, 5 (2015), 15411. 

[43]

Z. WangM. A. AndrewsZ.-X. WuL. Wang and C. T. Bauch, Coupled disease-behavior dynamics on complex networks: A review, Physics of Life Reviews, 15 (2015), 1-29.  doi: 10.1016/j.plrev.2015.07.006.

[44]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.  doi: 10.1016/j.mbs.2005.12.022.

[45]

Z. Yicang and L. Hanwu, Stability of periodic solutions for an SIS model with pulse vaccination, Mathematical and Computer Modelling, 38 (2003), 299-308.  doi: 10.1016/S0895-7177(03)90088-4.

[46]

M. T. Caserta, ed. (September 2013), http://www.merckmanuals.com/professional/pediatrics/miscellaneous-viral-infections-in-infants-and-children/measles, Merck Manual Professional. Merck Sharp and Dohme Corp. Retrieved 15 January 2017.

[47]

Government of Canada, http://healthycanadians.gc.ca/publications/healthy-living-vie-saine/4-canadian-immunization-guide-canadien-immunisation/index-eng.php?page=12 (accessed January 12,2017).

[48]

http://www.saskatchewan.ca/residents/health/diseases-and-conditions/measles, Government of Saskatchewan, Retrieved 15 January 2017.

[49]

http://www.who.int/mediacentre/factsheets/fs286/en/, November 2016, Retrieved 15 January 2017.

[50]

http://www.immune.org.nz/duration-protection-efficacy-and-effectiveness

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
ParametersValuesSources
$m$92.9 % [27]
$n$0.0Assumed
$\sigma$40 % Assumed
$\lambda$0.09091 per day[24]
$\delta$0.05Assumed
$\beta$0.96[4]
$\phi$1Assumed
$\mu$0.02755 per year[24]
$\nu$10 % Assumed
ParametersValuesSources
$m$92.9 % [27]
$n$0.0Assumed
$\sigma$40 % Assumed
$\lambda$0.09091 per day[24]
$\delta$0.05Assumed
$\beta$0.96[4]
$\phi$1Assumed
$\mu$0.02755 per year[24]
$\nu$10 % Assumed
Table 2.  Corresponding endemic steady state values for Table 1
$s^*$ $ i^* $ $v^*$ $p$
0.5330.2660.2000.214
$s^*$ $ i^* $ $v^*$ $p$
0.5330.2660.2000.214
Table 3.  Fixed parameter values
Parametersm $\sigma$ $\lambda$ $\delta$ $\beta$ $\phi$
Values0.80.60.60.050.963
Parametersm $\sigma$ $\lambda$ $\delta$ $\beta$ $\phi$
Values0.80.60.60.050.963
Table 4.  Parameter values satisfying proposition 3
CasesParameters $|\lambda|$
$L<1$ $\nu=0.2$ $|\lambda_{1}|=0.556$
$\mu=0.05$ $|\lambda_{2}|=0.714$
$n=0.6$
$1<L<2$ $\nu=0.4$ $|\lambda_{1}|=0.921$
$\mu=0.05$ $|\lambda_{2}|=0.549$
$n=0.6$
$L>2$ $\nu=0.8$ $|\lambda_{1}|=1.426$
$\mu=0.6$ $|\lambda_{2}|=0.183$
$n=0.7$
CasesParameters $|\lambda|$
$L<1$ $\nu=0.2$ $|\lambda_{1}|=0.556$
$\mu=0.05$ $|\lambda_{2}|=0.714$
$n=0.6$
$1<L<2$ $\nu=0.4$ $|\lambda_{1}|=0.921$
$\mu=0.05$ $|\lambda_{2}|=0.549$
$n=0.6$
$L>2$ $\nu=0.8$ $|\lambda_{1}|=1.426$
$\mu=0.6$ $|\lambda_{2}|=0.183$
$n=0.7$
Table 5.  Corresponding endemic steady state values
$s^*$ $i^*$ $v^*$ $c^*$ $p$
0.190.1940.6168.800.663
$s^*$ $i^*$ $v^*$ $c^*$ $p$
0.190.1940.6168.800.663
Table 6.  Corresponding endemic steady state values
$s^*$ $i^*$ $v^*$ $c^*$ $p$
0.1350.0590.8079.2610.282
$s^*$ $i^*$ $v^*$ $c^*$ $p$
0.1350.0590.8079.2610.282
Table 7.  Parameter values for bifurcation analysis
Parameter $\mu$ $\nu$nm $\sigma$ $\delta$ $\beta$ $\phi$
Value0.050.50.60.50.60.050.963
Parameter $\mu$ $\nu$nm $\sigma$ $\delta$ $\beta$ $\phi$
Value0.050.50.60.50.60.050.963
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