October  2018, 23(8): 3137-3151. doi: 10.3934/dcdsb.2017211

Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources

Faculty of Statistical Studies, Complutense University of Madrid, Ciudad Universitaria, 28040 Madrid, Spain

Received  March 2017 Published  September 2017

The paper deals with a stochastic SEIR model with nonlinear incidence rate and limited resources for a treatment. We focus on a long term study of two measures for the severity of an epidemic: the total number of cases of infection and the maximum of individuals simultaneously infected during an outbreak of the communicable disease. Theoretical and computational results are numerically illustrated.

Citation: Julia Amador, Mariajesus Lopez-Herrero. Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3137-3151. doi: 10.3934/dcdsb.2017211
References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Pearson Education, INC. , Upper Saddle River, NJ, 2003. Google Scholar

[2]

E. AlmarazA. Gómez-Corral and M. T. Rodríguez-Bernal, On the time to reach a critical number of infections in epidemic models with infective and susceptible immigrants, Biosystems, 144 (2016), 68-77.   Google Scholar

[3]

S. Al-Sheikh, Modeling and Analysis of an SEIR epidemic model with a limited resource for treatment, GJSFR-F, 12 (2012), 57-66.   Google Scholar

[4]

J. Amador, The stochastic SIRA model for computer viruses, Appl. Math. Comput., 232 (2014), 1112-1124.  doi: 10.1016/j.amc.2014.01.125.  Google Scholar

[5]

J. Amador and J. R. Artalejo, Stochastic modeling of computer virus spreading with warning signals, J. Franklin Inst., 350 (2013), 1112-1138.  doi: 10.1016/j.jfranklin.2013.02.008.  Google Scholar

[6]

J. Amador and J. R. Artalejo, Modelling computer virus with the BSDE approach, Comp. Networks, 57 (2013), 302-316.   Google Scholar

[7]

J. R. Artalejo and M. J. Lopez-Herrero, Stochastic epidemic models: New behavioral indicators of the disease spreading, Appl. Math. Model., 38 (2014), 4371-4387.  doi: 10.1016/j.apm.2014.02.017.  Google Scholar

[8]

J. R. ArtalejoA. Economou and M. J. Lopez-Herrero, The maximum number of infected individuals in SIS epidemic models: Computational techniques and quasi-stationary distributions, J. Comput. Appl. Math., 233 (2010), 2563-2574.  doi: 10.1016/j.cam.2009.11.003.  Google Scholar

[9]

J. R. ArtalejoA. Economou and M. J. Lopez-Herrero, The stochastic SEIR model before extinction: Computational approaches, Appl. Math, Comput., 265 (2015), 1026-1043.  doi: 10.1016/j.amc.2015.05.141.  Google Scholar

[10]

F. Ball, A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. Appl. Probab., 18 (1986), 289-310.  doi: 10.2307/1427301.  Google Scholar

[11]

M. V. Barbarossa, A. Denes, G. Kiss, Y. Nakata, G. Rö st and Z. Vizi, Transmission dynamics and final epidemic size of Ebola virus disease outbreaks with varying interventions, PLoS one, (2015), http://dx.doi.org/10.1371/journal.pone.0131398 Google Scholar

[12]

A. J. Black and J. V. Ross, Computation of epidemic final size distributions, J. Theor. Biol., 367 (2015), 159-165.   Google Scholar

[13]

A. J. Black, N. Geard, J. M. McCaw, J. McVernon and J. V. Ross, Characterising pandemic severity and transmissibility from data collected during first few hundred studies, Epidemics, 19 (2017), 61–73. http://dx.doi.org/10.1016/j.epidem.2017.01.004 Google Scholar

[14]

V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97 Springer, New York, 1993. doi: 10.1007/978-3-540-70514-7.  Google Scholar

[15]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic model, Math. Biosci., 42 (1978), 41-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[16]

F. CaponeV. De Cataldis and R. De Luca, On the nonlinear stability of an epidemic SEIR reaction-diffusion model, Ricerche Mat., 62 (2013), 161-181.  doi: 10.1007/s11587-013-0151-y.  Google Scholar

[17]

S. Cui and M. Bai, Mathematical analysis of population migration and its effect to spread of epidemics, Discrete Cont. Dyn.-B, 20 (2015), 2819-2858.  doi: 10.3934/dcdsb.2015.20.2819.  Google Scholar

[18]

D. J. Daley and J. Gani, Epidemic Modelling: An Introduction, Cambridge Studies in Mathematical Biology, Vol. 15. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511608834.  Google Scholar

[19]

M. De la SenS. Alonso-Quesada and I. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270 (2015), 953-976.  doi: 10.1016/j.amc.2015.08.099.  Google Scholar

[20]

Z. Feng, Applications of Epidemiological Models to Public Health Policy making. The Role of Heterogeneity in Model Predictions, World Scientific Publishing, Singapore, 2014. Google Scholar

[21]

A. Gómez-Corral and M. López-García, Modeling host-parasitoid interactions with correlated events, Appl. Math.Model., 37 (2013), 5452-5463.   Google Scholar

[22]

A. Gómez-Corral and M. López-García, On SIR epidemic models with generally distributed infectious periods: Number of secondary cases and probability of infection, Int. J. Biomath. , 10 (2017), 1750024. Google Scholar

[23]

E. Grigorieva, E. Khailov and A. Korobeinikov, Optimal control for an epidemic in populations of varying size, in "Dynamical Systems, Differential Equations and Applications" (eds. M. de Leon, W. Feng, Z. Feng, J. Lopez-Gomez, X. Lu, J. M. Martell, J. Parcet, D. Peralta-Salas and W. Ruan), AIMS Proceedings, (2015), 549-561. doi: 10.3934/proc.2015.0549.  Google Scholar

[24]

P. Guo, X. Yang and Z. Yang, Dynamical behaviors of an SIRI epidemic model with nonlinear incidence and latent period, Adv. Differ. Equ-NY, 2014 (2014), 18pp. doi: 10.1186/1687-1847-2014-164.  Google Scholar

[25]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.  doi: 10.1007/BF00160539.  Google Scholar

[26]

T. House, J. V. Ross and D. Sirl, How big is an outbreak likely to be? Methods for epidemic final size calculation, Proc. R. Soc. Lond. A, 469 (2013), article 20120436, 22pp. doi: 10.1098/rspa.2012.0436.  Google Scholar

[27]

H. F. Huo and M. X. Zou, Modelling effects of treatment at home on tuberculosis transmission dynamics, App. Math. Model., 40 (2016), 9474-9484.  doi: 10.1016/j.apm.2016.06.029.  Google Scholar

[28]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.   Google Scholar

[29]

Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal. Real, 14 (2013), 1286-1299.  doi: 10.1016/j.nonrwa.2012.09.016.  Google Scholar

[30]

Q. LiuD. JiangN. ShiT. Hayat and A. Alsaedi, Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence, Physica A, 462 (2016), 870-882.  doi: 10.1016/j.physa.2016.06.095.  Google Scholar

[31]

M. J. Lopez-Herrero, Epidemic transmission on SEIR stochastic models with nonlinear incidence rate, Math. Methods Appl. Sci., 40 (2017), 2532-2541.  doi: 10.1002/mma.4179.  Google Scholar

[32]

M. F. Neuts and J. M. Li, An algorithmic study of S-I-R stochastic epidemic models, in: Lecture Notes in Statistics, 114, (eds. C. C. Heyde, Yu V. Prohorov, R. Pyke, S. T. Rachev), Athens Conference on Applied Probability and Time Series. Springer-Verlag, Heidelberg (1996), 295-306. doi: 10.1007/978-1-4612-0749-8_21.  Google Scholar

[33]

J. M. Ponciano and M. A. Capistrán, First principles modeling of nonlinear incidence rates in seasonal epidemics, PLoS Comput. Biol. , 7 (2011), e1001079, 14pp. doi: 10.1371/journal.pcbi.1001079.  Google Scholar

[34]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[35]

H. ShuD. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. Real, 13 (2012), 1581-1592.  doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar

[36]

P. StoneH. Wilkinson-Herbots and V. Isham, A stochastic model for head-lice infections, J. Math. Biol., 56 (2008), 743-763.  doi: 10.1007/s00285-007-0136-0.  Google Scholar

[37]

S. Tipsri and W. Chinviriyasit, The effect of time delay on the dynamics of an SEIR model with nonlinear incidence, Chaos Soliton Fract., 75 (2015), 153-172.  doi: 10.1016/j.chaos.2015.02.017.  Google Scholar

[38]

H. Wan and J. Cui, Rich dynamics of an epidemic model with saturation recovery, J. Appl. Math. , 2013 (2013), Article ID 314958, 9pp.  Google Scholar

[39]

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

[40]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072.  Google Scholar

[41]

N. YiQ. ZhangK. MaoD. Yang and Q. Li, Analysis and control of an SEIR epidemic system with nonlinear transmission rate, Math. Comput. Model., 50 (2009), 1498-1513.  doi: 10.1016/j.mcm.2009.07.014.  Google Scholar

[42]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[43]

J. Zhang, J. Jia and X. Song, Analysis of an SEIR epidemic model with saturated incidence and saturated treatment function, Sci. World J. , (2014), Article ID 910421, http://dx.doi.org/10.1155/2014/910421 doi: 10.1016/j.aml.2013.11.002.  Google Scholar

[44]

X. Zhou and J. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci., 16 (2011), 4438-4450.  doi: 10.1016/j.cnsns.2011.03.026.  Google Scholar

show all references

References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Pearson Education, INC. , Upper Saddle River, NJ, 2003. Google Scholar

[2]

E. AlmarazA. Gómez-Corral and M. T. Rodríguez-Bernal, On the time to reach a critical number of infections in epidemic models with infective and susceptible immigrants, Biosystems, 144 (2016), 68-77.   Google Scholar

[3]

S. Al-Sheikh, Modeling and Analysis of an SEIR epidemic model with a limited resource for treatment, GJSFR-F, 12 (2012), 57-66.   Google Scholar

[4]

J. Amador, The stochastic SIRA model for computer viruses, Appl. Math. Comput., 232 (2014), 1112-1124.  doi: 10.1016/j.amc.2014.01.125.  Google Scholar

[5]

J. Amador and J. R. Artalejo, Stochastic modeling of computer virus spreading with warning signals, J. Franklin Inst., 350 (2013), 1112-1138.  doi: 10.1016/j.jfranklin.2013.02.008.  Google Scholar

[6]

J. Amador and J. R. Artalejo, Modelling computer virus with the BSDE approach, Comp. Networks, 57 (2013), 302-316.   Google Scholar

[7]

J. R. Artalejo and M. J. Lopez-Herrero, Stochastic epidemic models: New behavioral indicators of the disease spreading, Appl. Math. Model., 38 (2014), 4371-4387.  doi: 10.1016/j.apm.2014.02.017.  Google Scholar

[8]

J. R. ArtalejoA. Economou and M. J. Lopez-Herrero, The maximum number of infected individuals in SIS epidemic models: Computational techniques and quasi-stationary distributions, J. Comput. Appl. Math., 233 (2010), 2563-2574.  doi: 10.1016/j.cam.2009.11.003.  Google Scholar

[9]

J. R. ArtalejoA. Economou and M. J. Lopez-Herrero, The stochastic SEIR model before extinction: Computational approaches, Appl. Math, Comput., 265 (2015), 1026-1043.  doi: 10.1016/j.amc.2015.05.141.  Google Scholar

[10]

F. Ball, A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Adv. Appl. Probab., 18 (1986), 289-310.  doi: 10.2307/1427301.  Google Scholar

[11]

M. V. Barbarossa, A. Denes, G. Kiss, Y. Nakata, G. Rö st and Z. Vizi, Transmission dynamics and final epidemic size of Ebola virus disease outbreaks with varying interventions, PLoS one, (2015), http://dx.doi.org/10.1371/journal.pone.0131398 Google Scholar

[12]

A. J. Black and J. V. Ross, Computation of epidemic final size distributions, J. Theor. Biol., 367 (2015), 159-165.   Google Scholar

[13]

A. J. Black, N. Geard, J. M. McCaw, J. McVernon and J. V. Ross, Characterising pandemic severity and transmissibility from data collected during first few hundred studies, Epidemics, 19 (2017), 61–73. http://dx.doi.org/10.1016/j.epidem.2017.01.004 Google Scholar

[14]

V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97 Springer, New York, 1993. doi: 10.1007/978-3-540-70514-7.  Google Scholar

[15]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic model, Math. Biosci., 42 (1978), 41-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[16]

F. CaponeV. De Cataldis and R. De Luca, On the nonlinear stability of an epidemic SEIR reaction-diffusion model, Ricerche Mat., 62 (2013), 161-181.  doi: 10.1007/s11587-013-0151-y.  Google Scholar

[17]

S. Cui and M. Bai, Mathematical analysis of population migration and its effect to spread of epidemics, Discrete Cont. Dyn.-B, 20 (2015), 2819-2858.  doi: 10.3934/dcdsb.2015.20.2819.  Google Scholar

[18]

D. J. Daley and J. Gani, Epidemic Modelling: An Introduction, Cambridge Studies in Mathematical Biology, Vol. 15. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511608834.  Google Scholar

[19]

M. De la SenS. Alonso-Quesada and I. Ibeas, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Appl. Math. Comput., 270 (2015), 953-976.  doi: 10.1016/j.amc.2015.08.099.  Google Scholar

[20]

Z. Feng, Applications of Epidemiological Models to Public Health Policy making. The Role of Heterogeneity in Model Predictions, World Scientific Publishing, Singapore, 2014. Google Scholar

[21]

A. Gómez-Corral and M. López-García, Modeling host-parasitoid interactions with correlated events, Appl. Math.Model., 37 (2013), 5452-5463.   Google Scholar

[22]

A. Gómez-Corral and M. López-García, On SIR epidemic models with generally distributed infectious periods: Number of secondary cases and probability of infection, Int. J. Biomath. , 10 (2017), 1750024. Google Scholar

[23]

E. Grigorieva, E. Khailov and A. Korobeinikov, Optimal control for an epidemic in populations of varying size, in "Dynamical Systems, Differential Equations and Applications" (eds. M. de Leon, W. Feng, Z. Feng, J. Lopez-Gomez, X. Lu, J. M. Martell, J. Parcet, D. Peralta-Salas and W. Ruan), AIMS Proceedings, (2015), 549-561. doi: 10.3934/proc.2015.0549.  Google Scholar

[24]

P. Guo, X. Yang and Z. Yang, Dynamical behaviors of an SIRI epidemic model with nonlinear incidence and latent period, Adv. Differ. Equ-NY, 2014 (2014), 18pp. doi: 10.1186/1687-1847-2014-164.  Google Scholar

[25]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.  doi: 10.1007/BF00160539.  Google Scholar

[26]

T. House, J. V. Ross and D. Sirl, How big is an outbreak likely to be? Methods for epidemic final size calculation, Proc. R. Soc. Lond. A, 469 (2013), article 20120436, 22pp. doi: 10.1098/rspa.2012.0436.  Google Scholar

[27]

H. F. Huo and M. X. Zou, Modelling effects of treatment at home on tuberculosis transmission dynamics, App. Math. Model., 40 (2016), 9474-9484.  doi: 10.1016/j.apm.2016.06.029.  Google Scholar

[28]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.   Google Scholar

[29]

Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal. Real, 14 (2013), 1286-1299.  doi: 10.1016/j.nonrwa.2012.09.016.  Google Scholar

[30]

Q. LiuD. JiangN. ShiT. Hayat and A. Alsaedi, Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence, Physica A, 462 (2016), 870-882.  doi: 10.1016/j.physa.2016.06.095.  Google Scholar

[31]

M. J. Lopez-Herrero, Epidemic transmission on SEIR stochastic models with nonlinear incidence rate, Math. Methods Appl. Sci., 40 (2017), 2532-2541.  doi: 10.1002/mma.4179.  Google Scholar

[32]

M. F. Neuts and J. M. Li, An algorithmic study of S-I-R stochastic epidemic models, in: Lecture Notes in Statistics, 114, (eds. C. C. Heyde, Yu V. Prohorov, R. Pyke, S. T. Rachev), Athens Conference on Applied Probability and Time Series. Springer-Verlag, Heidelberg (1996), 295-306. doi: 10.1007/978-1-4612-0749-8_21.  Google Scholar

[33]

J. M. Ponciano and M. A. Capistrán, First principles modeling of nonlinear incidence rates in seasonal epidemics, PLoS Comput. Biol. , 7 (2011), e1001079, 14pp. doi: 10.1371/journal.pcbi.1001079.  Google Scholar

[34]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[35]

H. ShuD. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. Real, 13 (2012), 1581-1592.  doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar

[36]

P. StoneH. Wilkinson-Herbots and V. Isham, A stochastic model for head-lice infections, J. Math. Biol., 56 (2008), 743-763.  doi: 10.1007/s00285-007-0136-0.  Google Scholar

[37]

S. Tipsri and W. Chinviriyasit, The effect of time delay on the dynamics of an SEIR model with nonlinear incidence, Chaos Soliton Fract., 75 (2015), 153-172.  doi: 10.1016/j.chaos.2015.02.017.  Google Scholar

[38]

H. Wan and J. Cui, Rich dynamics of an epidemic model with saturation recovery, J. Appl. Math. , 2013 (2013), Article ID 314958, 9pp.  Google Scholar

[39]

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

[40]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072.  Google Scholar

[41]

N. YiQ. ZhangK. MaoD. Yang and Q. Li, Analysis and control of an SEIR epidemic system with nonlinear transmission rate, Math. Comput. Model., 50 (2009), 1498-1513.  doi: 10.1016/j.mcm.2009.07.014.  Google Scholar

[42]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[43]

J. Zhang, J. Jia and X. Song, Analysis of an SEIR epidemic model with saturated incidence and saturated treatment function, Sci. World J. , (2014), Article ID 910421, http://dx.doi.org/10.1155/2014/910421 doi: 10.1016/j.aml.2013.11.002.  Google Scholar

[44]

X. Zhou and J. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci., 16 (2011), 4438-4450.  doi: 10.1016/j.cnsns.2011.03.026.  Google Scholar

Figure 1.  Final size distribution for several values of $ \sigma $
Figure 2.  Peak prevalence mass function for several latency rates $\sigma $
Figure 3.  Box plot for $M$
Figure 4.  $P(M\leq I_{0})$, for $I_{0}=25$ units
Table 1.  Numerical descriptors of Z for several incidence functions
Mass ActionInhibitory EffectReaction-Diffusion
$P(Z=1)$$0.168067$$0.287769$$0.091743$
$P(Z=100)$$0.403300$$2.110\times 10^{-7}$$0.898751$
$Q_{1}$$97$$1$$100$
$Q_{2}$$99$$66$$100$
$Q_{3}$$100$$78$$100$
$E(Z)$$79.303911$$44.601683$$89.989081$
$\sigma (Z)$$39.426192$$37.099482$$29.828126$
Mass ActionInhibitory EffectReaction-Diffusion
$P(Z=1)$$0.168067$$0.287769$$0.091743$
$P(Z=100)$$0.403300$$2.110\times 10^{-7}$$0.898751$
$Q_{1}$$97$$1$$100$
$Q_{2}$$99$$66$$100$
$Q_{3}$$100$$78$$100$
$E(Z)$$79.303911$$44.601683$$89.989081$
$\sigma (Z)$$39.426192$$37.099482$$29.828126$
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