# American Institute of Mathematical Sciences

## 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, 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. [2] E. Almaraz, A. 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. [3] S. Al-Sheikh, Modeling and Analysis of an SEIR epidemic model with a limited resource for treatment, GJSFR-F, 12 (2012), 57-66. [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. [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. [6] J. Amador and J. R. Artalejo, Modelling computer virus with the BSDE approach, Comp. Networks, 57 (2013), 302-316. [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. [8] J. R. Artalejo, A. 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. [9] J. R. Artalejo, A. 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. [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. [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 [12] A. J. Black and J. V. Ross, Computation of epidemic final size distributions, J. Theor. Biol., 367 (2015), 159-165. [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. [14] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97 Springer, New York, 1993. [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. [16] F. Capone, V. 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. [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. [18] D. J. Daley and J. Gani, Epidemic Modelling: An Introduction, Cambridge Studies in Mathematical Biology, Vol. 15. Cambridge University Press, Cambridge, 1999. [19] M. De la Sen, S. 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. [20] Z. Feng, Applications of Epidemiological Models to Public Health Policy making. The Role of Heterogeneity in Model Predictions, World Scientific Publishing, Singapore, 2014. [21] A. Gómez-Corral and M. López-García, Modeling host-parasitoid interactions with correlated events, Appl. Math.Model., 37 (2013), 5452-5463. [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. [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. [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. [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. [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. [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. [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. [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. [30] Q. Liu, D. Jiang, N. Shi, T. 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. [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. [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. [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. [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. [35] H. Shu, D. 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. [36] P. Stone, H. 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. [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. [38] H. Wan and J. Cui, Rich dynamics of an epidemic model with saturation recovery, J. Appl. Math. , 2013 (2013), Article ID 314958, 9pp. [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. [40] Q. Yang, D. Jiang, N. 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. [41] N. Yi, Q. Zhang, K. Mao, D. 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. [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. [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 [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.

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##### References:
 [1] L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Pearson Education, INC. , Upper Saddle River, NJ, 2003. [2] E. Almaraz, A. 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. [3] S. Al-Sheikh, Modeling and Analysis of an SEIR epidemic model with a limited resource for treatment, GJSFR-F, 12 (2012), 57-66. [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. [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. [6] J. Amador and J. R. Artalejo, Modelling computer virus with the BSDE approach, Comp. Networks, 57 (2013), 302-316. [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. [8] J. R. Artalejo, A. 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. [9] J. R. Artalejo, A. 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. [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. [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 [12] A. J. Black and J. V. Ross, Computation of epidemic final size distributions, J. Theor. Biol., 367 (2015), 159-165. [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. [14] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97 Springer, New York, 1993. [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. [16] F. Capone, V. 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. [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. [18] D. J. Daley and J. Gani, Epidemic Modelling: An Introduction, Cambridge Studies in Mathematical Biology, Vol. 15. Cambridge University Press, Cambridge, 1999. [19] M. De la Sen, S. 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. [20] Z. Feng, Applications of Epidemiological Models to Public Health Policy making. The Role of Heterogeneity in Model Predictions, World Scientific Publishing, Singapore, 2014. [21] A. Gómez-Corral and M. López-García, Modeling host-parasitoid interactions with correlated events, Appl. Math.Model., 37 (2013), 5452-5463. [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. [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. [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. [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. [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. [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. [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. [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. [30] Q. Liu, D. Jiang, N. Shi, T. 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. [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. [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. [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. [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. [35] H. Shu, D. 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. [36] P. Stone, H. 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. [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. [38] H. Wan and J. Cui, Rich dynamics of an epidemic model with saturation recovery, J. Appl. Math. , 2013 (2013), Article ID 314958, 9pp. [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. [40] Q. Yang, D. Jiang, N. 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. [41] N. Yi, Q. Zhang, K. Mao, D. 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. [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. [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 [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.
Final size distribution for several values of $\sigma$
Peak prevalence mass function for several latency rates $\sigma$
Box plot for $M$
$P(M\leq I_{0})$, for $I_{0}=25$ units
Numerical descriptors of Z for several incidence functions
 Mass Action Inhibitory Effect Reaction-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 Action Inhibitory Effect Reaction-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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