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August 2018, 23(6): 2153-2176. doi: 10.3934/dcdsb.2018229

On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections

1. 

Department of Statistics and Operations Research, School of Mathematical Sciences, Complutense University of Madrid, 28040 Madrid, Spain

2. 

Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Calle Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain

* Corresponding author: A. Gómez-Corral

Received  June 2016 Revised  April 2018 Published  July 2018

Fund Project: The authors are supported by the Ministry of Economy and Competitiveness (Government of Spain), Project MTM2014-58091-P

A Markov-modulated framework is presented to incorporate correlated inter-event times into the stochastic susceptible-infectious-recovered (SIR) epidemic model for a closed finite community. The resulting process allows us to deal with non-exponential distributional assumptions on the contact process between the compartment of infectives and the compartment of susceptible individuals, and the recovery process of infected individuals, but keeping the dimensionality of the underlying Markov chain model tractable. The variability between SIR-models with distinct level of correlation is discussed in terms of extinction times, the final size of the epidemic, and the basic reproduction number, which is defined here as a random variable rather than an expected value.

Citation: E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229
References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011.

[2]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson Education, Inc., New Jersey, 2007.

[3]

L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology (eds. F. Brauer, P. van der Driessche and J. Wu), Lecture Notes in Mathematics, Vol. 1945, Springer-Verlag, Berlin Heidelberg, (2008), 81–130. doi: 10.1007/978-3-540-78911-6_3.

[4]

L. J. S. Allen and A. M. Burgin, Comparison of determistic and stochastic SIS and SIR models in discrete time, Mathematical Biosciences, 163 (2000), 1-33. doi: 10.1016/S0025-5564(99)00047-4.

[5]

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. doi: 10.1016/j.biosystems.2016.04.007.

[6]

J. Amador and J. R. Artalejo, Modeling computer virus with the BSDE approach, Computer Networks, 57 (2013), 302-316. doi: 10.1016/j.comnet.2012.09.014.

[7]

R. M. Anderson and R. M. May, Infectious Diseases of Humans; Dynamics and Control, Oxford University Press, Oxford, 1991.

[8]

H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Lecture Notes in Statistics, Vol. 151, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1158-7.

[9]

J. R. ArtalejoA. Economou and M. J. López-Herrero, On the number of recovered individuals in the SIS and SIR stochastic epidemic models, Mathematical Biosciences, 228 (2010), 45-55. doi: 10.1016/j.mbs.2010.08.006.

[10]

J. R. Artalejo and A. Gómez-Corral, A state-dependent Markov-modulated mechanism for generating events and stochastic models, Mathematical Methods in the Applied Sciences, 33 (2010), 1342-1349. doi: 10.1002/mma.1252.

[11]

J. R. ArtalejoA. Gómez-Corral and Q. M. He, Markovian arrivals in stochastic modeling: A survey and some new results, SORT - Statistics and Operations Research Transactions, 34 (2010), 101-144.

[12]

J. R. Artalejo and M. J. López-Herrero, Quasi-stationary and ratio of expectations distributions: A comparative study, Journal of Theoretical Biology, 266 (2010), 264-274. doi: 10.1016/j.jtbi.2010.06.030.

[13]

J. R. Artalejo and M. J. López-Herrero, The SIS and SIR stochastic epidemic models: A maximum entropy approach, Theoretical Population Biology, 80 (2011), 256-264. doi: 10.1016/j.tpb.2011.09.005.

[14]

J. R. Artalejo and M. J. López-Herrero, On the exact measure of disease spread in stochastic epidemic models, Bulletin of Mathematical Biology, 75 (2013), 1031-1050. doi: 10.1007/s11538-013-9836-3.

[15]

J. R. Artalejo and M. J. López-Herrero, Stochastic epidemic models: New behavioral indicators of the disease spreading, Applied Mathematical Modelling, 38 (2014), 4371-4387. doi: 10.1016/j.apm.2014.02.017.

[16]

N. T. L. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Charles Griffin and Company, London, 1975.

[17]

F. G. Ball, A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Journal of Applied Probability, 18 (1986), 289-310. doi: 10.2307/1427301.

[18]

F. Ball and P. Neal, A general model for stochastic SIR epidemics with two levels of mixing, Mathematical Biosciences, 180 (2002), 73-102. doi: 10.1016/S0025-5564(02)00125-6.

[19]

N. G. Becker, Analysis of Infectious Disease Data, Chapman and Hall, London, 1989.

[20]

A. J. Black and J. V. Ross, Computation of epidemic final size distributions, Journal of Theoretical Biology, 367 (2015), 159-165. doi: 10.1016/j.jtbi.2014.11.029.

[21]

T. Britton, Stochastic epidemic models: A survey, Mathematical Biosciences, 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.

[22]

P. Buchholz, J. Kriege and I. Felko, Input Modeling with Phase-Type Distributions and Markov Models. Theory and Applications, Springer, Dordrecht, 2014. doi: 10.1007/978-3-319-06674-5.

[23]

D. Clancy, SIR epidemic models with general infectious period distribution, Statistics and Probability Letters, 85 (2014), 1-5. doi: 10.1016/j.spl.2013.10.017.

[24]

O. DiekmannM. C. M. de Jong and J. A. J. Metz, A deterministic epidemic model taking account of repeated contacts between the same individuals, Journal of Applied Probability, 35 (1989), 448-462. doi: 10.1239/jap/1032192860.

[25]

H. El MaroufyL. Omari and Z. Taib, Transition probabilities for generalized SIR epidemic model, Stochastic Models, 28 (2012), 15-28. doi: 10.1080/15326349.2011.614201.

[26]

H. El MaroufyD. Kiouach and Z. Taib, Final outcome probabilities for SIR epidemic model, Communications in Statistics-Theory and Methods, 45 (2016), 2426-2437. doi: 10.1080/03610926.2014.881494.

[27]

J. Gani and P. Purdue, Matrix-geometric methods for the general stochastic epidemic, Mathematical Medicine and Biology, 1 (1984), 333-342. doi: 10.1093/imammb/1.4.333.

[28]

A. Gómez-Corral and M. López García, Modeling host-parasitoid interactions with correlated events, Applied Mathematical Modelling, 37 (2014), 5452-5463. doi: 10.1016/j.apm.2012.10.035.

[29]

A. Gómez-Corral and M. López-García, On SIR epidemic models with generally distributed infectious periods: Number of secondary cases and probability of infection, International Journal of Biomathematics, 10 (2017), 1750024 (13 pages). doi: 10.1142/S1793524517500243.

[30]

L. F. GordilloS. A. MarionA. Martin-Löf and P. E. Greenwood, Bimodal epidemic size distributions for near-critical SIR with vaccination, Bulletin of Mathematical Biology, 70 (2008), 589-602. doi: 10.1007/s11538-007-9269-y.

[31]

J. Grasman, Stochastic epidemics: The expected duration of the endemic period in higher dimensional models, Mathematical Biosciences, 152 (1998), 13-27. doi: 10.1016/S0025-5564(98)10020-2.

[32]

Q. M. He and M. F. Neuts, Markov chains with marked transitions, Stochastic Processes and Their Applications, 74 (1998), 37-52. doi: 10.1016/S0304-4149(97)00109-9.

[33]

Q. M. He, Fundamentals of Matrix-Analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5.

[34]

V. Isham, Stochastic models for epidemics with special reference to AIDS, The Annals of Applied Probability, 3 (1993), 1-27. doi: 10.1214/aoap/1177005505.

[35]

V. Isham, Stochastic models for epidemics, in Celebrating Statistics: Papers in Honour of Sir David Cox on his 80th Birthday (eds. A. C. Davison, Y. Dodge and N. Wermuth), Oxford University Press, Oxford, 33 (2005), 27–54. doi: 10.1093/acprof:oso/9780198566540.003.0002.

[36]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, 2008.

[37]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A, 115 (1927), 700-721.

[38]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM, Philadelphia, 1999. doi: 10.1137/1.9780898719734.

[39]

M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981.

[40]

M. F. Neuts and J. M. Li, An algorithmic study of SIR stochastic epidemic models, in Athens Conference on Applied Probability and Time Series Analysis, Lecture Notes in Statistics, Vol. 114, Springer, New York, (1996), 295–306. doi: 10.1007/978-1-4612-0749-8_21.

[41]

P. Picard and C. Lefévre, A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes, Advances in Applied Probability, 22 (1990), 269-294. doi: 10.2307/1427536.

[42]

J. V. Ross, Invasion of infectious diseases in finite homogeneous populations, Journal of Theoretical Biology, 289 (2011), 83-87. doi: 10.1016/j.jtbi.2011.08.035.

[43]

I. W. Saunders, A model for myxomatosis, Mathematical Biosciences, 48 (1980), 1-15. doi: 10.1016/0025-5564(80)90012-7.

[44]

T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic, Journal of Applied Probability, 20 (1983), 390-394. doi: 10.2307/3213811.

[45]

G. Streftaris and G. J. Gibson, Non-exponential tolerance to infection in epidemic systems-modeling, inference, and assessment, Biostatistics, 13 (2012), 580-593. doi: 10.1093/biostatistics/kxs011.

show all references

References:
[1]

L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second edition. CRC Press, Boca Raton, FL, 2011.

[2]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson Education, Inc., New Jersey, 2007.

[3]

L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology (eds. F. Brauer, P. van der Driessche and J. Wu), Lecture Notes in Mathematics, Vol. 1945, Springer-Verlag, Berlin Heidelberg, (2008), 81–130. doi: 10.1007/978-3-540-78911-6_3.

[4]

L. J. S. Allen and A. M. Burgin, Comparison of determistic and stochastic SIS and SIR models in discrete time, Mathematical Biosciences, 163 (2000), 1-33. doi: 10.1016/S0025-5564(99)00047-4.

[5]

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. doi: 10.1016/j.biosystems.2016.04.007.

[6]

J. Amador and J. R. Artalejo, Modeling computer virus with the BSDE approach, Computer Networks, 57 (2013), 302-316. doi: 10.1016/j.comnet.2012.09.014.

[7]

R. M. Anderson and R. M. May, Infectious Diseases of Humans; Dynamics and Control, Oxford University Press, Oxford, 1991.

[8]

H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Lecture Notes in Statistics, Vol. 151, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1158-7.

[9]

J. R. ArtalejoA. Economou and M. J. López-Herrero, On the number of recovered individuals in the SIS and SIR stochastic epidemic models, Mathematical Biosciences, 228 (2010), 45-55. doi: 10.1016/j.mbs.2010.08.006.

[10]

J. R. Artalejo and A. Gómez-Corral, A state-dependent Markov-modulated mechanism for generating events and stochastic models, Mathematical Methods in the Applied Sciences, 33 (2010), 1342-1349. doi: 10.1002/mma.1252.

[11]

J. R. ArtalejoA. Gómez-Corral and Q. M. He, Markovian arrivals in stochastic modeling: A survey and some new results, SORT - Statistics and Operations Research Transactions, 34 (2010), 101-144.

[12]

J. R. Artalejo and M. J. López-Herrero, Quasi-stationary and ratio of expectations distributions: A comparative study, Journal of Theoretical Biology, 266 (2010), 264-274. doi: 10.1016/j.jtbi.2010.06.030.

[13]

J. R. Artalejo and M. J. López-Herrero, The SIS and SIR stochastic epidemic models: A maximum entropy approach, Theoretical Population Biology, 80 (2011), 256-264. doi: 10.1016/j.tpb.2011.09.005.

[14]

J. R. Artalejo and M. J. López-Herrero, On the exact measure of disease spread in stochastic epidemic models, Bulletin of Mathematical Biology, 75 (2013), 1031-1050. doi: 10.1007/s11538-013-9836-3.

[15]

J. R. Artalejo and M. J. López-Herrero, Stochastic epidemic models: New behavioral indicators of the disease spreading, Applied Mathematical Modelling, 38 (2014), 4371-4387. doi: 10.1016/j.apm.2014.02.017.

[16]

N. T. L. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Charles Griffin and Company, London, 1975.

[17]

F. G. Ball, A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models, Journal of Applied Probability, 18 (1986), 289-310. doi: 10.2307/1427301.

[18]

F. Ball and P. Neal, A general model for stochastic SIR epidemics with two levels of mixing, Mathematical Biosciences, 180 (2002), 73-102. doi: 10.1016/S0025-5564(02)00125-6.

[19]

N. G. Becker, Analysis of Infectious Disease Data, Chapman and Hall, London, 1989.

[20]

A. J. Black and J. V. Ross, Computation of epidemic final size distributions, Journal of Theoretical Biology, 367 (2015), 159-165. doi: 10.1016/j.jtbi.2014.11.029.

[21]

T. Britton, Stochastic epidemic models: A survey, Mathematical Biosciences, 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.

[22]

P. Buchholz, J. Kriege and I. Felko, Input Modeling with Phase-Type Distributions and Markov Models. Theory and Applications, Springer, Dordrecht, 2014. doi: 10.1007/978-3-319-06674-5.

[23]

D. Clancy, SIR epidemic models with general infectious period distribution, Statistics and Probability Letters, 85 (2014), 1-5. doi: 10.1016/j.spl.2013.10.017.

[24]

O. DiekmannM. C. M. de Jong and J. A. J. Metz, A deterministic epidemic model taking account of repeated contacts between the same individuals, Journal of Applied Probability, 35 (1989), 448-462. doi: 10.1239/jap/1032192860.

[25]

H. El MaroufyL. Omari and Z. Taib, Transition probabilities for generalized SIR epidemic model, Stochastic Models, 28 (2012), 15-28. doi: 10.1080/15326349.2011.614201.

[26]

H. El MaroufyD. Kiouach and Z. Taib, Final outcome probabilities for SIR epidemic model, Communications in Statistics-Theory and Methods, 45 (2016), 2426-2437. doi: 10.1080/03610926.2014.881494.

[27]

J. Gani and P. Purdue, Matrix-geometric methods for the general stochastic epidemic, Mathematical Medicine and Biology, 1 (1984), 333-342. doi: 10.1093/imammb/1.4.333.

[28]

A. Gómez-Corral and M. López García, Modeling host-parasitoid interactions with correlated events, Applied Mathematical Modelling, 37 (2014), 5452-5463. doi: 10.1016/j.apm.2012.10.035.

[29]

A. Gómez-Corral and M. López-García, On SIR epidemic models with generally distributed infectious periods: Number of secondary cases and probability of infection, International Journal of Biomathematics, 10 (2017), 1750024 (13 pages). doi: 10.1142/S1793524517500243.

[30]

L. F. GordilloS. A. MarionA. Martin-Löf and P. E. Greenwood, Bimodal epidemic size distributions for near-critical SIR with vaccination, Bulletin of Mathematical Biology, 70 (2008), 589-602. doi: 10.1007/s11538-007-9269-y.

[31]

J. Grasman, Stochastic epidemics: The expected duration of the endemic period in higher dimensional models, Mathematical Biosciences, 152 (1998), 13-27. doi: 10.1016/S0025-5564(98)10020-2.

[32]

Q. M. He and M. F. Neuts, Markov chains with marked transitions, Stochastic Processes and Their Applications, 74 (1998), 37-52. doi: 10.1016/S0304-4149(97)00109-9.

[33]

Q. M. He, Fundamentals of Matrix-Analytic Methods, Springer, New York, 2014. doi: 10.1007/978-1-4614-7330-5.

[34]

V. Isham, Stochastic models for epidemics with special reference to AIDS, The Annals of Applied Probability, 3 (1993), 1-27. doi: 10.1214/aoap/1177005505.

[35]

V. Isham, Stochastic models for epidemics, in Celebrating Statistics: Papers in Honour of Sir David Cox on his 80th Birthday (eds. A. C. Davison, Y. Dodge and N. Wermuth), Oxford University Press, Oxford, 33 (2005), 27–54. doi: 10.1093/acprof:oso/9780198566540.003.0002.

[36]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, 2008.

[37]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A, 115 (1927), 700-721.

[38]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM, Philadelphia, 1999. doi: 10.1137/1.9780898719734.

[39]

M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981.

[40]

M. F. Neuts and J. M. Li, An algorithmic study of SIR stochastic epidemic models, in Athens Conference on Applied Probability and Time Series Analysis, Lecture Notes in Statistics, Vol. 114, Springer, New York, (1996), 295–306. doi: 10.1007/978-1-4612-0749-8_21.

[41]

P. Picard and C. Lefévre, A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes, Advances in Applied Probability, 22 (1990), 269-294. doi: 10.2307/1427536.

[42]

J. V. Ross, Invasion of infectious diseases in finite homogeneous populations, Journal of Theoretical Biology, 289 (2011), 83-87. doi: 10.1016/j.jtbi.2011.08.035.

[43]

I. W. Saunders, A model for myxomatosis, Mathematical Biosciences, 48 (1980), 1-15. doi: 10.1016/0025-5564(80)90012-7.

[44]

T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic, Journal of Applied Probability, 20 (1983), 390-394. doi: 10.2307/3213811.

[45]

G. Streftaris and G. J. Gibson, Non-exponential tolerance to infection in epidemic systems-modeling, inference, and assessment, Biostatistics, 13 (2012), 580-593. doi: 10.1093/biostatistics/kxs011.

Figure 1.  The expected number $E[S(T_{(i, s)})]$ of surviving susceptibles versus the basic reproduction number $\mathcal{R}_0$ for scenarios A, C and I, and initial state $(i, s) = (1, n)$ with $1+n = 50$
Figure 2.  The expected value $\overline{R}_{exact, 0}$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios A, E and G, and initial size $N = 1+n\in\{10, 50,100,150\}$
Figure 3.  The expected value $\overline{R}_{exact, 0}$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios B, F and H, and initial size $N = 1+n\in\{10, 50,100,150\}$
Figure 4.  The expected value $\overline{R}_{exact, 0}$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios C, D and I, and initial size $N = 1+n\in\{10, 50,100,150\}$
Figure 5.  The expected value $\overline{R}_p$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios A, E and G, and initial size $N = 1+n\in\{10, 50,100,150\}$
Figure 6.  The expected value $\overline{R}_p$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios B, F and H, and initial size $N = 1+n\in\{10, 50,100,150\}$
Figure 7.  The expected value $\overline{R}_p$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios C, D and I, and initial size $N = 1+n\in\{10, 50,100,150\}$
Table 1.  Stochastic transitions, events and rates in the basic SIR-model
TransitionsEvents $~$Rates
$i\to i+1, ~s\to s-1, ~r\to r, ~\hbox{for }i, s\in\mathbb{N}$A new infection $\lambda_{i, s}$
$i\to i-1, ~s\to s, ~r\to r+1, ~\hbox{for }i\in\mathbb{N}, s\in\mathbb{N}_0$A removal $\mu_i$
TransitionsEvents $~$Rates
$i\to i+1, ~s\to s-1, ~r\to r, ~\hbox{for }i, s\in\mathbb{N}$A new infection $\lambda_{i, s}$
$i\to i-1, ~s\to s, ~r\to r+1, ~\hbox{for }i\in\mathbb{N}, s\in\mathbb{N}_0$A removal $\mu_i$
Table 2.  Nine scenarios defined in terms of the matrices ${\bf C}^*_{1, (i, s)}$ and ${\bf D}^*_{1, (i, s)}$ for the occurrence of an infection and the removal of an infective, respectively, which are related to MAPs with positive and negative correlation (with characteristic matrices $({\bf E}^+_0, {\bf E}_1^+)$ and $({\bf E}^-_0, {\bf E}_1^-)$, respectively), and Poisson streams (with rates $\lambda_{i, s}$ and $\mu_i$)
ScenarioOccurrence of an infection (Matrices ${\bf C}_{1, (i, s)}^*$)Occurrence of a removal (Matrices ${\bf D}_{1, (i, s)}^*$)
A $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}^+_1$ $\mu_i$
B $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}^-_1$ $\mu_i$
C $\lambda_{i, s}$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$
D $\lambda_{i, s}$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$
E $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^+$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$
F $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^-$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$
G $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^+$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$
H $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^-$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$
I $\lambda_{i, s}$ $\mu_i$
ScenarioOccurrence of an infection (Matrices ${\bf C}_{1, (i, s)}^*$)Occurrence of a removal (Matrices ${\bf D}_{1, (i, s)}^*$)
A $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}^+_1$ $\mu_i$
B $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}^-_1$ $\mu_i$
C $\lambda_{i, s}$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$
D $\lambda_{i, s}$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$
E $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^+$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$
F $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^-$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$
G $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^+$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$
H $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^-$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$
I $\lambda_{i, s}$ $\mu_i$
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