Figure 1.
Flowchart presentation of the compartmental model (1) for Ebola
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April
2018, 14(2): 427-446.
doi: 10.3934/jimo.2017054
Ebola model and optimal control with vaccination constraints
| 1. | Departamento de Matemática Aplicada Ⅱ, E. E. Aeronáutica e do Espazo, Campus As Lagoas, Universidade de Vigo, 32004 Ourense, Spain |
| 2. | African Institute for Mathematical Sciences (AIMS-Cameroon), P.O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon |
| 3. | Departamento de Análise Matemática, Estatística e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain |
| 4. | Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
The Ebola virus disease is a severe viral haemorrhagic fever syndrome caused by Ebola virus. This disease is transmitted by direct contact with the body fluids of an infected person and objects contaminated with virus or infected animals, with a death rate close to 90% in humans. Recently, some mathematical models have been presented to analyse the spread of the 2014 Ebola outbreak in West Africa. In this paper, we introduce vaccination of the susceptible population with the aim of controlling the spread of the disease and analyse two optimal control problems related with the transmission of Ebola disease with vaccination. Firstly, we consider the case where the total number of available vaccines in a fixed period of time is limited. Secondly, we analyse the situation where there is a limited supply of vaccines at each instant of time for a fixed interval of time. The optimal control problems have been solved analytically. Finally, we have performed a number of numerical simulations in order to compare the models with vaccination and the model without vaccination, which has recently been shown to fit the real data. Three vaccination scenarios have been considered for our numerical simulations, namely: unlimited supply of vaccines; limited total number of vaccines; and limited supply of vaccines at each instant of time.
Keywords:
Ebola virus,
mathematical modelling,
transmission of Ebola,
control of the spread of the Ebola disease,
optimal control with vaccination constraints,
vaccination scenarios.
Mathematics Subject Classification:
Primary: 49J15, 92D30; Secondary: 34C60, 49N90.
Citation:
IvÁn Area, FaÏÇal NdaÏrou, Juan J. Nieto, Cristiana J. Silva, Delfim F. M. Torres. Ebola model and optimal control with vaccination constraints. Journal of Industrial & Management Optimization,
2018, 14
(2)
: 427-446.
doi: 10.3934/jimo.2017054
![]()
References:
| [1] |
M. D. Ahmad, M. Usman, A. Khan and M. Imran,
Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination, Infect. Dis. Poverty, 5 (2016), p72.
doi: 10.1186/s40249-016-0161-6. |
| [2] |
C. L. Althaus, Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in west Africa, PLOS Currents Outbreaks Edition 1,2014.
doi: 10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288. |
| [3] |
I. Area, H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh and A. Torres, On a fractional order Ebola epidemic model Adv. Difference Equ. , 2015 (2015), 12 pp.
doi: 10.1186/s13662-015-0613-5. |
| [4] |
I. Area, J. Losada, F. Ndaïrou, J. J. Nieto and D. D. Tcheutia,
Mathematical modeling of 2014 Ebola outbreak, Math. Method. Appl. Sci., (2015).
doi: 10.1002/mma.3794. |
| [5] |
A. Atangana and E. F. Doungmo Goufo, On the mathematical analysis of Ebola hemorrhagic fever: Deathly infection disease in west african countries BioMed Research International 2014 (2014), Art. ID 261383, 7 pp.
doi: 10.1155/2014/261383. |
| [6] |
M. H. A. Biswas, L. T. Paiva and M. R. de Pinho,
A SEIR model for control of infectious diseases with constraints, Math. Biosci. Eng., 11 (2014), 761-784.
doi: 10.3934/mbe.2014.11.761. |
| [7] |
M. A. Bwaka,
Ebola hemorrhagic fever in Kikwit, Democratic Republic of the Congo: Clinical observations in 103 patients, J. Infect. Dis., 179 (1999), S1-S7.
doi: 10.1086/514308. |
| [8] |
L. Cesari, Optimization Theory and Applications, Vol. 17 of Applications of Mathematics
(New York), Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
| [9] |
G. Chowell, N. W. Hengartner, C. Castillo-Chavez, P. W. Fenimore and J. M. Hyman,
The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda, J. Theor. Biol., 229 (2004), 119-126.
doi: 10.1016/j.jtbi.2004.03.006. |
| [10] |
G. Chowell and H. Nishiura,
Transmission dynamics and control of Ebola virus disease (EVD): A review, BMC Med., 12 (2014), p196.
doi: 10.1186/s12916-014-0196-0. |
| [11] |
F. Clarke,
Functional Analysis, Calculus of Variations and Optimal Control Vol. 264 of Graduate Texts in Mathematics, Springer, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
| [12] |
F. Clarke and M. R. de Pinho,
Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.
doi: 10.1137/090757642. |
| [13] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz,
On the definition and the computation of the basic reproduction ratio ${R}_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
| [14] |
S. F. Dowell,
Transmission of Ebola hemorrhagic fever: A study of risk factors in family members, Kikwit, Democratic Republic of the Congo, 1995, J. Infect. Dis., 179 (1999), S87-S91.
doi: 10.1086/514284. |
| [15] |
S. Duwal, S. Winkelmann, C. Schütte and M. von Kleist,
Optimal treatment strategies in the context of 'Treatment for Prevention' against HIV-1 in resource-poor settings, PLoS Comput. Biol., 11 (2015), e1004200.
doi: 10.1371/journal.pcbi.1004200. |
| [16] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin-New York, 1975.
|
| [17] |
S. K. Gire, A. Goba, K. G. Andersen, R. S. G. Sealfon, D. J. Park and L. Kanneh,
Genomic surveillance elucidates Ebola virus origin and transmission during the 2014 outbreak, Science, 345 (2014), 1369-1372.
doi: 10.1126/science.1259657. |
| [18] |
E. C. Hayden, World struggles to stop Ebola, Nature, 512 (2014), 355-356. |
| [19] |
D. Hincapie-Palacio, J. Ospina and D. F. M. Torres,
Approximated analytical solution to an Ebola optimal control problem, Int. J. Comput. Methods Eng. Sci. Mech., 17 (2016), 382-390.
doi: 10.1080/15502287.2016.1231236. |
| [20] |
J. Kaufman, S. Bianco and A. Jones, https://wiki.eclipse.org/Ebola_Models. |
| [21] |
A. Kaushik, S. Tiwari, R. D. Jayant, A. Marty and M. Nair,
Towards detection and diagnosis of Ebola virus disease at point-of-care, Biosensors and Bioelectronics, 75 (2016), 254-272.
doi: 10.1016/j.bios.2015.08.040. |
| [22] |
A. S. Khan,
The reemergence of Ebola hemorrhagic fever, Democratic Republic of the Congo, 1995, J. Infect. Dis., 179 (1999), S76-S86.
doi: 10.1086/514306. |
| [23] |
U. Ledzewicz and H. Schättler,
Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
| [24] |
J. Legrand, R. F. Grais, P. Y. Boelle, A. J. Valleron and A. Flahault,
Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610-621.
doi: 10.1017/S0950268806007217. |
| [25] |
P. E. Lekone and B. F. Finkenstädt,
Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62 (2006), 1170-1177.
doi: 10.1111/j.1541-0420.2006.00609.x. |
| [26] |
N. K. Martin, A. B. Pitcher, P. Vickerman, A. Vassall and M. Hickman, Optimal control of hepatitis C antiviral treatment programme delivery for prevention amongst a population of injecting drug users, PLoS One, 274 (2011), e22309, 17 pp. |
| [27] |
R. Miller Neilan and S. Lenhart, An introduction to optimal control with an application in
disease modeling, In: Modeling paradigms and analysis of disease transmission models, Vol. 75
of DIMACS Ser. Discrete Math. Theoret. Comput. Sci. Amer. Math. Soc., Providence, RI,
2010, 67-81. |
| [28] |
R. Ndambi,
Epidemiologic and clinical aspects of the Ebola virus epidemic in Mosango, Democratic Republic of the Congo, 1995, J. Infect. Dis., 179 (1999), S8-S10.
doi: 10.1086/514297. |
| [29] |
G. A. Ngwa and M. I. Teboh-Ewungkem, A mathematical model with quarantine states for the dynamics of Ebola virus disease in human populations Comput. Math. Methods Med., 2016 (2016), Art. ID 9352725, 29 pp.
doi: 10.1155/2016/9352725. |
| [30] |
A. Papa, K. Tsergouli, D. Çağlayık, S. Bino, N. Como and Y. Uyar,
Cytokines as biomarkers of Crimean-Congo hemorrhagic fever, J. Med. Virol., 88 (2016), 21-27.
doi: 10.1002/jmv.24312. |
| [31] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.
|
| [32] |
A. Rachah and D. F. M. Torres, Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa Discrete Dyn. Nat. Soc., 2015 (2015), Art. ID 842792, 9 pp.
doi: 10.1155/2015/842792. |
| [33] |
A. Rachah and D. F. M. Torres,
Predicting and controlling the Ebola infection, Math. Method. Appl. Sci., (2016).
doi: 10.1002/mma.3841. |
| [34] |
C. M. Rivers, E. T. Lofgren, M. Marathe, S. Eubank and B. L. Lewis, Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia Technical report, PLOS Currents Outbreaks, 2014.
doi: 10.1371/currents.outbreaks.4d41fe5d6c05e9df30ddce33c66d084c. |
| [35] |
A. K. Rowe,
Clinical, virologic, and immunologic follow-up of convalescent Ebola hemorrhagic fever patients and their household contacts, Kikwit, Democratic Republic of the Congo, J. Infect. Dis., 179 (1999), S28-S35.
doi: 10.1086/514318. |
| [36] |
C. J. Silva and D. F. M. Torres,
A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.
doi: 10.3934/dcds.2015.35.4639. |
| [37] |
P. D. Walsh, R. Biek and L. A. Real,
Wave-like spread of Ebola Zaire, PLoS Biology, 3 (2005), e371.
doi: 10.1371/journal.pbio.0030371. |
show all references
References:
| [1] |
M. D. Ahmad, M. Usman, A. Khan and M. Imran,
Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination, Infect. Dis. Poverty, 5 (2016), p72.
doi: 10.1186/s40249-016-0161-6. |
| [2] |
C. L. Althaus, Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in west Africa, PLOS Currents Outbreaks Edition 1,2014.
doi: 10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288. |
| [3] |
I. Area, H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh and A. Torres, On a fractional order Ebola epidemic model Adv. Difference Equ. , 2015 (2015), 12 pp.
doi: 10.1186/s13662-015-0613-5. |
| [4] |
I. Area, J. Losada, F. Ndaïrou, J. J. Nieto and D. D. Tcheutia,
Mathematical modeling of 2014 Ebola outbreak, Math. Method. Appl. Sci., (2015).
doi: 10.1002/mma.3794. |
| [5] |
A. Atangana and E. F. Doungmo Goufo, On the mathematical analysis of Ebola hemorrhagic fever: Deathly infection disease in west african countries BioMed Research International 2014 (2014), Art. ID 261383, 7 pp.
doi: 10.1155/2014/261383. |
| [6] |
M. H. A. Biswas, L. T. Paiva and M. R. de Pinho,
A SEIR model for control of infectious diseases with constraints, Math. Biosci. Eng., 11 (2014), 761-784.
doi: 10.3934/mbe.2014.11.761. |
| [7] |
M. A. Bwaka,
Ebola hemorrhagic fever in Kikwit, Democratic Republic of the Congo: Clinical observations in 103 patients, J. Infect. Dis., 179 (1999), S1-S7.
doi: 10.1086/514308. |
| [8] |
L. Cesari, Optimization Theory and Applications, Vol. 17 of Applications of Mathematics
(New York), Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
| [9] |
G. Chowell, N. W. Hengartner, C. Castillo-Chavez, P. W. Fenimore and J. M. Hyman,
The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda, J. Theor. Biol., 229 (2004), 119-126.
doi: 10.1016/j.jtbi.2004.03.006. |
| [10] |
G. Chowell and H. Nishiura,
Transmission dynamics and control of Ebola virus disease (EVD): A review, BMC Med., 12 (2014), p196.
doi: 10.1186/s12916-014-0196-0. |
| [11] |
F. Clarke,
Functional Analysis, Calculus of Variations and Optimal Control Vol. 264 of Graduate Texts in Mathematics, Springer, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
| [12] |
F. Clarke and M. R. de Pinho,
Optimal control problems with mixed constraints, SIAM J. Control Optim., 48 (2010), 4500-4524.
doi: 10.1137/090757642. |
| [13] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz,
On the definition and the computation of the basic reproduction ratio ${R}_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
| [14] |
S. F. Dowell,
Transmission of Ebola hemorrhagic fever: A study of risk factors in family members, Kikwit, Democratic Republic of the Congo, 1995, J. Infect. Dis., 179 (1999), S87-S91.
doi: 10.1086/514284. |
| [15] |
S. Duwal, S. Winkelmann, C. Schütte and M. von Kleist,
Optimal treatment strategies in the context of 'Treatment for Prevention' against HIV-1 in resource-poor settings, PLoS Comput. Biol., 11 (2015), e1004200.
doi: 10.1371/journal.pcbi.1004200. |
| [16] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin-New York, 1975.
|
| [17] |
S. K. Gire, A. Goba, K. G. Andersen, R. S. G. Sealfon, D. J. Park and L. Kanneh,
Genomic surveillance elucidates Ebola virus origin and transmission during the 2014 outbreak, Science, 345 (2014), 1369-1372.
doi: 10.1126/science.1259657. |
| [18] |
E. C. Hayden, World struggles to stop Ebola, Nature, 512 (2014), 355-356. |
| [19] |
D. Hincapie-Palacio, J. Ospina and D. F. M. Torres,
Approximated analytical solution to an Ebola optimal control problem, Int. J. Comput. Methods Eng. Sci. Mech., 17 (2016), 382-390.
doi: 10.1080/15502287.2016.1231236. |
| [20] |
J. Kaufman, S. Bianco and A. Jones, https://wiki.eclipse.org/Ebola_Models. |
| [21] |
A. Kaushik, S. Tiwari, R. D. Jayant, A. Marty and M. Nair,
Towards detection and diagnosis of Ebola virus disease at point-of-care, Biosensors and Bioelectronics, 75 (2016), 254-272.
doi: 10.1016/j.bios.2015.08.040. |
| [22] |
A. S. Khan,
The reemergence of Ebola hemorrhagic fever, Democratic Republic of the Congo, 1995, J. Infect. Dis., 179 (1999), S76-S86.
doi: 10.1086/514306. |
| [23] |
U. Ledzewicz and H. Schättler,
Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
| [24] |
J. Legrand, R. F. Grais, P. Y. Boelle, A. J. Valleron and A. Flahault,
Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610-621.
doi: 10.1017/S0950268806007217. |
| [25] |
P. E. Lekone and B. F. Finkenstädt,
Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62 (2006), 1170-1177.
doi: 10.1111/j.1541-0420.2006.00609.x. |
| [26] |
N. K. Martin, A. B. Pitcher, P. Vickerman, A. Vassall and M. Hickman, Optimal control of hepatitis C antiviral treatment programme delivery for prevention amongst a population of injecting drug users, PLoS One, 274 (2011), e22309, 17 pp. |
| [27] |
R. Miller Neilan and S. Lenhart, An introduction to optimal control with an application in
disease modeling, In: Modeling paradigms and analysis of disease transmission models, Vol. 75
of DIMACS Ser. Discrete Math. Theoret. Comput. Sci. Amer. Math. Soc., Providence, RI,
2010, 67-81. |
| [28] |
R. Ndambi,
Epidemiologic and clinical aspects of the Ebola virus epidemic in Mosango, Democratic Republic of the Congo, 1995, J. Infect. Dis., 179 (1999), S8-S10.
doi: 10.1086/514297. |
| [29] |
G. A. Ngwa and M. I. Teboh-Ewungkem, A mathematical model with quarantine states for the dynamics of Ebola virus disease in human populations Comput. Math. Methods Med., 2016 (2016), Art. ID 9352725, 29 pp.
doi: 10.1155/2016/9352725. |
| [30] |
A. Papa, K. Tsergouli, D. Çağlayık, S. Bino, N. Como and Y. Uyar,
Cytokines as biomarkers of Crimean-Congo hemorrhagic fever, J. Med. Virol., 88 (2016), 21-27.
doi: 10.1002/jmv.24312. |
| [31] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.
|
| [32] |
A. Rachah and D. F. M. Torres, Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa Discrete Dyn. Nat. Soc., 2015 (2015), Art. ID 842792, 9 pp.
doi: 10.1155/2015/842792. |
| [33] |
A. Rachah and D. F. M. Torres,
Predicting and controlling the Ebola infection, Math. Method. Appl. Sci., (2016).
doi: 10.1002/mma.3841. |
| [34] |
C. M. Rivers, E. T. Lofgren, M. Marathe, S. Eubank and B. L. Lewis, Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia Technical report, PLOS Currents Outbreaks, 2014.
doi: 10.1371/currents.outbreaks.4d41fe5d6c05e9df30ddce33c66d084c. |
| [35] |
A. K. Rowe,
Clinical, virologic, and immunologic follow-up of convalescent Ebola hemorrhagic fever patients and their household contacts, Kikwit, Democratic Republic of the Congo, J. Infect. Dis., 179 (1999), S28-S35.
doi: 10.1086/514318. |
| [36] |
C. J. Silva and D. F. M. Torres,
A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.
doi: 10.3934/dcds.2015.35.4639. |
| [37] |
P. D. Walsh, R. Biek and L. A. Real,
Wave-like spread of Ebola Zaire, PLoS Biology, 3 (2005), e371.
doi: 10.1371/journal.pbio.0030371. |
Figure 2.
(a) Cumulative confirmed cases: in dashed circle line the real data from WHO and in continuous line the values of $I(t) + R(t) + D(t) + H(t) + B(t) + C(t) - \mu(N - S(t) - E(t))$ from (1) with the parameter values from Table 1. (b) Cumulative confirmed cases given in (2), when available an unlimited supply of vaccines, also with the parameter values from Table 1
Figure 6.
(a) Cumulative confirmed cases. (b) Optimal control for the case of limited total number of vaccines. Dashed line for $W(90)\leq 10000$ and continuous line for $W(90)\leq 20000$
Figure 7.
Individuals $S(t)$ , $E(t)$ , $I(t)$ and $R(t)$ . The dashed line represents the case where $W(90) \leq 10000$ and the continuous line represents the case where $W(90) \leq 20000$
Figure 8.
Individuals $D(t)$ , $H(t)$ , $B(t)$ and $C(t)$ . The dashed line represents the case where $W(90) \leq 10000$ and the continuous line represents the case where $W(90) \leq 20000$
Figure 9.
Optimal control $u(t)$ and number of vaccines $W(t)$ for $W(90)\leq 10000$ , $W(90)\leq 11000$ , $W(90)\leq 13000$ , $W(90)\leq 15000$ , $W(90)\leq 16000$ , $W(90)\leq 18000$ and $W(90)\leq 20000$
Figure 10.
Optimal control $u(t)$ and number of vaccines $W(t)$ for $W(90)\leq 10000$ . In dashed line the case $w_2=50$ and in continuous line the case $w_2 = 500$
Figure 11.
(a) Cumulative confirmed cases, (b) completely recovered, (c) optimal control. In (a), (b) and (c) the following mixed constraints are considered: $S(t) u(t) \leq 1000$ for all $t \in [0, 10]$ , $S(t) u(t) \leq 1200$ for all $t \in [0, 15]$ , and $S(t) u(t) \leq 900$ for all $t \in [0, 16]$
| Symbol | Description | Value |
| | per capita rate at which exposed individuals become infectious | |
| | death rate | |
| | contact rate of infective and susceptible individuals | |
| | contact rate of infective and dead individuals | |
| | contact rate of infective and hospitalized individuals | |
| | contact rate of infective and asymptomatic individuals | |
| | per capita rate of progression of individuals from the infectious class | |
| to the asymptomatic class | | |
| | fatality rate | |
| | per capita rate of progression of individuals from the dead class | |
| to the buried class | | |
| | per capita rate of progression of individuals from the hospitalized class | |
| to the buried class | | |
| | per capita rate of progression of individuals from the hospitalized class | |
| to the asymptomatic class | | |
| | per capita rate of progression of individuals from the infectious class | |
| to the hospitalized class | | |
| | per capita rate of progression of individuals from the asymptomatic class | |
| to the completely recovered class | | |
| | incineration rate | |
| Symbol | Description | Value |
| | per capita rate at which exposed individuals become infectious | |
| | death rate | |
| | contact rate of infective and susceptible individuals | |
| | contact rate of infective and dead individuals | |
| | contact rate of infective and hospitalized individuals | |
| | contact rate of infective and asymptomatic individuals | |
| | per capita rate of progression of individuals from the infectious class | |
| to the asymptomatic class | | |
| | fatality rate | |
| | per capita rate of progression of individuals from the dead class | |
| to the buried class | | |
| | per capita rate of progression of individuals from the hospitalized class | |
| to the buried class | | |
| | per capita rate of progression of individuals from the hospitalized class | |
| to the asymptomatic class | | |
| | per capita rate of progression of individuals from the infectious class | |
| to the hospitalized class | | |
| | per capita rate of progression of individuals from the asymptomatic class | |
| to the completely recovered class | | |
| | incineration rate | |
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