# Modeling and Controlling Epidemic Outbreaks: The Role of Population Size, Model Heterogeneity and Fast Response in the Case of Measles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model

_{k}(t) individuals in day $k\in \left\{1,2,\dots ,10\right\}$ of the incubation period, I

_{k}(t) individuals in day $k\in \left\{1,2,\dots ,8\right\}$ of the infectious period and R(t) individuals who had recovered from an infection. Note that $S\left(t\right)+{{\displaystyle \sum}}_{k=1}^{10}{E}_{k}\left(t\right)+{{\displaystyle \sum}}_{k=1}^{8}{I}_{k}\left(t\right)+R\left(t\right)=vn$. We denoted the total number of infectious individuals on day t as T(t). On day 1, we had exactly one infected individual who was in the first day of the incubation period. The system dynamics are then given by Equations (1)–(9), where N(t) was the number of new infections on day t.

_{0}and divided by the length of the infectious period, 8, and the size of the population. This way, the first infectious individual created R

_{0}secondary cases in expectation if the entire population was susceptible.

^{x}(t) by considering both infections from the same subgroup and the other subgroup. In addition, we not only had a public health intervention stopping the spread in a subgroup ε days after the first reported case in that subgroup but also $\sigma $ days after the first reported case in the other subgroup. Table 1 below summarizes the notation used in this manuscript.

#### 2.2. Simulations

_{2}, when we assumed two community groups. Specifically, we examined the expected size of the outbreak, E[C], the probability the outbreak was at least five, P [C ≥ 5] and the probability that the outbreak spread beyond the initial community group, P [C

_{2}≥ 1]. In the post elimination era, the median number of outbreak cases is five; thus, five was the threshold outbreak size we used in our metric [22].

## 3. Results

#### 3.1. Theoretical Results

#### 3.1.1. Convergence as Population Size Increased

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Proof of Theorem 1.**

#### 3.1.2. Outbreaks Are Longer with Two Subgroups

**Theorem**

**2.**

**Proof.**

#### 3.1.3. Heterogeneity in Infection Risk

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

#### 3.2. Numerical Results

#### 3.2.1. The Impact of Population Size

#### 3.2.2. The Impact of Heterogeneity

#### 3.2.3. The Impact of Public Health Interventions

_{2}≥ 1]). While the benefits of increasing the detection probability were significant, we saw diminishing returns for greater increases. Decreasing the lag between detection and response also decreased the outcome measures (except for the probability of the second subgroup being infected, which changed little) but only by a small amount. By the time a case was detected, the outbreak had already spread to the other subgroup. Thus, a few days of delay did not make much difference and the change in P[C

_{2}≥ 1] was even less.

_{2}≥ 1] now denoting the probability that the outbreak spread from the elementary school to the high school. The results had the same trend as those in the case of a population with two equal-sized subgroups we discussed previously. As the vaccinated fraction of the population, the mixing parameter or the response lag increased, the expected number of infected cases, the probability of having at least five infected cases and the probability of having at least one infected case in the high school when the public health authorities took action decreased.

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- DeRoo, S.S.; Pudalov, N.J.; Fu, L.Y. Planning for a COVID-19 Vaccination Program. JAMA
**2020**, 323, 2458–2459. [Google Scholar] [CrossRef] - Measles Origin Finding Could Inform COVID-19 Research. Available online: https://medicalxpress.com/news/2020-06-measles-covid-.html (accessed on 18 October 2020).
- Sidiq, K.R.; Sabir, D.K.; Ali, S.M.; Kodzius, R. Does early childhood vaccination protect against COVID-19? Front. Mol. Biosci.
**2020**, 7, 120. [Google Scholar] [CrossRef] - Franklin, R.; Young, A.; Neumann, B.; Fernandez, R.; Joannides, A.; Reyahi, A.; Modis, Y. Homologous protein domains in SARS-CoV-2 and measles, mumps and rubella viruses: Preliminary evidence that MMR vaccine might provide protection against COVID-19. medRxiv
**2020**. [Google Scholar] [CrossRef] [Green Version] - Deshpande, S.; Balji, S. MMR Vaccine and Covid-19: A Myth or a Low Risk-High Reward Preventive Measure? Indian Pediatrics
**2020**, 57, 773. [Google Scholar] [CrossRef] - Shanker, V. Measles immunization: Worth considering containment Strategy for SARS-CoV-2 global outbreak. Indian Pediatrics
**2020**, 57, 380. [Google Scholar] [CrossRef] - Fidel, P.L.; Noverr, M.C. Could an unrelated live attenuated vaccine serve as a preventive measure to dampen septic inflammation associated with COVID-19 Infection? mBio
**2020**, 11, e00907-20. [Google Scholar] [CrossRef] - Sajuni, S. Vaksinasi Measles, Mumps, dan Rubella (MMR) Sebagai Prophylaxis Terhadap COVID-19. Keluwih J. Kesehat. Kedokt.
**2020**, 1, 25–28. [Google Scholar] - Saad, M.E.; Elsalamony, R.A. Measles vaccines may provide partial protection against COVID-19. Int. J. Cancer Biomed. Res.
**2020**, 5, 14–19. [Google Scholar] [CrossRef] [Green Version] - Salman, S.; Salem, M.L. Routine childhood immunization may protect against COVID-19. Med. Hypotheses
**2020**, 140, 109689. [Google Scholar] [CrossRef] - Larenas-Linnemann, D.E.; Rodrigues-Monroy, F. Thirty-six COVID-19 cases preventively vaccinated with mumps-measles-rubella vaccine: All mild course. Allergy
**2020**. [Google Scholar] [CrossRef] - Zhu, H.; Li, Y.; Jin, X.; Hunag, J.; Liu, X.; Tan, J. Transmission dynamics and control methodology of COVID-19: A modeling study Transmission dynamics and control methodology of COVID-19: A modeling study. Appl. Math. Model.
**2020**. [Google Scholar] [CrossRef] - Chen, X.; Yu, B. First two months of the 2019 Coronavirus Disease (COVID-19) epidemic in China: Realtime surveillance and evaluation with a second derivative model. Glob. Health Res. Policy
**2020**, 5, 7. [Google Scholar] [CrossRef] [PubMed] - World Health Organization. Progress towards regional measles elimination—worldwide, 2000–2017 Wkly. Epidemiol. Rec.
**2018**, 93, 649–660. [Google Scholar] - WHO. Measles, rubella and CRS: Disease description, epidemiology and diagnosis. In Surveillance Guidelines for Measles, Rubella and Congenital Rubella Syndrome in the WHO European Region; World Health Organization: Geneva, Switzerland, 2012. Available online: https://www.ncbi.nlm.nih.gov/books/NBK143257/ (accessed on 17 October 2020).
- Miller, D.L. Frequency of complications of measles, 1963. Br. Med. J.
**1964**, 2, 75–78. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gindler, J.; Tinker, S.; Markowitz, L.; Atkinson, W.; Dales, L.; Papania, M.J. Acute measles mortality in the United States, 1987–2002. J. Infect. Dis.
**2004**, 189 (Suppl. 1), 69–77. [Google Scholar] [CrossRef] [Green Version] - Blumberg, S.; Enaroria, W.T.A.; Lyoyd-Smith, J.O.; Lietman, T.M.; Porco, T.C. Identifying post-elimination trends for the introduction and transmissibility of measles in the United States. Am. J. Epidemiol.
**2014**, 79, 1375–1382. [Google Scholar] [CrossRef] - Patel, M.; Lee, A.D.; Redd, S.B.; Clemmons, N.S.; McNall, R.J.; Cohn, A.C.; Gastañaduy, P.A. Increase in measles cases—United States, January 1–26 April 2019. Morb. Mortal. Wkly. Rep.
**2019**, 68, 402–404. [Google Scholar] [CrossRef] - Graham, M.K.; Winter, A.K.; Ferrari, M.; Grenfell, B.; Moss, W.J.; Azman, A.W.C.; Metcalf, J.E.; Lessler, J. Measles and the canonical path to elimination. Science
**2019**, 364, 584–587. [Google Scholar] [CrossRef] [Green Version] - Health Protection Agency. Laboratory confirmed cases of measles, mumps and rubella in England and Wales: Update to end June 2012. Health Protection Report. Available online: http://www.hpa.org.uk/hpr/archives/2012/hpr3412.pdf (accessed on 24 August 2012).
- Fiebelkorn, A.P.; Redd, S.B.; Gallagher, K.; Rota, P.A.; Rota, J.; Bellini, W.; Seward, J. Measles in the United States during the post-elimination era. J. Infect. Dis.
**2010**, 202, 1520–1528. [Google Scholar] [CrossRef] - The Lancet Child Adolescent Health. Vaccine hesitancy: A generation at risk. Lancet Child. Adolesc. Health
**2019**, 3, 281. [Google Scholar] [CrossRef] - Arede, M.; Bravo-Araya, M.; Milie Bouchard, É.; Singh Gill, G.; Plajer, V.; Shehraj, A.; Shuaib, Y.A. Combating Vaccine Hesitancy: Teaching the Next Generation to Navigate Through the Post Truth Era. Front. Public Health
**2019**, 14, 381–387. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shen, S.; Dubey, V. Addressing vaccine hesitancy. Clinical guidance for primary care physicians working with parents. Can. Fam. Physician
**2019**, 65, 175–181. [Google Scholar] [PubMed] - Marcus, B. A Nursing Approach to the Largest Measles Outbreak in Recent U.S. History: Lessons Learned Battling Homegrown Vaccine Hesitancy. Online J. Issues Nurs.
**2020**, 25. [Google Scholar] [CrossRef] - Banerjee, E.; Griffith, J.; Kenyon, C.; Christianson, B.; Strain, A.; Martin, K.; McMahon, M.; Bagstad, E.; Hardy, K.; Grilli, G.; et al. Containing a measles outbreak in Minnesota, 2017: Methods and challenges. Perspect. Public Health
**2020**, 140, 162–171. [Google Scholar] [CrossRef] [PubMed] - Gastanaduy, P.A.; Banerjee, E.; DeBolt, C.; Bravo-Alcantara, P.; Samad, S.A.; Pastor, D.; Rota, P.A.; Patel, M.; Crowcroft, N.S.; Durrheim, D.N. Public health responses during measles outbreaks in elimination settings: Strategies and challenges. Hum. Vaccines Immunother.
**2018**, 14, 2222–2238. [Google Scholar] [CrossRef] - Centers for Disease Control and Prevention. CDC Measles—United States, 24 August 2012. Morb. Mortal. Wkly. Rep.
**2012**, 61, 647–652. [Google Scholar] - Gastañaduy, P.A.; Redd, S.B.; Fiebelkorn, A.P.; Rota, J.S.; Rota, P.A.; Bellini, W.J.; Seward, J.F.; Wallace, G.S. CDC Measles—United States, 1 January–23 May 2014. Morb. Mortal. Wkly. Rep.
**2014**, 63, 496–499. [Google Scholar] - CDC Measles Cases. Available online: http://www.cdc.gov/measles/cases-outbreaks.html (accessed on 12 August 2020).
- Glass, K.; Kappey, J.; Grenfell, B.T. The effect of heterogeneity in measles vaccination on population immunity. Epidemiol. Infect.
**2004**, 132, 675–683. [Google Scholar] [CrossRef] - Barlett, M.S. Measles periodicity and community Size. J. R. Stat. Soc. A
**1957**, 120, 48–70. [Google Scholar] [CrossRef] - Barlett, M.S. The critical community size for measles in the United States. J. R. Stat. Soc. A
**1960**, 123, 37–44. [Google Scholar] [CrossRef] - Black, F. Measles endemicity in insular populations: Critical community size and its evolutionary implication. J. Theor. Biol.
**1966**, 11, 207–211. [Google Scholar] [CrossRef] - Keeling, M.J.; Grenfell, B.T. Disease extinction and community size: Modeling the persistence of measles. Science
**1997**, 275, 65–67. [Google Scholar] [CrossRef] - Anderson, R.M.; May, R.M. Infectious Diseases in Humans. Dynamics and Control; Oxford University Press: Oxford, UK, 1992. [Google Scholar]
- Measles History. Available online: https://www.cdc.gov/measles/about/history.html (accessed on 17 October 2020).
- Murray, G.D.; Cliff, A.D. A stochastic model for measles epidemics in a multi-region setting. Trans. Inst. Br. Geogr.
**1977**, 2, 158–174. [Google Scholar] [CrossRef] - Bolker, B.; Grenfell, B.T. Space, persistence and dynamics of measles epidemics. Phil. Trans. R. Soc. Lond.
**1995**, B251, 75–81. [Google Scholar] - Hethcote, H.W.; Van Ark, J.W. Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation and immunization programs. Math. Biosci.
**1987**, 75, 205–227. [Google Scholar] [CrossRef] - Castillo-Chavez, C.; Hethcote, H.W.; Andreasen, V.; Levin, S.A.; Liu, W.M. Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol.
**1989**, 27, 233–258. [Google Scholar] [CrossRef] [PubMed] - Hethcote, H.W. Modeling heterogeneous mixing in infectious disease dynamics. In Models for Infectious Human Diseases: Their Structure and Relation to Data; Isham, V., Medley, G., Eds.; Cambridge University Press: Cambridge, UK, 1996; pp. 215–238. [Google Scholar]
- Llyod, A.L.; May, R.M. Spatial heterogeneity in epidemic models. J. Theor. Biol.
**1996**, 179, 1–11. [Google Scholar] [CrossRef] - Sattenspiel, L.; Dietz, K. A structured epidemic model incorporating geographic mobility among regions. Math. Biosci.
**1995**, 128, 71–91. [Google Scholar] [CrossRef] - Allen, L.J.S.; Jones, M.A.; Martin, C.F. A discrete-time model with vaccination for a measles epidemic. Math. Biosci.
**1991**, 105, 111. [Google Scholar] [CrossRef] - Bonačić Marinović, A.A.; Swaan, C.; Wichmann, O.; Steenbergen, J.V.; Kretzschmar, M. Effectiveness and timing of vaccination during school measles outbreak. Emerg. Infect. Dis.
**2009**, 18, 1405–1413. [Google Scholar] [CrossRef] - Ejima, K.; Omori, R.; Aihara, K.; Nishiura, H. Real-time investigation of measles epidemics with estimate of vaccine efficacy. Int. J. Biol. Sci.
**2012**, 8, 620. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Babad, H.R.; Nokes, D.J.; Gay, N.J.; Miller, E.; Morgan-Capner, P.; Anderson, R.M. Predicting the impact of measles vaccination in England and Wales: Model validation and analysis of policy options. Epidemiol. Infect.
**1995**, 114, 319–344. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lessler, J.; Metcalf, C.J.E.; Grais, R.F.; Luquero, F.J.; Cummings, D.A.; Grenfell, B.T. Measuring the Performance of Vaccination Programs Using Cross-Sectional Surveys: A Likelihood Framework and Retrospective Analysis. PLoS Med.
**2011**, 8, e1001110. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Public Health Agency of Canada. Measles. Available online: http://www.phac-aspc.gc.ca/im/vpd-mev/measles-rougeole-eng.php (accessed on 12 July 2020).
- Perez, L.; Dragicevic, S. An Agent-based Approach for Modeling Dynamics of Contagious Disease Spread. Int. J. Health Geogr.
**2009**, 8, 50. [Google Scholar] [CrossRef] [Green Version] - Glasser, J.W.; Feng, Z.; Omer, S.B.; Smith, P.J.; Rodewald, L.E. The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: A modelling study. Lancet Infect. Dis.
**2016**, 16, 599–605. [Google Scholar] [CrossRef] [Green Version] - Wallinga, J.; Heijne, J.C.M.; Kretzschmar, M. A Measles Epidemic Threshold in a Highly Vaccinated Population. PLoS Med.
**2005**, 2, e316. [Google Scholar] [CrossRef] [PubMed] - Kutty, P.; Rota, J.; Bellini, W. Manual for the Surveillance of Vaccine-Preventable Diseases, 6th ed.; Centers for Disease Control and Prevention: Atlanta, GA, USA, 2013; Chapter 7 Measles. Available online: http://www.cdc.gov/vaccines/pubs/surv-manual/chpt07-measles.html (accessed on 12 April 2020).
- Marx, G.E.; Chase, J.; Jasperse, J.; Stinson, K.; McDonald, C.E.; Runfola, J.K.; Jaskunas, J.; Hite, D.; Barnes, M.; Askenazi, M.; et al. Public Health Economic Burden Associated with Two Single Measles Case Investigations—Colorado, 2016–2017. Morb. Mortal. Wkly. Rep.
**2017**, 66, 1272–1275. [Google Scholar] [CrossRef] [Green Version] - National Center for Education. Available online: https://nces.ed.gov/programs/digest/d19/tables/dt19_214.40.asp (accessed on 17 October 2020).
- Malone, K.M.; Hinman, A.R. Vaccination mandates: The public health imperative and individual rights. In Law in Public Health Practice, 2nd ed.; Goodman, R.A., Hoofman, R.E., Lopez, W., Matthews, G.W., Rothstein, M.A., Foster, K.L., Eds.; Oxford University Press: Oxford, NY, USA, 2007; pp. 338–360. [Google Scholar]
- Woudenberg, T.; Woonink, F.; Kerkhof, J.; Cox, K.; Ruijs, W.L.M.; van Binnendijk, R.; de Melker, H.; Hahné, S.J.M.; Wallinga, J. The tip of the iceberg: Incompleteness of measles reporting during a large outbreak in The Netherlands in 2013–2014. Epidemiol. Infect.
**2018**, 147, e23. [Google Scholar] [CrossRef] [Green Version] - Parent du Châtelet, I.; Waku-Kouomou, D.; Freymuth, F.; Maine, C.; Lévy-Bruhl, D. La rougeole en France en 2008: Bilan de la déclaration obligatoire. [Measles in France in 2008: Results of the mandatory declaration]. French. Institut de Veille Sanitaire. Bull. Epidemiol. Heb.
**2009**, 39–40, 415–419. Available online: http://www.invs.sante.fr/beh/2009/39_40/beh_39_40_2009.pdf (accessed on 16 October 2020). - CDC. Measles, Mumps, and Rubella (MMR) Vaccination Coverage Among Children 19–35 Months by State, HHS Region, and the United States, National Immunization Survey-Child (NIS-Child), 1995 through 2017. Available online: https://www.cdc.gov/vaccines/imz-managers/coverage/childvaxview/data-reports/mmr/trend/index.html (accessed on 16 October 2020).
- Azimi, P.; Keshavarz, Z.; Cedeno Laurent, J.G.; Allen, J.G. Estimating the nationwide transmission risk of measles in US schools and impacts of vaccination and supplemental infection control strategies. BMC Infect. Dis.
**2020**, 20, 497. [Google Scholar] [CrossRef]

**Figure 1.**The average number of infected cases (E[C],

**left**,

**blue**) and the probability of a large outbreak (P[C ≥ 5],

**right**,

**green**) as we varied the size of the population. The dashed lines are for the case of a homogenous population while the solid lines are for the case where the population was divided into two equal-sized subgroups. There were 1000 replications and the standard error was less than 0.11 for E[C] and 0.02 for P[C ≥ 5].

**Figure 2.**Histogram of the total number of infected cases in a homogenous population of 1000 or a population with two subgroups each of 500 individuals. There were 1000 replications.

**Figure 3.**The average number of infected cases (E[C],

**left**,

**blue**) and the probability of a large outbreak (P[C ≥ 5],

**right**,

**green**) as the mixing parameter changed in a population with two subgroups each of 500 individuals. The mixing parameter was the ratio of the probability of infecting a susceptible in the other subgroup to the probability of infecting a susceptible in the same subgroup. There were replications and the standard error was less than 0.04 for E[C] and 0.02 for P[C ≥ 5].

**Figure 4.**The average number of infected cases (E[C],

**left**,

**blue**) and the probability of a large outbreak (P[C ≥ 5],

**right**,

**green**) as we varied the fraction vaccinated. The dashed lines are for the case of a homogenous population of 1000 while the solid lines are for the case where the population was divided into two equal-sized subgroups of 500. The red line indicates the deterministic herd immunity threshold. There were 1000 replications and the standard error was less than 0.22 for E[C] and 0.02 for P[C ≥ 5].

Population, n |

Fraction unvaccinated, v |

Susceptibles, S(t) |

Individuals in day k of the incubation period, E_{k}(t) |

Individuals in day k of the infectious period, I_{k}(t) |

Recovered individuals, R(t) |

Total number of infectious individuals, T(t) |

New infections, N(t) |

Probability a susceptible individual is infected, p(t) |

Basic reproduction number, ${R}_{0}$ |

Number of reported individuals, Z(t) |

Probability an infectious case is reported each day, q |

Lag between the first reported case in same subgroup and stopping, ε |

Stopping time, τ |

Mixing parameter, $\phi $ |

Lag between the first reported case in the other subgroup and stopping, $\sigma $ |

Parameter | Value | Source |
---|---|---|

Population, n (people) | 1000 | [57] |

Basic reproduction number, ${R}_{0}$ | 18 | [37] |

Fraction unvaccinated, v (%) | 10% | [58] |

Incubation period (days) | 10 | [51] |

Infectious period (days) | 8 | [52] |

Mixing parameter, $\phi $ (%) | 0.5 | |

Probability of the case being reported each contagious day, q (%) | 0.1 | [54,59,60] |

Lag between the first reported case in the subgroup stopping, ε (days) | 3 | [56] |

Lag between the first reported case in the other subgroup and stopping, σ (days) | 5 |

**Table 3.**Additional sensitivity analysis. Here C

_{2}is the number of infected cases in the other subgroup (not the one with the initial infection). The number of replications was 1000. The standard error was less than 0.18 for E[C] and 0.02 for P [C ≥ 5] and P [C

_{2}≥ 1].

Two Community Groups | One Community | |||||
---|---|---|---|---|---|---|

E[C] | $\mathit{P}\left[\mathit{C}\ge 5\right]$ | $\mathit{P}\left[{\mathit{C}}_{2}\ge 1\right]$ | E[C] | $\mathit{P}\left[\mathit{C}\ge 5\right]$ | ||

Daily reporting probability, q | 0.05 | 5.9 | 0.465 | 0.650 | 5.2 | 0.403 |

0.10 | 3.9 | 0.287 | 0.554 | 3.4 | 0.257 | |

0.15 | 3.2 | 0.199 | 0.533 | 2.8 | 0.159 | |

0.20 | 2.7 | 0.116 | 0.499 | 2.6 | 0.108 | |

Lag between first case reported and simulation stopped,ε | 0 | 3.1 | 0.212 | 0.558 | 2.8 | 0.185 |

1 | 3.4 | 0.247 | 0.531 | 3.0 | 0.192 | |

2 | 3.7 | 0.262 | 0.567 | 3.4 | 0.239 | |

3 | 3.7 | 0.251 | 0.551 | 3.5 | 0.242 |

**Table 4.**Sensitivity analysis for two unequal-sized groups, an elementary school of 500 and a high school of 2000 individuals. The initial infection was in the elementary school and C

_{2}is the number of cases in the high school. There were 1000 replications. The standard error was less than 0.23 for E[C] and 0.02 for P[C ≥ 5] and P[C

_{2}≥ 1].

E[C] | $\mathit{P}\left(\mathit{C}\ge 5\right)$ | $\mathit{P}\left({\mathit{C}}_{2}\ge 1\right)$ | ||
---|---|---|---|---|

Fraction vaccinated, %, (1-v) | 80 | 8.3 | 0.621 | 0.823 |

85 | 5.9 | 0.464 | 0.717 | |

90 | 4.0 | 0.308 | 0.579 | |

95 | 2.1 | 0.076 | 0.306 | |

Mixing parameter,$\text{}\mathbf{\phi}$ | 0 | 3.4 | 0.237 | 0.000 |

0.25 | 3.8 | 0.285 | 0.451 | |

0.5 | 3.9 | 0.297 | 0.570 | |

0.75 | 3.9 | 0.279 | 0.659 | |

1 | 4.1 | 0.315 | 0.677 | |

Daily reporting probability, q | 0.05 | 5.9 | 0.460 | 0.639 |

0.1 | 3.8 | 0.276 | 0.552 | |

0.15 | 3.1 | 0.200 | 0.525 | |

0.2 | 2.8 | 0.128 | 0.492 | |

Lag between first case reported and simulation stopped,ε | 0 | 3.1 | 0.211 | 0.537 |

1 | 3.6 | 0.261 | 0.572 | |

2 | 3.8 | 0.286 | 0.556 | |

3 | 3.9 | 0.307 | 0.572 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yagci Sokat, K.; Armbruster, B.
Modeling and Controlling Epidemic Outbreaks: The Role of Population Size, Model Heterogeneity and Fast Response in the Case of Measles. *Mathematics* **2020**, *8*, 1892.
https://doi.org/10.3390/math8111892

**AMA Style**

Yagci Sokat K, Armbruster B.
Modeling and Controlling Epidemic Outbreaks: The Role of Population Size, Model Heterogeneity and Fast Response in the Case of Measles. *Mathematics*. 2020; 8(11):1892.
https://doi.org/10.3390/math8111892

**Chicago/Turabian Style**

Yagci Sokat, Kezban, and Benjamin Armbruster.
2020. "Modeling and Controlling Epidemic Outbreaks: The Role of Population Size, Model Heterogeneity and Fast Response in the Case of Measles" *Mathematics* 8, no. 11: 1892.
https://doi.org/10.3390/math8111892