ArticlePDF Available

FITTING TWO SIMPLE PROBABILISTIC MODELS FOR COVID-19 WITH DATA FROM ECUADOR

Article

FITTING TWO SIMPLE PROBABILISTIC MODELS FOR COVID-19 WITH DATA FROM ECUADOR

Abstract

Background: In this study, we use public data of COVID-19 diagnosed cases from Guayas province (Ecuador) and use them to evaluate the forecasting ability of some probabilistic models commonly used in the epidemiological and mathematical literature. We highlight the precision of the models in predicting COVID-19 cases, and the evolution of the epidemic curve until 2021. Those models are helpful to estimate a possible stabilization date for the epidemic. Methods: We use Gompertz and Hill functions to provide robust predictions of the COVID-19 diagnosed cases and the stabilization date of the epidemic curve. We simulate the model errors for predictions by including a probabilistic error term from a Gamma distribution. Non-linear least square method is performed trough the Gauss-Newton algorithm with 5000 iterations to estimate each parameter in the models. The Coefficient of Determination, the Akaike Information Criteria and the Root Mean Squared Error are used as the metrics to evaluate the model performances. Conclusions: The accuracy of our mathematical models is high compared to diagnosed cases until December 9, 2020. Medium terms predictions indicate an increase of cumulative COVID-19 cases until 2021. Since to date, the diagnosed cases will be increasing until the stabilization is achieved. According to the Hill model the stabilization will be achieved on February, 2021, while for the Gompertz model the stabilization will be on December, 2020.
FITTING TWO SIMPLE PROBABILISTIC MODELS FOR
COVID–19 WITH DATA FROM ECUADOR
FELIPE BENAVIDES AND JHOANA P. ROMERO–LEITON
Abstract. Background: In this study, we use public data of COVID–19
diagnosed cases from Guayas province (Ecuador) and use them to evaluate
the forecasting ability of some probabilistic models commonly used in the
epidemiological and mathematical literature. We highlight the precision of
the models in predicting COVID–19 cases, and the evolution of the epidemic
curve until 2021. Those models are helpful to estimate a possible stabilization
date for the epidemic. Methods: We use Gompertz and Hill functions to
provide robust predictions of the COVID–19 diagnosed cases and the stabi-
lization date of the epidemic curve. We simulate the model errors for pre-
dictions by including a probabilistic error term from a Gamma distribution.
Non–linear least square method is performed trough the Gauss–Newton al-
gorithm with 5000 iterations to estimate each parameter in the models. The
Coefficient of Determination, the Akaike Information Criteria and the Root
Mean Squared Error are used as the metrics to evaluate the model perfor-
mances. Conclusions: The accuracy of our mathematical models is high
compared to diagnosed cases until December 9, 2020. Medium terms predic-
tions indicate an increase of cumulative COVID–19 cases until 2021. Since to
date, the diagnosed cases will be increasing until the stabilization is achieved.
According to the Hill model the stabilization will be achieved on February,
2021, while for the Gompertz model the stabilization will be on December,
2020.
1. Introduction
In December (2019), the Wuhan Municipal Health Commission (Hubei Province,
China) informed to World Health Organization (WHO) about a group of 27 cases
of unknown etiology pneumonia, who were commonly exposed to a wet market
in Wuhan City. It was also noticed that seven of these patients were critically
serious. The symptoms of the first case began on December 8, 2019. On January
7, 2020, Chinese authorities identified a new type of family virus as the agent
causing the pneumonia. The causative agent of this pneumonia was identified as
a new virus in the Coronaviridae family that was officially named SARS–CoV–2,
more broadly known as COVID–19. The clinical picture associated with this virus
has been named COVID–19. On March 11, WHO declared the global pandemic.
Since the beginning of the epidemic, to December 9, 2020 have been reported more
than 70,522,778 cases and more than 1,584,795 deaths in the world [6].
2000 Mathematics Subject Classification. 62PXX, 62KXX.
Key words and phrases. SARS–Cov–2, Gompertz model, Hill model, Predictions.
1
Journal of Mathematical Control Science and Applications
Vol. 6 No. 2 (July-December, 2020)
Submitted: 19th November 2020 Revised: 10th December 2020 Accepted: 15th December 2020
143
2 FELIPE BENAVIDES AND JHOANA P. ROMERO–LEITON
Coronaviruses are a family of viruses that cause infection in humans and some
animals. Diseases by coronavirus are zoonotic, that is, they can be transmitted
from animals to humans. Coronaviruses that infect humans (HCoV) can produce
clinical symptoms from the common cold to serious ones like those caused by the
Severe Acute Respiratory Syndrome (SARS) viruses and Middle East Respiratory
Syndrome (MERS–CoV) [7]. The transmission mechanisms of SARS–COV–2 are
animal–human and human–human. The first one is still unknown, but but some
researchers claim that it could be driven by respiratory secretions and/or feces
[8]. The second one, is considered similar for other coronaviruses through the
secretions of infected people, mainly by direct contact with respiratory drops and
hands or fomites contaminated with these secretions, followed by contact with the
mucosa of the mouth, nose or eyes [8].
The COVID–19 pandemic in Ecuador is an expansion of the COVID–19 out-
break that started in China. This fact alerted all countries worldwide from the be-
ginning of 2020 and at the end of February it reached Latin America. On February
29, Ecuador reported a first case of COVID–19, corresponding to an Ecuadorian
woman who had arrived from Spain on February 14 but with no symptoms. On
March 13, the first death due to COVID–19 was reported in the country, who was
the first infected to arrive from Spain. By December 9, 2020, Ecuador reported
200,379 cases of COVID–19 and more than 13,850 deaths [2]. According to data
from the Johns Hopkins Coronavirus Resource Center [1], to date, Ecuador is one
of the most affected Latin American countries and it ranks second in number of in-
fections and deaths after Brazil, even though its population is twelve times smaller
while its territory is 30 times smaller.
The Financial Times, an international British daily newspaper, claims that
Guayas province in Ecuador, is one of the cities most affected by the COVID–19
pandemic in the world and the province of Ecuador most affected by this disease,
both in the number of infections and deaths. In fact, Civil Registry Office of
Guayas Province, reported that at least 8,600 people died for different reasons
during April (2020), while the average number of deaths in Guayas the past years
was 2,000 deaths, this means a death increase of 488%. In fact, the number of
diagnosed cases until December 9, 2020 was reported in 25,706 (see Figure 1).
According to the previous scenario and the global epidemic situation, it is im-
portant for the scientific community to be informed about the behaviour of the
disease in short and medium term. In the early stages of an infectious disease
outbreak, it is important to comprehend its transmission dynamics and to esti-
mate the changes in transmission over time and space. This can provide better
understanding about the epidemiological situation and determine whether social
control measures implemented are being effective or not. Such type of analysis
can forecast about potential future growth and guide us to design suitable control
interventions. To date, many researchers around the world have focused their in-
terests on understanding the transmission dynamics of COVID–19 disease using
mathematical and statistical models, see e.g., [3, 4, 12, 9, 10, 13].
In this study, we focus on the implications of two simple empirical functions and
probabilistic models, namely, Gompertz and Hill functions, to estimate the future
trends of COVID–19 disease in the Guayas province in Ecuador. We estimate
144
TWO SIMPLE PROBABILISTIC MODELS FOR COVID–19 3
the unknown parameters using public data of diagnosed cases, and compare the
results with dataset from February 29 to December 9 (2020). We also attempt to
forecast the evolution of number of patients until 800 days after first case. Finally,
we compare the precision and the results obtained with our proposed models in
order to make some useful recommendations on public health issues.
0
5000
10000
15000
20000
25000
30000
feb-29
mar- 06
mar- 12
mar- 18
mar- 24
mar- 30
apr 5
apr 11
apr 17
apr 23
apr 29
may- 04
may- 10
may- 16
may- 22
may- 28
jun-03
jun-09
jun-15
jun-21
jun-27
jul-0 3
jul-0 9
jul-1 5
jul-2 1
jul-2 7
agt 2
agt 8
agt 14
agt 20
agt 26
sept-02
sept-08
sept-14
sept-20
sept-26
oct-02
oct-08
oct-14
oct-20
oct-26
nov-01
nov-07
nov-13
nov-19
nov-25
dic-01
dic-07
COVID-19 diagnos ed cases in Guayas, Ecuado r
Figure 1. Diagnosed cases of COVID–19 in Guayas province,
Ecuador from February 29 to December 9, 2020.
2. Methods
Gompertz and Hill functions are two empirical mathematical models highly
reliable to provide robust estimations of total cases, active cases, deaths, peak and
stabilization times in epidemic dynamics. During the COVID–19 pandemic, these
models have showed very accurate estimates of cumulative cases in countries such
as Italy, Spain, Austria, Switzerland, Norway, South Korea, and other countries
that are already reaching the stabilization stage of the COVID-19 epidemic curve
[5, 11]. In this work, these models are used for the following two purposes: (a) to fit
models to data of COVID–19 diagnosed cases provided by the Ministerio de Salud
ublica of Ecuador for Guayas province from February 29 to December 9, 2020 and
(b) to predict its future behaviour at medium–term assuming that the quarantine
and confinement measures are kept during almost all 2020. Here, it is important
highlight that data are provided with variable error (See Figure 2), whose sources
could be several and from different nature, for instance, delays in testing, delays in
results, and low and unstable daily sampling rates, among others. We simulated
these errors for the model predictions by including a probabilistic error term using
a Gamma distribution. Gamma distribution was chosen among others (normal,
lognormal,Poisson, etc.) because it is only positive and allow high skewness
as those observed in data from infectious cases. Furthermore, it captures the
145
4 FELIPE BENAVIDES AND JHOANA P. ROMERO–LEITON
increasing error variance in observed data as the epidemic advances. This approach
allowed us to consider a reasonable measure of uncertainty in predictions, based
on the uncertainty of the reported data.
Let Nrepresenting the predicted number of diagnosed cases in a moment t.
Thus, the formulation of the Gompertz and Hill models is as follows
N(t) = αeβeRt +γ, Gompertz model
N(t) = d+αd
1+(t
c)R+γ, Hill model.
(2.1)
In above formulation, αrepresents the maximum number of diagnosed cases at
the end of the epidemic, βand crepresent parameters controlling the horizontal
displacement (time to stabilization) of the curve, Ris the epidemic growth rate
of the population , dis a correction factor and γis an error term from a Gamma
distribution. The probability density function of the Gamma distribution was
estimated to build the error terms
γ(x) = λeλx (λx)k1
Γ(k),
where kand λare parameters greater than zero and Γ is the gamma function.
Non–linear least square method was performed trough the Gauss–Newton algo-
rithm with 5000 iterations to estimate the values of the parameters involved in the
models. The Coefficient of Determination (R2), the Akaike Information Criteria
(AIC) and the Root Mean Squared Error (RMSE) were used as the metrics to
evaluate the models performance (See Table 1).
Parameters λand kwere set to 0.1 and 0.001 respectively and 10,000 random
simulations of gamma distributions with these fixed parameters were ran to build
a distribution of 10,000 error terms (γ) for each model. This was performed both
for the observed data (Feb 29 to December 9, 2020) as well as for the prediction
time window (800 days after the first case). Gompertz and Hill models were fitted
with every different γto produce a distribution of predictions aimed to capture
uncertainties in the future behaviour of cumulative diagnosed cases, as much as
possible. Therefore, final predictions of COVID–19 diagnosed cases 800 days after
the first case are presented with a region of uncertainty according with these errors
(See Figure 2).
3. Results
The results of the modelling procedures are shown in Table 1 and Figure 2.
The fit of both models to the observed data was good (95%–97%), and their
performances were similar during the first 100 days after the first case, whereas
after it, the predicted cases with the Gompertz model are greater with the Hill
model. This suggests that for the first stage of the epidemic both models could be
reliable in their predictions, agreeing with Yang et al [13]. The Gompertz model
predicted a minimum of 18,000 and a maximum of 30,000 cases in Guayas province
at the end of 2021, and the stabilization of the curve around days 300–400 after the
first case (December, 2020). The Hill model predicted a minimum of 16,000 and
146
TWO SIMPLE PROBABILISTIC MODELS FOR COVID–19 5
a maximum of 28,000 cases by the end of 2021 and the stabilization around days
400–500 after the first case (February, 2021). The predictions of the Gompertz
model probably underestimate the potential number of cases by December, 2020.
We must stress the fact that these results could dramatically change if increased
social interaction (even if small) led to an increase of parameter Rin the models.
Parameters and model
performance metrics Gompertz Hill
α2.24×104±235.9 -2.08×103±416.8
β2.94 ×104±416.8 —
R1.84 ×102±0.69 ×1031.32 ±0.077
c— 1.001×102±4.93
d— 2.94×104±1.07×103
R295% 97%
AIC 5017.95 4839.9
RMSE 1540.2 1124.4
Table 1. Parameters used for the Gompertz and Hill models.
These parameters were estimated by non–linear least squares
from the observed data of cumulative COVID–19 cases in Guayas
province (Ecuador) from February 29 to December 9 (2020) and
used for future predictions. Parameter values are shown ±stan-
dard errors.
4. Conclusions
In this study, we used two empirical functions to assess the predictive accuracy
of COVID–19 diagnosed cases in Guayas province (Ecuador) since February 29,
2020. By comparing the models outcomes with the diagnosed cases, it has been
observed that our estimated values have good correspondence with the diagnosed
cases. If the current pattern is going on then according to our estimate, following
the Hill function there will be a maximum of 28,000 positive cases by the end of
2021, and the stabilization of the cases will be achieved approximately on February
(2021). Following the Gompertz function there will be a maximum of 30,000
cases by the end of 2021, and the stabilization will be achieved approximately on
December (2020). Taking into account the previous scenario, it is recommended to
the government and health authorities of Guayas province, to maintain isolation
and quarantine measures, at least until the stabilization of the epidemic curve has
been achieved, that is, at least until February, 2021.
Although these models are simple, they are very helpful in designing control
measures; however, the results of this work are conditioned to the quality and
truthfulness of the epidemic data from the province. The initial conditions and
the parameter regions may also affect the results significantly.
147
6 FELIPE BENAVIDES AND JHOANA P. ROMERO–LEITON
Gompertz
Hill
Observed data
Figure 2. Results of the Gompertz and Hill models for predict-
ing COVID–19 cases in Guayas province (Ecuador). Shaded ar-
eas represent the uncertainty in predictions according to Gamma
error terms. The vertical black dotted lines represent the stabi-
lization for each model.
Susceptibles
S(t)
Infectious
I(t)
Recovereds
R(t)
Births
Contacts
Natural deaths
Natural deaths
Deaths due to Covid-19
Recoveries
Natural deaths
Prevention
strategies
Figure 3. Compartmental SIR–type model for COVID–19 disease.
Since the epidemic process is compartmental (see e.g., Figure 3), in addition to
our two functions, other epidemic models such as differential equations and time–
delayed equations should be considered in future researches to cope with problems
such like control strategies, the low testing rates which may cause underestimation
148
TWO SIMPLE PROBABILISTIC MODELS FOR COVID–19 7
of cases and the delays from infection to diagnose (10–15 days in average). For
instance, it would be interesting to contrast the predictions of this research with
the predictions given by a ODEs system with quarantine (ν) as a control strategy
such as
dS(t)
dt = Λ βI(t)S(t)
a+νI(t)µS(t)
dI(t)
dt =βI(t)S(t)
a+νI(t)(α+µ+δ)I(t)
dR(t)
dt =αI(t)µR(t)
(S(0), I(0), R(0)) = (S0, I0, R0),
where N(t) represents the total population of Guayas at time t, which is partitioned
into susceptible (S), infectious (I) and recovered (R) individuals at time t, and
(S(0), I(0), R(0)) = (S0, I0, R0) is the initial state of population just as the first
infected individual enters to the country (February 29, 2020), and other parameters
involved in the model (Λ , µ,α,β,δ,aand ν) represent rates as born, death,
recovery, transmission among others.
Acknowledgments. The authors thank the support of Fundaci´on Ceiba,
Colombia.
References
1. Johns Hopkins Coronavirus Resource Center, https://coronavirus.jhu.edu/, 2020, Acces 05-
10-2020.
2. Worldometer Coronavirus Updates, https://www.worldometers.info/, 2020, Acces 05-10-
2020.
3. Tian-Mu Chen, Jia Rui, Qiu-Peng Wang, Ze-Yu Zhao, Jing-An Cui, and Ling Yin, A math-
ematical model for simulating the phase–based transmissibility of a novel coronavirus, Infec-
tious Diseases of Poverty 9(2020), no. 1, 1–8.
4. Sunhwa Choi and Moran Ki, Estimating the reproductive number and the outbreak size
of novel coronavirus disease (COVID–19) using mathematical model in republic of korea,
Epidemiology and Health (2020), 123–145.
5. John L Haybittle, The use of the gompertz function to relate changes in life expectancy
to the standardized mortality ratio, International journal of epidemiology 27 (1998), no. 5,
885–889.
6. Laura D´ıez Izquierdo et al., Informe t´ecnico nuevo coronavirus 2019–ncov, Ph.D. thesis,
Instituto de Salud Carlos III, 2020.
7. Catharine I Paules, Hilary D Marston, and Anthony S Fauci, Coronavirus infections more
than just the common cold, Jama 323 (2020), no. 8, 707–708.
8. LJ Saif, Animal coronavirus vaccines: lessons for SARS., Developments in biologicals 119
(2004), 129–140.
9. Pedro V Savi, Marcelo A Savi, and Beatriz Borges, A mathematical description of the
dynamics of coronavirus disease (COVID–10): A case study of Brazil, arXiv preprint
arXiv:2004.03495 (2020).
10. Amjad S Shaikh, Iqbal N Shaikh, and Kottakkaran Sooppy Nisar, A mathematical model
of COVID–19 using fractional derivative: Outbreak in india with dynamics of transmission
and control, Preprints (2020).
149
8 FELIPE BENAVIDES AND JHOANA P. ROMERO–LEITON
11. Kathleen MC Tjørve and Even Tjørve, The use of gompertz models in growth analyses,
and new gompertz–model approach: An addition to the unified–richards family, PloS one 12
(2017), no. 6.
12. Chayu Yang and Jin Wang, A mathematical model for the novel coronavirus epidemic in
wuhan, china, Mathematical Biosciences and Engineering 17 (2020), no. 3, 2708–2724.
13. Wuyue Yang, Dongyan Zhang, Liangrong Peng, Changjing Zhuge, and Liu Hong, Rational
evaluation of various epidemic models based on the covid–19 data of china, arXiv preprint
arXiv:2003.05666 (2020).
Instituto de Estudios del Pac
´
ıfico, Universidad Nacional de Colombia, Colombia
Email address:pipeben@gmail.com
Facultad de Ingenier
´
ıa, Universidad Cesmag, Colombia
Email address:jpatirom3@gmail.com
150
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Since the first case of 2019 novel coronavirus disease (COVID-19) detected on 30 January, 2020, in India, the number of cases rapidly increased to 3819 cases including 106 deaths as of 5 April, 2020. Taking this into account, in the present work, we have analysed a Bats-Hosts-Reservoir-People transmission fractional-order COVID-19 model for simulating the potential transmission with the thought of individual response and control measures by the government. The real data available about number of infected cases from 14 March, 2000 to 26 March, 2020 is analysed and, accordingly, various parameters of the model are estimated or fitted. The Picard successive approximation technique and Banach's fixed point theory have been used for verification of the existence and stability criteria of the model. Further, we conduct stability analysis for both disease-free and endemic equilibrium states. On the basis of sensitivity analysis and dynamics of the threshold parameter, we estimate the effectiveness of preventive measures, predicting future outbreaks and potential control strategies of the disease using the proposed model. Numerical computations are carried out utilising the iterative Laplace transform method and comparative study of different fractional differential operators is done. The impacts of various biological parameters on transmission dynamics of COVID-19 is investigated. Finally, we illustrate the obtained results graphically.
Article
Full-text available
Objectives: Since the first novel coronavirus disease(COVID-19) patient was diagnosed on 20-Jan, about 30 patients were diagnosed in Korea until 17-Feb. However, 5,298 more patient were confirmed until 4-Mar. The purpose is to estimate and evaluate the effectiveness of preventive measures using mathematical modeling. Methods: Deterministic mathematical model(SEIHR) has been established to suit the Korean outbreak. The number of confirmed patients in Daegu and North Gyeongsang Province(Daegu/NGP), the main area of outbreak, were used. The first patient's symptom onset date was assumed on 22-Jan. We estimate the reproduction number(R), and the effect of preventive measures, assuming that the effect has been shown from 29-Feb. or 5-Mar. Results: The estimated R in Hubei Province was 4.2655, while the estimated initial R in Korea was 0.5555, but later in Daegu/NGP, the value was between 3.4721 and 3.5428. When the transmission period decreases from 4 days to 2 days, the outbreak finished early, but the peak of the epidemic has increased, and the total number of patients has not changed much. If transmission rate decreases about 90% or 99%, the outbreak finished early, and the size of the peak and the total number of patients also decreased. Conclusion: To early end of the COVID-19 epidemic, efforts to reduce the spread of the virus such as social distancing and mask wearing are absolutely crucial with the participation of the public, along with the policy of reducing the transmission period by finding and isolating patients as quickly as possible through efforts by the quarantine authorities.
Article
Full-text available
Background: As reported by the World Health Organization, a novel coronavirus (2019-nCoV) was identified as the causative virus of Wuhan pneumonia of unknown etiology by Chinese authorities on 7 January, 2020. The virus was named as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) by International Committee on Taxonomy of Viruses on 11 February, 2020. This study aimed to develop a mathematical model for calculating the transmissibility of the virus. Methods: In this study, we developed a Bats-Hosts-Reservoir-People transmission network model for simulating the potential transmission from the infection source (probably be bats) to the human infection. Since the Bats-Hosts-Reservoir network was hard to explore clearly and public concerns were focusing on the transmission from Huanan Seafood Wholesale Market (reservoir) to people, we simplified the model as Reservoir-People (RP) transmission network model. The next generation matrix approach was adopted to calculate the basic reproduction number (R0) from the RP model to assess the transmissibility of the SARS-CoV-2. Results: The value of R0 was estimated of 2.30 from reservoir to person and 3.58 from person to person which means that the expected number of secondary infections that result from introducing a single infected individual into an otherwise susceptible population was 3.58. Conclusions: Our model showed that the transmissibility of SARS-CoV-2 was higher than the Middle East respiratory syndrome in the Middle East countries, similar to severe acute respiratory syndrome, but lower than MERS in the Republic of Korea.
Article
Change in life expectancy may be more readily appreciated by a lay person as a measure of risk than the standardized mortality ratio (SMR). The linear increase in the logarithm of the age-specific mortality rates with age (the Gompertz function) is used to deduce formulae connecting SMR with change in life expectancy. Their validity is checked by a comparison between the 1992 and 1952 mortality data for England and Wales, and between smokers and non-smokers in the American Cancer Society's second Cancer Prevention Study. It is shown that the Gompertz function is a good fit to mortality data for England and Wales from age 30 years upwards. Changes in life expectancy at ages 15, 25, 45 and 65 are presented for values of SMR from 0.5 to 3. A very simple formula connecting the two is valid at ages 15 and 25, and provides a reasonable approximation at age 45. The Gompertz relationship can be used to calculate the change in life expectancy corresponding to a particular SMR over a greater range than have previous methods, and, although subject to some uncertainties, can provide a quick method of judging the change in life expectancy that is associated with a given SMR value.
Article
Severe acute respiratory syndrome (SARS) emerged in China and spread globally as a human pandemic. It is caused by a new coronavirus (CoV) of suspect animal origin. The emergence of SARS stunned medical scientists, but veterinary virologists had previously recognized CoVs as causing fatal respiratory or enteric disease in animals with interspecies transmission and wildlife reservoirs. Because of its public health impact, major efforts are focused on development of SARS vaccines. Occurrence of CoV disease at mucosal surfaces necessitates the stimulation of local immunity, having an impact on the vaccine type, delivery and adjuvant needed to achieve mucosal immunity. Such immunity is often short-lived, requires frequent boosting and may not prevent re-infection, all factors complicating CoV vaccine design. SARS vaccine efforts should be enhanced by understanding the correlates of protection and reasons for the success or failure of animal CoV vaccines. This review will focus on studies of immunity and protection in swine to the enteric CoV, transmissible gastroenteritis (TGEV) versus the respiratory variant, porcine respiratory CoV (PRCV), comparing live, inactivated and subunit vaccines, various vaccine vectors, routes and adjuvants. In addition avian infectious bronchitis CoV (IBV) vaccines targeted for protection of the upper respiratory tract of chickens are discussed. Unfortunately, despite long-term efforts, effective vaccines to prevent enteric CoV infections remain elusive, and generally live, but not killed vaccines, have induced the most consistent protection against animal CoVs. Confirmation of the pathogenesis of SARS in humans or animals models that mimic SARS may further aid in vaccine design and evaluation.
Informe técnico nuevo coronavirus 2019-ncov
  • Laura Díez Izquierdo
Laura Díez Izquierdo et al., Informe técnico nuevo coronavirus 2019-ncov, Ph.D. thesis, Instituto de Salud Carlos III, 2020.
Coronavirus infections more than just the common cold
  • Hilary D Catharine I Paules
  • Anthony S Marston
  • Fauci
Catharine I Paules, Hilary D Marston, and Anthony S Fauci, Coronavirus infections more than just the common cold, Jama 323 (2020), no. 8, 707-708.
  • V Pedro
  • Marcelo A Savi
  • Beatriz Savi
  • Borges
Pedro V Savi, Marcelo A Savi, and Beatriz Borges, A mathematical description of the dynamics of coronavirus disease (COVID-10): A case study of Brazil, arXiv preprint arXiv:2004.03495 (2020).