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# Prediction of the Epidemic Peak of Coronavirus Disease in Japan, 2020

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The first case of coronavirus disease 2019 (COVID-19) in Japan was reported on 15 January 2020 and the number of reported cases has increased day by day. The purpose of this study is to give a prediction of the epidemic peak for COVID-19 in Japan by using the real-time data from 15 January to 29 February 2020. Taking into account the uncertainty due to the incomplete identification of infective population, we apply the well-known SEIR compartmental model for the prediction. By using a least-square-based method with Poisson noise, we estimate that the basic reproduction number for the epidemic in Japan is R 0 = 2.6 (95%CI, 2.4-2.8) and the epidemic peak could possibly reach the early-middle summer. In addition, we obtain the following epidemiological insights: (1) the essential epidemic size is less likely to be affected by the rate of identification of the actual infective population; (2) the intervention has a positive effect on the delay of the epidemic peak; (3) intervention over a relatively long period is needed to effectively reduce the final epidemic size.
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Journal of
Clinical Medicine
Article
Prediction of the Epidemic Peak of Coronavirus
Disease in Japan, 2020
Toshikazu Kuniya
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan;
tkuniya@port.kobe-u.ac.jp
Received: 24 February 2020; Accepted: 10 March 2020; Published: 13 March 2020


Abstract:
The ﬁrst case of coronavirus disease 2019 (COVID-19) in Japan was reported on 15 January
2020 and the number of reported cases has increased day by day. The purpose of this study is to give a
prediction of the epidemic peak for COVID-19 in Japan by using the real-time data from 15 January to
29 February 2020. Taking into account the uncertainty due to the incomplete identiﬁcation of infective
population, we apply the well-known SEIR compartmental model for the prediction. By using a
least-square-based method with Poisson noise, we estimate that the basic reproduction number for
the epidemic in Japan is
R0=
2.6 (95%CI, 2.4–2.8) and the epidemic peak could possibly reach the
early-middle summer. In addition, we obtain the following epidemiological insights: (1) the essential
epidemic size is less likely to be affected by the rate of identiﬁcation of the actual infective population;
(2) the intervention has a positive effect on the delay of the epidemic peak; (3) intervention over a
relatively long period is needed to effectively reduce the ﬁnal epidemic size.
Keywords: COVID-19; SEIR compartmental model; basic reproduction number
1. Introduction
In December 2019, the ﬁrst case of respiratory disease caused by a novel coronavirus was
identiﬁed in Wuhan City, Hubei Province, China. The outbreak of the disease is ongoing worldwide
and the World Health Organization named it coronavirus disease 2019 (COVID-19) on 11 February
2020 [
1
]. In Japan, the ﬁrst case was reported on 15 January 2020 and the number of reported
laboratory-conﬁrmed COVID-19 cases per week has increased day by day (see Table 1).
Table 1. Number of newly reported COVID-19 cases in Japan until 1 March 2020 [2].
Week Number of Newly Reported Cases Number of Accumulated Cases
12 January–18 January 1 1
19 January–25 January 2 3
26 January–1 February 14 17
2 February–8 February 8 25
9 February–16 February 28 53
17 February–23 February 79 132
24 February–1 March 107 239
As seen in Table 1, the number of newly reported cases per week has increased and a serious
outbreak in Japan is a realistic outcome. One of the greatest public concerns is whether the epidemic
continues until summer so that it affects the Summer Olympics, which is planned to be held in
Tokyo. The purpose of this study is to give a prediction of the epidemic peak of COVID-19 in Japan,
which might help us to act appropriately to reduce the epidemic risk.
J. Clin. Med. 2020,9, 789; doi:10.3390/jcm9030789 www.mdpi.com/journal/jcm
J. Clin. Med. 2020,9, 789 2 of 7
The epidemic data as shown in Table 1would have mainly twofold uncertainty. The ﬁrst one is
due to the fact that asymptomatic infected people could spread the infection [
3
]. The second one is
due to the lack of opportunity for the diagnostic test as sufﬁciently simple diagnostic test kits have
not been developed yet and the diagnosis in the early stage in Japan was mainly restricted to people
who visited Wuhan [
4
]. In this study, taking into account such uncertainty, we apply a simple and
well-known mathematical model for the prediction. More precisely, we assume that only
p
(0
<p
1)
fraction of infective individuals can be identiﬁed by diagnosis.
2. Methods
2.1. Model
We apply the following well-known SEIR compartmental model (see, e.g., [
5
]) for the prediction.
(S0(t) = βS(t)I(t),E0(t) = βS(t)I(t)εE(t),
I0(t) = εE(t)γI(t),R0(t) = γI(t),t>0, (1)
where
S(t)
,
E(t)
,
I(t)
and
R(t)
denote the susceptible, exposed, infective and removed populations at
time
t
, respectively.
β
,
ε
and
γ
denote the infection rate, the onset rate and the removal rate, respectively.
Note that 1
/ε
and 1
/γ
imply the average incubation period and the average infectious period, respectively.
Let the unit time be 1 day. Based on the previous studies [
6
,
7
], we fix 1
/ε=
5, and thus,
ε=
0.2 and
γ=
0.1, respectively. We fix
S+E+I+R
to be 1 so that each population implies the proportion
to the total population. We assume that one infective person is identified at time
t=
0 among total
N=1.26 ×108number of people in Japan [8]. That is, Y(0) = pI(0)×1.26 ×108=1, where
Y(t) = p I(t)×1.26 ×108
denotes the number of infective individuals who are identiﬁed at time
t
. Thus, we obtain
I(
0
) =
1
/(p×
1.26
×
10
8)
. We assume that there is no exposed and removed populations at
t=
0, that is,
E(0) = R(0) = 0, and hence,
S(0) = 1E(0)I(0)R(0) = 11
p×1.26 ×108.
It was estimated in [9] that 77 cases were conﬁrmed among the possible 940 infected population
in February in Hokkaido, Japan. Based on this report, we assume that
p
ranges from 0.01 to 0.1.
The basic reproduction number
R0
, which means the expected value of secondary cases produced by
one infective individual [
10
], is calculated as the maximum eigenvalue of the next generation matrix
FV1[11], where
F="0βS(0)
0 0 #,V="ε0
ε γ #.
Thus, we obtain
R0=βS(0)
γ=β
γ11
p×1.26 ×108. (2)
2.2. Sensitivity of the Basic Reproduction Number
It is obvious that the basic reproduction number
R0
is independent from the onset rate
ε
.
The sensitivity of R0to other parameters β,γand pare calculated as follows:
Aβ=β
R0
R0
∂β =1, Aγ=γ
R0
R0
∂γ =1, Ap=p
R0
R0
p=1
p×1.26 ×1081, (3)
J. Clin. Med. 2020,9, 789 3 of 7
where
Aβ
,
Aγ
and
Ap
denote the normalized sensitivity indexes with respect to
β
,
γ
and
p
, respectively.
We see from Equation
(3)
that the
k
time’s increase in
β
(resp.
γ
) results in the
k
(resp.
k1
) time’s
increase in
R0
. In particular, we see from the third equation in Equation
(3)
that
Ap
0 if
p
1.0
×
10
6
.
This implies that the identiﬁcation rate pin a realistic range almost does not affect the size of R0.
2.3. Estimation of the Infection Rate
Let
y(t)
,
t=
0, 1,
. . .
, 45 be the number of daily reported cases of COVID-19 in Japan from 15
January (t=0) to 29 February (t=45) 2020. We perform the following least-square-based procedure
with Poisson noise to estimate the infection rate β.
Description 1.
(P1) Fix β>0and calculate the numerical value of Y(t), t =0, 1, . . . , 45 by using model Equation (1).
(P2) Calculate
˜
Y(t) = Y(t) + qY(t)e(t) = Y(t) + (Poisson noise),t=0, 1, . . . , 45,
where
e(t)
,
t=
0, 1,
. . .
, 45 denote random variables from a normal distribution with mean zero and variance 1[
12
].
(P3) Calculate J (β) = 45
t=0[y(t)˜
Y(t)]2.
(P4) Run (P1)–(P3) for 0.2 β0.4 and ﬁnd βsuch that J(β) = min0.2β0.4 J(β).
(P5) Repeat (P1)–(P4) 10, 000 times and obtain the distribution of β.
(P6) Approximate the distribution of βby a normal distribution and obtain a 95% conﬁdence interval.
Note that for the reason stated above, the value of 0.01
p
0.1 does not affect this estimation
procedure. By (P1)–(P6), we obtain a normal distribution with mean 0.26 and standard derivation 0.01.
Thus, we obtain an estimation of
β
as 0.26 (95%CI, 0.24–0.28) (see Figure 1). Moreover, by Equation
(2)
,
we obtain an estimation of R0as 2.6 (95%CI, 2.4–2.8) (see Table 2).
Figure 1.
Comparison of
Y(t)
with the estimated infection rate
β
and the number of daily reported
cases of COVID-19 in Japan from 15 January (t=0) to 29 February (t=45).
Table 2. Parameter values for model Equation (1).
Parameter Description Value Reference
βInfection rate 0.26 (95%CI, 0.24–0.28) Estimated
R0Basic reproduction number 2.6 (95%CI, 2.4–2.8) Estimated
εOnset rate 0.2 [6]
γRemoval rate 0.1 [7]
NTotal population in Japan 1.26 ×108[8]
pIdentiﬁcation rate 0.01–0.1 [9]
J. Clin. Med. 2020,9, 789 4 of 7
3. Results
3.1. Peak Prediction
We deﬁne the epidemic peak
t
by the time such that
Y
attains its maximum in 1 year, that is,
Y(t) = max0t365 Y(t)
. We ﬁrst set
p=
0.1. In this case, we obtain the following ﬁgure on the long
time behavior of Y(t)for β=0.28, 0.26 and 0.24.
We see from Figure 2that the estimated epidemic peak is
t=
208 (95%CI, 191–229). That is,
starting from 15 January (
t=
0), the estimated epidemic peak is 10 August (
t=
208) and the uncertainty
range is from 24 July (t=191) to 31 August (t=229).
(a)β=0.28 (t=191) (b)β=0.26 (t=208) (c)β=0.24 (t=229)
Figure 2.
Time variation of the number
Y(t)
of infective individuals who are identiﬁed at time
t
(0 t365) for p=0.1. The dot lines represent the epidemic peak t.
We next set p=0.01. In this case, we obtain the following ﬁgure.
We see from Figure 3that the estimated epidemic peak is
t=
179 (95%CI, 165–197). That is,
starting from January 15 (t=0), the estimated epidemic peak is July 12 (t=179) and the uncertainty
range is from June 28 (
t=
165) to July 30 (
t=
197). In contrast to
R0
, the epidemic peak and the
(apparent) epidemic size are sensitive to the identiﬁcation rate
p
. Note that the essential epidemic size,
which is characterized by R0, is almost the same in both of p=0.1 and p=0.01.
(a)β=0.28 (t=165) (b)β=0.26 (t=179) (c)β=0.24 (t=197)
Figure 3.
Time variation of the number
Y(t)
of infective individuals who are identiﬁed at time
t
(0 t365) for p=0.01. The dot lines represent the epidemic peak t.
3.2. Possible Effect of Intervention
We next discuss the effect of intervention. In Japan, school closure has started in almost all
prefectures from the beginning of March [
13
] and many social events have been cancelled off to reduce
the contact risk. However, the exact effect of such social efforts is unclear and might be limited as the
proportion of young people to the whole infected people of COVID-19 seems not so high (2% of 72, 314
reported cases in China [
14
]). In this simulation, we assume that such social efforts successfully reduce
the infection rate
β=
0.26 to 75% during a period from 1 March (
t=
46) to a planned day (
t=T
47).
In what follows, we ﬁx p=0.01.
J. Clin. Med. 2020,9, 789 5 of 7
First, we set
T=
77, that is, the intervention is carried out for 1 month (from 1 March to 1 April).
In this case, the epidemic peak
t
is delayed from 179 (12 July) to 190 (23 July). However, the epidemic
size is almost the same. On the other hand, if
T=
220, that is, the intervention is carried out for 6
months (from 1 March to 1 September), then the epidemic peak
t
is delayed from 179 (12 July) to 243
(14 September) and the epidemic size is effectively reduced (see Figure 4).
Figure 4.
Time variation of the number
Y(t)
of infective individuals who are identiﬁed at time
t
(0
t
365) for
p=
0.01 and no intervention, 1 month intervention (
T=
77) and 6 months
intervention (T=220). The dot lines represent the epidemic peak.
More precisely, we see from Figure 5a that the epidemic peak
t
is delayed almost linearly for
47 T239 and ﬁxed to t=237 for T240.
(a) (b)
Figure 5.
The relation between the planned ﬁnal day for intervention
T
and (
a
) the epidemic peak
t
;
(b) the number of accumulated cases at time t=365: pR(365)×1.26 ×108.
This implies that the intervention has a positive effect on the delay of the epidemic peak, which
would contribute to improve the medical environment utilizing the extra time period. On the other
hand, we see from Figure 5b that the number of accumulated cases at
t=
365, which is calculated as
pR(
365
)×
1.26
×
10
8
, is monotonically decreasing and converges to 0.99
×
10
6
as
T
increases. However,
it almost does not change for small
T
180. This implies that the intervention over a relatively long
duration is required to effectively reduce the ﬁnal epidemic size.
4. Discussion
In this study, by applying the SEIR compartmental model to the daily reported cases of COVID-19
in Japan from 15 January to 29 February, we have estimated that the basic reproduction number
R0
is
2.6 (95%CI, 2.4–2.8) and the epidemic peak could possibly reach the early-middle summer. Of course,
this kind of long range peak prediction would contain the essential uncertainty due to the possibility
of some big changes in the social and natural (climate) situations. Nevertheless, our result suggests
that the epidemic of COVID-19 in Japan would not end so quickly. This might be consistent with the
WHO’s statement on 6 March 2020 that it is a false hope that COVID-19 will disappear in the summer
like the ﬂu [15].
J. Clin. Med. 2020,9, 789 6 of 7
The estimated value of the basic reproduction number
R0
in this study is not so different from
early estimations: 2.6 (95%CI, 1.5–3.5) [
16
], 2.90 (95%CI, 2.32–3.63) [
17
], 3.11 (95%CI, 2.39–4.13) [
18
],
3.58 (95%CI, 2.89–4.39) [
19
] and 3.28 (average of estimations in 12 studies) [
20
study, we have obtained the following epidemiological insights:
The essential epidemic size, which is characterized by
R0
, would not be affected by the
identiﬁcation rate pin a realistic parameter range 0.01–0.1, in particular, p1.0 ×106.
The intervention exactly has a positive effect on the delay of the epidemic peak, which would
contribute to improve the medical environment utilizing the extra time period.
Intervention over a relatively long period is needed to effectively reduce the ﬁnal epidemic size.
The ﬁrst statement implies that underestimation of the actual infective population would not
contribute to the reduction of the essential epidemic risk. Correct information based on an adequate
diagnosis system would be desired for people to act appropriately.
Funding:
This research was funded by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant
number 19K14594.
Acknowledgments:
The author would like to thank the associate editor and the anonymous reviewers for their
Conﬂicts of Interest: The author declares no conﬂict of interest.
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... A natural inclination for modelers attempting to predict demand for COVID-19 ICU beds consists in using variations of the so-called susceptible-infectious-recovered (SIR) model (Keeling and Danon, 2009;Papo et al., 2020;Rahimi et al., 2021) or the susceptible-exposed-infectious-removed (SEIR) model (Aron and Schwartz, 1984;Kuniya, 2020;Liu et al., 2021;Rahimi et al., 2021;Tagliazucchi et al., 2020). However, as different authors have remarked (Gitto et al., 2021;Manca et al., 2020), while SIR/SEIR-type models can be useful for running parametric predictive scenarios, they are based on many assumptions and hypotheses that are quite sensitive to the selection of proper values of the adaptive parameters and functional description, which might not be as appropriate when real-time data are incomplete. ...
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In Chile, due to the explosive increase of new Coronavirus disease 2019 (COVID‐19) cases during the first part of 2021, the ability of health services to accommodate new incoming cases was jeopardized. It has become necessary to be able to manage intensive care unit (ICU) capacity, and for this purpose, monitoring both the evolution of new cases and the demand for ICU beds has become urgent. This paper presents short‐term forecast models for the number of new cases and the number of COVID‐19 patients admitted to ICUs in the Metropolitan Region in Chile.
... The model can handle disordered raw data better by converting it to sequential data using a differential equation. The Grey model has been developed and applied across many fields such as medicine (Kuniya [9], Saxena [10], Chutiman et al. [11]), agriculture (Busababodhin and Chiangpradit [12]), environment and water allocation (Shi [13], Shao et al. [14], Shirisha [15]), ecology (Chen and Wang [16]), meteorology (Salookolaei [17]), and engineering (Zhou et al. [18], Wang et al. [19], Wu et al. [20], Shaheen et al. [21], Liu and Wu [22]). Therefore, the researcher would like to present the Grey Theory to develop a model for forecasting daily maximum rainfall affected by tropical cyclones in Northeastern Thailand during August and September so that relevant agencies can use it as a guideline for planning water management in the region efficiently. ...
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This research aims to develop a model for forecasting daily maximum rainfall caused by tropical cyclones over Northeastern Thailand during August and September 2022 and 2023. In the past, the ARIMA or ARIMAX method to forecast rainfall was used in research. It is a short-term rainfall prediction. In this research, the Grey Theory was applied as it is an approach that manages limited and discrete data for long-term forecasting. The Grey Theory has never been used to forecast rainfall that is affected by tropical cyclones in Northeastern Thailand. The Grey model GM(1,1) was analyzed with the highest daily cumulative rainfall data during the August and September tropical cyclones of the years 2018–2021, from the weather stations in Northeastern Thailand in 17 provinces. The results showed that in August 2022 and 2023, only Nong Bua Lamphu province had a highest daily rainfall forecast of over 100 mm, while the other provinces had values of less than 70 mm. For September 2022 and 2023, there were five provinces with the highest daily rainfall forecast of over 100 mm. The average of mean absolute percentage error (MAPE) of the maximum rainfall forecast model in August and September is approximately 20 percent; therefore, the model can be applied in real scenarios. Doi: 10.28991/CEJ-2022-08-08-02 Full Text: PDF
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The contact and interaction of human is considered to be one of the important factors affecting the epidemic transmission, and it is critical to model the heterogeneity of individual activities in epidemiological risk assessment. In digital society, massive data makes it possible to implement this idea on large scale. Here, we use the mobile phone signaling to track the users’ trajectories and construct contact network to describe the topology of daily contact between individuals dynamically. We show the spatiotemporal contact features of about 7.5 million mobile phone users during the outbreak of COVID-19 in Shanghai, China. Furthermore, the individual feature matrix extracted from contact network enables us to carry out the extreme event learning and predict the regional transmission risk, which can be further decomposed into the risk due to the inflow of people from epidemic hot zones and the risk due to people close contacts within the observing area. This method is much more flexible and adaptive, and can be taken as one of the epidemic precautions before the large-scale outbreak with high efficiency and low cost.
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By proposing a varying coefficient Susceptible-Infected-Removal model (vSIR), we track the epidemic of COVID-19 in 30 provinces in China and 15 cities in Hubei province, the epicenter of the outbreak. It is found that the spread of COVID-19 has been significantly slowing down within the two weeks from January 27 to February 10th with 87.0% and 84.3% reductions in the reproduction number R0 among the 30 provinces and 15 Hubei cities, respectively. This suggests the extreme control measures implemented since January 23, which include cutting off Wuhan and many other cities and towns, a great public awareness and high level of self isolation at home, have contributed to a substantial decline in the reproductivity of the COVID-19 in China. We predict that Hubei province will reach its peak between February 20 and 22, 2020, and if the removal rate can be increased to 0.1, the epidemic outside Hubei province will end in May 2020, and inside Hubei in early June.
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Teaser: Our review found the average R0 for 2019-nCoV to be 3.28, which exceeds WHO estimates of 1.4 to 2.5.
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Background: Since December 29, 2019, pneumonia infection with 2019-nCoV has rapidly spread out from Wuhan, Hubei Province, China to most others provinces and other counties. However, the transmission dynamics of 2019-nCoV remain unclear. Methods: Data of confirmed 2019-nCoV cases before January 23, 2020 were collected from medical records, epidemiological investigations or official websites. Data of severe acute respiratory syndrome (SARS) cases in Guangdong Province during 2002-2003 were obtained from Guangdong Provincial Center for Disease Control and Prevention (GDCDC). Exponential Growth (EG) and maximum likelihood estimation (ML) were applied to estimate the reproductive number (R) of 2019-nCoV and SARS. Findings: As of January 23, 2020, a total of 830 confirmed 2019-nCoV cases were identified across China, and 9 cases were reported overseas. The average incubation duration of 2019-nCoV infection was 4.8days. The average period from onset of symptoms to isolation of 2019-nCoV and SARS cases were 2.9 and 4.2 days, respectively. The R values of 2019-nCoV were 2.90 (95%CI: 2.32-3.63) and 2.92 (95%CI: 2.28-3.67) estimated using EG and ML respectively, while the corresponding R values of SARS-CoV were 1.77 (95%CI: 1.37-2.27) and 1.85 (95%CI: 1.32-2.49). We observe a decreasing trend of the period from onset to isolation and R values of both 2019-nCoV and SARS-CoV. Interpretation: The 2019-nCoV may have a higher pandemic risk than SARS broken out in 2003. The implemented public-health efforts have significantly decreased the pandemic risk of 2019-nCoV. However, more rigorous control and prevention strategies and measures to contain its further spread.
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In December 2019, a novel coronavirus (2019-nCoV) is thought to have emerged into the human population in Wuhan, China. The number of identified cases in Wuhan has increased rapidly since, and cases have been identified in other Chinese cities and other countries (as of 23 January 2020). We fitted a transmission model to reported case information up to 21 January to estimate key epidemiological measures, and to predict the possible course of the epidemic, as the potential impact of travel restrictions into and from Wuhan. We estimate the basic reproduction number of the infection (R_0) to be 3.8 (95% confidence interval, 3.6-4.0), indicating that 72-75% of transmissions must be prevented by control measures for infections to stop increasing. We estimate that only 5.1% (95%CI, 4.8-5.5) of infections in Wuhan are identified, and by 21 January a total of 11,341 people (prediction interval, 9,217-14,245) had been infected in Wuhan since the start of the year. Should the epidemic continue unabated in Wuhan, we predict the epidemic in Wuhan will be substantially larger by 4 February (191,529 infections; prediction interval, 132,751-273,649), infection will be established in other Chinese cities, and importations to other countries will be more frequent. Our model suggests that travel restrictions from and to Wuhan city are unlikely to be effective in halting transmission across China; with a 99% effective reduction in travel, the size of the epidemic outside of Wuhan may only be reduced by 24.9% on 4 February. Our findings are critically dependent on the assumptions underpinning our model, and the timing and reporting of confirmed cases, and there is considerable uncertainty associated with the outbreak at this early stage. With these caveats in mind, our work suggests that a basic reproductive number for this 2019-nCoV outbreak is higher compared to other emergent coronaviruses, suggesting that containment or control of this pathogen may be substantially more difficult.
Book
This book is the first one in which basic demographic models are rigorously formulated by using modern age-structured population dynamics, extended to study real-world population problems. Age structure is a crucial factor in understanding population phenomena, and the essential ideas in demography and epidemiology cannot be understood without mathematical formulation; therefore, this book gives readers a robust mathematical introduction to human population studies. In the first part of the volume, classical demographic models such as the stable population model and its linear extensions, density-dependent nonlinear models, and pair-formation models are formulated by the McKendrick partial differential equation and are analyzed from a dynamical system point of view. In the second part, mathematical models for infectious diseases spreading at the population level are examined by using nonlinear differential equations and a renewal equation. Since an epidemic can be seen as a nonlinear renewal process of an infected population, this book will provide a natural unification point of view for demography and epidemiology. The well-known epidemic threshold principle is formulated by the basic reproduction number, which is also a most important key index in demography. The author develops a universal theory of the basic reproduction number in heterogeneous environments. By introducing the host age structure, epidemic models are developed into more realistic demographic formulations, which are essentially needed to attack urgent epidemiological control problems in the real world. © Springer Science+Business Media Singapore 2017. All rights are reserved.
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A precise definition of the basic reproduction number, R o , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R o < 1, then the disease free equilibrium is locally asymptotically stable; whereas if R o > 1, then it is unstable. Thus, R o is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super-and sub-threshold endemic equilibria for R o near one. This criterion, together with the definition of R o , is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.