1 Introduction

A major challenge in the application of epidemic models is the determination of model parameters.The challenge is made difficult by the lack of complete data available in the course of a typical epidemic disease. In the United States, typical epidemic data consist of Morbidity and Mortality Weekly Reports (MMWR) published by the Centers for Disease Control and Prevention. For many diseases, such as seasonal influenza, these data are very incomplete, and record only a small fraction of total cases. In general, reported cases are not representative of all cases, with respect to symptoms severity, infectiousness level, demographic classification, and social context. Identifying the fraction of unreported cases requires that the unknown becomes known. Mathematical models of an epidemic’s evolution provide a way to make known this unknown. The distinction can be incorporated into the model parameterization, and the known case data can be related to the number of unknown cases through the transmission dynamics in the model construction.

We assume that reported case data provide a time-sequential record of the number of cases reported in some specified formulation by public health officials. We assume that these cases are removed from the infected class and that they no longer have capability to infect susceptible individuals (due to mortality, hospitalization, isolation, social instruction, or other reasons). We assume that a typical susceptible individual has the potential to be infected by an infected individual independently of whether or not the infected individual will ultimately be reported.

We note several examples of case data for specific epidemics. In Biggerstaff and Balluz (2011) data for the 2010–2011 influenza season in the United States were obtained by landline telephone survey of approximately 90,000 people. Of these 8.9% of adults and 33.9% of children reported influenza like symptoms. In Reed et al. (2009) a statistical estimator model was used to estimate the ratio of unreported to reported cases for the H1N1 influenza epidemic in the United States from April to July 2009 as 79 to 1, and the ratio of total cases to confirmed cases as 140 to 1. In Shrestha et al. (2011) it is estimated that approximately 60.8 million cases occurred in the United States during this epidemic, with the population approximately 300 million during this time period.

In Sect. 2 we present the SIR model and formulate a system of equations connecting its parameters to reported case data. In Sect. 3 we solve this system of equations and identify the model parameters. In Sect. 4 we present three examples of epidemics with reported case data, and determine the fraction of their unreported cases. In Sect. 5 we discuss our results and further work.

2 The deterministic SIR epidemic model and the identification of its parameters

The first SIR deterministic epidemic models were developed by Bernoulli (1760), Dietz and Heesterbeek (2000, 2002), Ross (1910), Kermack and McKendrick (1927, 1932, 1933), and Macdonald (1957). Investigations of these models and their extensions have been developed by many researchers, including Anderson and May (1991), Bailey (1957), Brauer et al. (2008), Brauer and Castillo-Chavez (2000), Busenberg and Cooke (1993), Diekmann et al. (2013), Hethcote (1976, 2000), Keeling and Rohani (2007), Murray (1993), and Thieme (2003). The parameter identification problem for SIR model has been investigated by many researchers, including Andreasen (2011), Arino et al. (2007), Capistran et al. (2009), Chowell et al. (2003, 2007), Diekmann et al. (1990), Evans et al. (2005), Grassly and Fraser (2006), Hadeler (2011a, b), Hethcote (1996), Hooker et al. (2011), Hsieh et al. (2010), Lange (2016), Li and Lou (2016), Ma and Earn (2006), Mummert (2013), Pellis et al. (2009), Pollicott et al. (2012), Roeger et al. (2009), and Driessche and Watmough (2002).

The SIR model we analyze is the following: at time t, let S(t) be the number of susceptible, I(t) the number of infected, and R(t) the number of removed and permanently immune. The equations for the SIR model are

$$\begin{aligned} \frac{d}{d t} S(t)= & {} -\tau I(t) S(t) \end{aligned}$$
(2.1)
$$\begin{aligned} \frac{d}{d t} I(t)= & {} \tau I(t) S(t) - (\nu _1+\nu _2) I(t) \end{aligned}$$
(2.2)
$$\begin{aligned} \frac{d}{d t} R(t)= & {} (\nu _1+\nu _2) I(t) \end{aligned}$$
(2.3)

with initial conditions \( S(0)=S_0> 0, \, I(0)=I_0 > 0, \, R(0)=R_0 \ge 0.\) Here \(\tau \) is the transmission rate, \(\nu _1\) is the removal rate of reported infected individuals, and \(\nu _2\) is the removal rate of infected individuals due to all other unreported causes such as mortality, recovery, or other reasons. It is assumed that a reported case, known to medical care-givers or public health authorities, produces no further cases.

Although the analysis of (2.1), (2.2), (2.3) is well-known, we provide details here for the sake of completeness. First, S(t) and I(t) are nonnegative when \(S_0 > 0\) and \(I_0 > 0\), which implies S(t) is decreasing. The basic reproduction number \(\mathbb {R}_0\) is defined as \(\tau S_0 / (\nu _1 + \nu _2)\). If \(\mathbb {R}_0 < 1\), then I(t) is decreasing. If \(\mathbb {R}_0 > 1\), then I(t) is initially increasing and then decreasing. Add (2.1) and (2.2) to obtain

$$\begin{aligned} S(t) + I(t) + (\nu _1 + \nu _2) \int _0^t I(s) ds = S_0 + I_0, \end{aligned}$$
(2.4)

which implies \(I(t) < \infty \) for \(t \ge 0\), and \(\int _0^{\infty } I(t) dt < \infty .\) Also, (2.1) implies

$$\begin{aligned} \log \bigg (\frac{S(t)}{S_0}\bigg ) = - \tau \int _0^t I(s) ds \Longleftrightarrow S(t) = S_0 \exp \bigg (-\tau \int _0^t I(s) ds \bigg ). \end{aligned}$$
(2.5)

Since S(t) is decreasing and \(\int _0^{\infty } I(t) dt < \infty \) by (2.4), \(\lim _{t \rightarrow \infty } S(t) = S_{\infty } > 0\). Then (2.4) implies \(\lim _{t \rightarrow \infty } I(t) = 0\). Further, (2.4) and (2.5) imply

$$\begin{aligned} S_{\infty } = S_0 + I_0 + \frac{\nu _1 + \nu _2}{\tau } \log \bigg (\frac{S_{\infty }}{S_0} \bigg ). \end{aligned}$$
(2.6)

It is possible to use the data of cumulative reported cases to determine the total number of cases \(S(0) - S_{\infty }\) over the course of the epidemic, as well as the parameters \(\tau \), \(\nu _1\), and \(\nu _2\). The cumulative number of reported cases at time t is

$$\begin{aligned} CR(t) = \nu _1 \int _0^t I(s) ds = \frac{\nu _1}{\tau } \log \bigg (\frac{S_0}{S(t)} \bigg ), \end{aligned}$$

and the total cumulative number of reported cases at the end of the epidemic is

$$\begin{aligned} CR_{\infty } = \lim _{t \rightarrow \infty } CR(t) = \frac{\nu _1}{\tau } \log \bigg (\frac{S_0}{S_{\infty }} \bigg ), \end{aligned}$$
(2.7)

which implies

$$\begin{aligned} S_{\infty } = S_0 \exp \bigg [- \frac{\tau }{\nu _1} CR_{\infty }\bigg ]. \end{aligned}$$
(2.8)

Then (2.6) implies

$$\begin{aligned} S_0 \exp \bigg [- \frac{\tau }{\nu _1} CR_{\infty }\bigg ] = S_0 + I_0 - \frac{\nu _1 + \nu _2}{\nu _1} CR_{\infty }. \end{aligned}$$
(2.9)

The cumulative number of both reported and unreported cases at time t is \(C(t) = S_0+I_0 - S(t)\), and the cumulative number of unreported cases at time t is \(CU(t) = C(t) - CR(t)\). Since the total number of cases (the epidemic final size) is

$$\begin{aligned} CR_{\infty } + CU_{\infty } = S_0 + I_0 - S_{\infty }, \end{aligned}$$

the total number of unreported cases is

$$\begin{aligned} CU_{\infty } = S_0 + I_0 - S_{\infty } - CR_{\infty }. \end{aligned}$$

The epidemic attack ratio (defined as the fraction of the susceptible population that becomes infected) is

$$\begin{aligned} (S_0 + I_0 - S_{\infty }) / (S_0 + I_0) = (CR_{\infty } + CU_{\infty })/ (S_0+ I_0). \end{aligned}$$

By symmetry, the ratio of reported cases to unreported cases is

$$\begin{aligned} \frac{CR(t)}{CU(t)} = \frac{\nu _1}{\nu _2}, \, \, \text {since} \, \, CR(t)=\nu _1 \int _0^t I(s) ds \text { and } CU(t)=\nu _2 \int _0^t I(s) ds. \end{aligned}$$

Assume that the reported cases have a maximum at time tp, which we call the turning point. Since (2.2) implies \( \nu _1 I^{\prime }(tp) = \nu _1 (\tau S(tp) - (\nu _1+\nu _2)) I(tp) = 0\), we obtain \(S(tp) = (\nu _1+\nu _2) / \tau \). From (2.4 and (2.5) we obtain

$$\begin{aligned}&CR(tp) = \frac{\nu _1}{\tau } \log \bigg (\frac{\tau S_0}{\nu _1+\nu _2} \bigg ) \Longleftrightarrow \frac{\nu _1+\nu _2}{\tau }=S_0 \exp \bigg [- \frac{\tau }{\nu _1} CR(tp) \bigg ], \nonumber \\ \end{aligned}$$
(2.10)
$$\begin{aligned}&S_0 + I_0 - \bigg (\nu _1 I(tp)\bigg )\bigg ( \frac{1}{\nu _1}\bigg ) = \frac{\nu _1+ \nu _2}{\tau } \bigg (1 - \log \bigg (\frac{\nu _1+ \nu _2}{\tau S_0}\bigg ) \bigg ). \end{aligned}$$
(2.11)

3 Analysis of Eqs. (2.9), (2.10), (2.11)

The Eqs. (2.9), (2.10), (2.11) provide an algorithm for identifying the parameters \(S_0,\) \(I_0\) and \(\tau \), \(\nu _1\) and \(\nu _2\). Assume that \(tp, CR^{\prime }(tp), CR(tp)\), and \(CR_{\infty }\) are known. The algorithm is based on the following equation:

$$\begin{aligned} S_0 + I_0 = S_0 \exp \bigg [- \frac{\tau }{\nu _1} CR_{\infty }\bigg ]+ \frac{\tau }{\nu _1} S_0 \exp \bigg [- \frac{\tau }{\nu _1} CR(tp)\bigg ]CR_{\infty }. \end{aligned}$$
(3.1)

Equation (3.1) is obtained by using Eq. (2.10) to obtain

$$\begin{aligned} \nu _1+\nu _2 =\tau S_0 \exp \bigg (- \frac{\tau }{\nu _1} CR(tp) \bigg ), \end{aligned}$$

and then substituting \(\nu _1 + \nu _2\) into Eq. (2.9). Re-write (3.1) as

$$\begin{aligned} \exp \bigg [- \frac{\tau }{\nu _1} CR_{\infty }\bigg ] + \frac{\tau }{\nu _1} \exp \bigg [- \frac{\tau }{\nu _1} CR(tp)\bigg ]CR_{\infty } -1 = \frac{I_0}{S_0}. \end{aligned}$$
(3.2)

Equation (3.2) is equivalent to

$$\begin{aligned} F(X) =\dfrac{I_0}{S_0}, \, \text { where } F(X) = e^{- c X} + c \, X e^{ - r c X} - 1, \, X \ge 0, \end{aligned}$$
(3.3)

with \(X = \dfrac{\tau }{\nu _1}\), and \(c = CR_{\infty }\), and \(r = \dfrac{CR(tp)}{CR_{\infty }}.\)

Proposition 3.1

Let \(c>0\) and \(r\in (0,1)\). We have the following alternative:

  1. i)

    If \(2\, r \ge 1\), then \(F(X)\le 0, \forall X\ge 0.\)

  2. ii)

    If \(2\, r < 1\), then \(Y_{\max }:= \sup _{X \ge 0} F(X)>0.\) Moreover there exists a unique \(X_{max}>0\) such that \(Y_{max}:=F(X_{max})\), and F is strictly increasing on \((0,X_{max})\) and strictly decreasing on \((X_{max},+\infty )\). Furthermore, for each \(Y_0 \in (0, Y_{max})\) there exists \(X^*\in (0,X_{max})\) and \(X^{**} \in (X_{max},+\infty )\) such that \(F(X^*)=F(X^{**})=Y_0. \)

Proof

Observe that \(F(0) = 0 \text { and } \lim _{X \rightarrow +\infty } F(X) = -1.\) Moreover

$$\begin{aligned} \begin{array}{ll} F^{\prime }(X)&{} = - \,c \,e^{-c \,X} + c\, e^{- c \, r \, X} - c^2 \, r \,X\, e^{- c \, r \, X}\\ &{}=c\, e^{-c\,r \,X} \left( 1- e^{-c \, \left[ 1-r \right] \, X}-c\,r\,X \right) , \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ll} F^{\prime \prime }(X)&{} = \,c^2 \,e^{-c \,X} - 2 c^2\,r \, e^{- c \, r \, X} - c^3 \, r^2 \,X\, e^{- c \, r \, X}\\ &{}=c^2\, e^{-c\,r \,X} \left( e^{-c \, \left( 1-r \right) \, X}-2 \,r+ c\,r^2\,X \right) . \end{array} \end{aligned}$$

Therefore,

$$\begin{aligned} F^{\prime }(0) = 0 \text { and } F^{\prime \prime }(0) =c^2( 1-2\, r). \end{aligned}$$
(3.4)

Define \(G(X):=1-e^{-c \, \left[ 1-r \right] \, X}- c\,r\,X,\) and we then have

$$\begin{aligned} F^{\prime }(X)=c\, e^{-c\,r \,X} G(X), \end{aligned}$$
(3.5)

and \(F^{\prime }(X) = 0 \Longleftrightarrow G(X)=0.\) Thus, \( G(0) = 0, \, \lim _{X \rightarrow +\infty } G(X) = -\infty , \) and \(G^{\prime }(X) =c \, \left[ 1-r \right] e^{-c \, \left[ 1-r \right] \, X}-c\, r.\) By assumption \(r < 1\), so the mapping \(X \rightarrow G^\prime (X)\) is strictly decreasing, \(G^{\prime }(0)=c\, \left( 1-2 \,r \right) \), and \(\lim _{X \rightarrow +\infty } G^\prime (X) =-c\, r<0.\)

Proof of i). Assume that \(1-2 \,r \le 0\). Then \(G^{\prime }(0)=c\, (1-2 \, r) \le 0\), and since \(G^{\prime }(x)\) is strictly decreasing, \(G^{\prime }(X) < 0, \forall X>0\). But \(G(X)=0\), therefore \( G(X) < 0, \forall X>0\). By using (3.5), we deduce that \(F^{\prime }(X) < 0, \forall X>0\) and i) follows from the fact that \(F(0)=0\).

Proof of ii). Assume that \(1-2 \,r > 0\). Then \(G^{\prime }(0)=c \, (1-2 \,r) >0\), and the mapping G has exactly one point \(X_{\max }>0\) such that \(G(X_{\max })=0\), and

$$\begin{aligned} G(X)>0,\forall X \in (0,X_{\max }), \text { and } G(X)<0,\forall X> X_{\max }. \end{aligned}$$
(3.6)

By using again (3.5), we deduce that \(X_{\max }>0\) is a critical point of F(X) (i.e. \(F^{\prime }(X_{\max }) = 0\)), and ii) follows from (3.5) and (3.7). \(\square \)

Remark 3.2

Proposition 3.1 implies that the cumulative reported case data CR(tp) and \(CR_{\infty }\) are compatible with the solution of model (2.1), (2.2) only if \(CR(tp) / CR_{\infty } < 1/2\). The condition \(2 \, CR(tp) < CR_{\infty }\) means that more than half of the reported cases will be reported after the turning point. The multiplier \(\nu _1\) in the definition of \(CR(t) = \nu _1 \int _0^t I(s) ds\) can be replaced by \(\nu _2\) or \(\nu _1 + \nu _2\), and the turning point is unchanged. Then, the cumulative unreported cases satisfy \(2 \, CU(tp) < CU_\infty \) (where \(CU(t) = \nu _2 \int _0^t I(s) ds\)), and all the cumulative cases satisfy \(2 \, CT(tp) < CT_\infty \) (where \(CT(t) = (\nu _1 + \nu _2) \int _0^t I(s) ds\)). Thus, \(CR(t), \, CU(t), \, CT(t)\) all have the same the turning point tp. Illustrations of Proposition 3.1 are given in Figs. 1 and 2.

Fig. 1
figure 1

The graph of F(X) (red), and the two solutions \(X^*\) and \(X^{* *}\) of \(F(X) = I_0 / S_0\). One of the two solutions provides a correct numerical simulation of (2.9), (2.10) for the reported case data and their turning point tp (color figure online)

Full size image
Fig. 2
figure 2

The graph of all cases (both reported and unreported) I(t) (red) and the graph of cumulative reported cases CR(t) (black), and their relationship at the turning point tp. \(I^{\prime }(tp) = 0\) and \(CR^{\prime \prime }(tp) = 0\) (color figure online)

Full size image

The algorithm to determine the initial conditions \(S_0, I_0\) and the parameters \(\tau , \nu _1, \nu _2\) from the reported case data is as follows:

  1. (1)

    Assume the reported case data \(\nu _1I(t)\) and the turning point tp are known, and \(2 CR(t) < CR_{\infty }\). Define \(F(X) = I_0/S_0\) as in (3.3) (where F depends only on CR(tp), \(CR_{\infty }\), and not on \(S_0\), \(I_0\), tp).

  2. (2)

    The value of the ratio \(I_0 / S_0\) in (3.3) must be adjusted so that the model simulation of (2.1), (2.2) gives agreement with the reported case data and its turning point tp. Let trial values of \(S_0\) and \(I_0\) be given. Then \(F(X) = I_0/S_0\) has two positive solutions \(X = X^*\) and \(X^{**}\), provided \(I_0/S_0 < \sup _{X \ge 0}F(X)\).

  3. (3)

    Set either \(X^*\) or \(X^{**}\) to \(\tau / \nu _1\), and use (2.10), (2.11) to obtain \(\nu _1\), \(\tau \), and \(\nu _2\), for each from

    $$\begin{aligned}&\nu _1 = \frac{CR^{\prime }(tp)}{S_0 + I_0 - S_0 \exp \bigg (- \frac{\tau }{\nu _1} CR(tp)\bigg ) \bigg (1+ \frac{\tau }{\nu _1} CR(tp)\bigg )}, \end{aligned}$$
    (3.7)
    $$\begin{aligned}&\tau =\left( \dfrac{\tau }{\nu _1} \right) \nu _1, \end{aligned}$$
    (3.8)
    $$\begin{aligned}&\nu _2 = \tau S_0 \exp \bigg (- \frac{\tau }{\nu _1} CR(tp)\bigg )- \nu _1. \end{aligned}$$
    (3.9)
  4. (4)

    One value of \(X^{*}\) or \(X^{**}\) for \(\tau / \nu _1\) will yield parameters \(\tau , \nu _1, \nu _2\) such that the corresponding numerical simulation of (2.1), (2.2) has a graph with the shape of the reported case data, but possibly not the same turning point tp of the data. The value of the ratio \(I_0 / S_0\) must be adjusted so that the value of \(X^{*}\) or \(X^{**}\) for \(\tau / \nu _1\) will yield parameters \(\tau , \nu _1, \nu _2\) such that the corresponding numerical simulation has a graph with the shape of the reported case data and the same turning point as the data.

Remark 3.3

We note that when \(I_0\) is small relative to \(S_0\), the turning point of the model simulation can be approximated as a decreasing function of the ratio \(I_0/S_0\) according to the following formula:

$$\begin{aligned} tp(k) \approx tp \, \frac{1 + \log (k I_0 / S_0)}{1 + \log (I_0 / S_0)}, \end{aligned}$$

where tp is the turning point obtained by the algorithm for \(I_0 / S_0\) and tp(k) is the turning point obtained by the algorithm for \(k \, I_0 / S_0\) with \(k > 0\).

Remark 3.4

For \(I_0 / S_0 = (c I_0) / (c S_0), \, c >0\) given, the algorithm yields parameters \(\tau (c), \nu _1(c), \nu _2(c)\) that depend on the scaling factor c, but with turning point independent of c. Further, \(S_0 c \nu _1(c)\), \(S_0 c \tau (c)\), \(\nu _1(c) + \nu _2(c)\), and \(\mathbb {R}_0(c)\) are independent of c. More precisely, assume that \(S_0=c\hat{S}_0\) and \(I_0=c\hat{I}_0\) for some constant \(c>0\), where \(\hat{S}_0>0\), \(\hat{I}_0>0\), and CR(t) are given. Then, by solving (3.2), we deduce that \(\dfrac{\tau (c)}{\nu _1(c)}\) is independent of c. By using (3.7), we deduce that \(\nu _1(c)= \dfrac{\hat{\nu }_1}{c}\), where

$$\begin{aligned} \hat{\nu }_1 := \frac{CR^{\prime }(tp)}{\hat{S}_0 + \hat{I}_0 - \hat{S}_0 \exp \bigg (- \frac{\tau (c)}{\nu _1(c)} CR(tp)\bigg ) (1+ \frac{\tau (c)}{\nu _1(c)} CR(tp))} \end{aligned}$$

is independent of c. By (3.8) \(\tau (c)= \dfrac{\tau (c)}{\nu _1(c)} \nu _1(c)\), and we obtain \(\tau (c)=\dfrac{\hat{\tau }}{c}\) (where \(\hat{\tau }:=\dfrac{\tau (c)}{\nu _1(c)} \hat{\nu }_1\) is independent of c). Thus, \(\tau S_0=\hat{\tau } \hat{S}_0\) is independent of c, and by using (3.9) we deduce that \(\nu _2(c)= \eta -\dfrac{\hat{\nu }_1}{c}\) with \(\eta :=\tau S_0 \exp (- \frac{\tau }{\nu _1} CR(tp))\), is independent of c. Now by replacing the values for \(\tau (c), \nu _1(c),\nu _2(c)\) with these formulas in (2.1), (2.2), we obtain

$$\begin{aligned} \left\{ \begin{array}{l} S^{\prime }(t)=-\dfrac{\hat{\tau }}{c} S(t)I(t)\\ I^{\prime }(t)=\dfrac{\hat{\tau }}{c} S(t)I(t)-\eta I(t)\\ S(0)=c\hat{S}_0, I(0)=c\hat{I}_0. \end{array} \right. \end{aligned}$$
(3.10)

By setting \(\hat{S}(t):=\dfrac{S(t)}{c}\) and \(\hat{I}(t):=\dfrac{I(t)}{c}\) we obtain

$$\begin{aligned} \left\{ \begin{array}{l} \hat{S}^{\prime }(t)=-\hat{\tau } \hat{S}(t)\hat{I}(t)\\ \hat{I}^{\prime }(t)=\hat{\tau } \hat{S}(t)\hat{I}(t)-\eta \hat{I}(t)\\ \hat{S}(0)=\hat{S}_0, \hat{I}(0)=\hat{I}_0. \end{array} \right. \end{aligned}$$
(3.11)

Moreover, since \(\hat{I}(t)=\dfrac{I(t)}{c}\), we obtain \(\hat{I}^\prime (t)=0 \Leftrightarrow I^\prime (t)=0\). Since (3.10) is independent of c, we deduce that the turning point of (3.10) is the same as the turning point of (3.11), which is independent of c.

From (2.10) \((\nu _1(c) + \nu _2(c)) / (S_0 c \, \tau (c)) = \mathbb {R}_0(c) = \exp [- \tau (c) CR(tp) / \nu _1(c) ]\), independently of c. Thus, \(\mathbb {R}_0(c)\) is independent of c. Further, (2.11) (divided by \(S_0 c\)) implies

$$\begin{aligned} 1 + \frac{I_0 c}{S_0 c} - \frac{\nu _1(c) I(tp)}{S_0 c \, \nu _1(c)} = \bigg ( \frac{\nu _1(c) + \nu _2(c)}{S_0 \, c \, \tau (c)} \bigg ) \, \bigg ( 1 - \log \bigg (\frac{\nu _1(c) + \nu _2(c)}{S_0 \, c \, \tau (c)}\bigg ) \bigg ), \end{aligned}$$

which yields \(S_0 c \, \nu _1(c)\) independent of c. Then, \(S_0 c \, \tau (c) = S_0 c \, \nu _1(c) \, \tau (c) / \nu _1(c)\) and \(\nu _1(c) + \nu _2(c) = S_0 c \, \tau (c) \mathbb {R}_0(c)\) are independent of c.

Remark 3.5

The total number of initial cases \(I_0\) may be known, both reported and unreported, but the initial number of susceptible individuals \(S_0\) may not be known. This is the case if a significant fraction of the demographic population has acquired immunity from a prior infection. The value of \(S_0\) can be obtained from the algorithm by varying \(S_0\) and comparing the model output to the reported case data and its turning point tp. The turning point in the model simulation, with \(I_0\) fixed, increases approximately linearly with \(\log (S_0)\), so the correct value of \(S_0\) can be matched to the turning point in the model and the data. An illustration is given in Fig. 3.

Fig. 3
figure 3

The relationship of \(I_0 / S_0\) for the simulation of the 2016–2017 seasonal influenza epidemic in Puerto Rico with model (2.1), (2.2). The algorithm yields the correct turning point \(tp = 14\) for the data with \(I_0 / S_0 = 2.28 \times 10^{-5}\). Left side: \(I_0 = 57\), tp increases linearly with \(\log (S_0)\) for \(I_0 / S_0 = 2.28 \times 10^{-5}\). Right side: \(S_0 = 2{,}500{,}000\), tp decreases linearly with \(\log (I_0)\) for \(I_0 / S_0 = 2.28 \times 10^{-5}\). The green graphs correspond to \(S_0 = 2{,}500{,}000\) and \(I_0 = 57\). For \(c I_0 / c S_0\), \(c \ge 1\), \(R_0 \approx 1.73\) independently of c, and \(\nu _1 / \nu _2 \approx 38.7 \, c-1.0\) (color figure online)

Full size image

Remark 3.6

The initial number of reported cases may be known from the data, but the initial number of all cases \(I_0\), including unreported cases, may not be known. If \(S_0\) is known, however, the algorithm can be extended to determine \(I_0\): vary \(I_0\) in the algorithm and compare the model output to the reported case data and its turning point tp. The turning point in the model simulation, with \(S_0\) fixed, decreases approximately linearly with \(\log (I_0)\), so the correct value of \(I_0\) can be matched to the turning point of the data. An illustration is given in Fig. 3.

Remark 3.7

In practice neither \(I_0\) nor \(S_0\) may be known. The algorithm can be used in this case by varying the ratio \(I_0/S_0\). As \(I_0/S_0\) increases, the turning point in the model simulation decreases. The correct value of the ratio \(I_0/S_0\) can be identified by matching the model turning point to the data and its turning point tp. For this value of \(I_0/S_0\) any scaling \(c I_0/c S_0\) of this ratio will also match the data and its turning point tp. For c given, \(\mathbb {R}_0\) is independent of c, and the ratio \(\nu _2(c) / \nu _1(c)\) is a linearly increasing function of c. Thus, the scaling factor c can be varied and the values of \(c S_0, c I_0, \tau (c), \nu _1(c), \nu _2(c)\) can be compared to known information about the epidemic to obtain a realistic value of c.

4 Examples of the algorithm for identifying the model parameters

Example 1

The Hong Kong influenza epidemic in New York City in 1968–1969.

We apply the parameter identification algorithm to the Hong Kong influenza epidemic in New York City in 1968–1969. The reported case data consist of weekly reported mortality cases in excess of typical reported mortality (assumed to be due to the influenza epidemic) for the same time period in previous years (Smith and Moore 2004). The cumulative mortality is obtained by adding these weekly values over the 13 week duration of the epidemic. The time units are weeks. We take \(S(0) = 7{,}900{,}000\) (the population of New York City in the 1970 census) and \(I(0) = 15{,}000\). The weekly reported data and the cumulative reported excess mortality data are illustrated by the black dots in Fig. 4. From these data we estimate \(CR_{\infty } \approx 1080\), \(tp \approx 6.15\) weeks, \(CR(tp) \approx 500\), and \(\nu _1 I(tp) = 190\). Notice that \(2 \, CR(tp) < CR_{\infty }\).

Fig. 4
figure 4

Hong Kong influenza epidemic in New York City in 1968–1969. The weekly reported mortality case data and cumulative reported case data (black dots), and the model output graphs of \(\nu _1 I(t)\) and CR(t) (blue) (color figure online)

Full size image

The two solutions of \(F(X) = I_0/S_0\), with F(X) given in (3.3), are \(X = 0.000276\) and \(X = 0.000728\). The solution that gives agreement with the data is \(X = 0.000728\). F(X) has its maximum at \(F(0.000524) \approx 0.00332\). If \(I_0 > 26{,}220\), then \(I_0 / S_0 > 0.00332\), and the reported case data are not consistent with these initial conditions. For \(X = 0.000728\), the numerical solutions of Eqs.  (2.9), (2.10), (2.11) are \(\tau \approx 3.24 \times 10^{-7}\), \(\nu _1 \approx 0.00044\), \(\nu _2 \approx 1.78\), which corresponds to an average infectious period in non-fatal cases of \(\approx 1/1.78\) weeks \(\approx 4\) days, and a ratio of unreported to reported cases of \(\approx 4045\) to 1. The graphs of the reported data and model reported cases output are given in Fig. 4. The epidemic peak size is \(I(tp) \approx 198{,}000\) at \(tp \approx 6.15\) weeks. From model output we obtain \(S_{\infty } \approx 3{,}620{,}000\). The epidemic final size is \(S_0 + I_0 - S_{\infty } \approx 4{,}295{,}000\). The epidemic attack ratio is \((S_0 + I_0 - S_{\infty }) / (S_0 + I_0) \approx 54\%\). The basic reproduction number \(\mathbb {R}_0 \approx 1.44\). The graphs of S(t), I(t), R(t) are given in Fig. 5, and the graph of F(X) is given in Fig. 6a. The value of \(\nu _1 I(t)\) obtained from the numerical solution of model, with parameters obtained from \(X = 0.000276\) as the solution of \(F(X) = I_0 / S_0\), is graphed in Fig. 6b, and is not in agreement with the reported data.

Fig. 5
figure 5

Hong Kong influenza epidemic in New York City in 1968–1969. The model output graphs of S(t) (blue), I(t) (black), and R(t) (dashed). The vertical lines are at the epidemic turning point \(tp \approx 6.15\) weeks (color figure online)

Full size image
Fig. 6
figure 6

Hong Kong influenza epidemic in New York City in 1968–1969. a The graphs of F(X) (yellow) and \(I_0 / S_0\) (blue). There exists two solutions \(X = 0.000276\) and \(X = 0.000728\) of \(F(X) = I_0 / S_0\).The solution corresponding to the data is \(X = 0.000728\). b The graph of the model output of the cumulative weekly reported case data for the solution \(X = 0.000276\), which does not match the epidemic data (color figure online)

Full size image
Fig. 7
figure 7

Bubonic plague epidemic in Bombay, India in 1906. a The weekly reported mortality case data (black dots) and model output graph of \(\nu _1 I(t)\) (blue). b The weekly reported cumulative mortality case data (black dots) and model output graph of CR(t) (blue). The vertical lines are at the epidemic turning point \(tp \approx 13.5\) weeks (color figure online)

Full size image

The case fatality ratio \(CR_{\infty } / (S_0 + I_0 - S_{\infty }) = 1080 / 4{,}295{,}000 \approx 0.00025 = 0.025 \%\) is lower than commonly claimed values for seasonal influenza, typically 0.1–0.2\(\%\) (Li et al. 2008). If the initial susceptible population \(S_0\) is reduced, then the fatality ratio is increased proportionately.

Example 2

The plague epidemic in Bombay, India in 1906.

We apply the parameter identification algorithm to the bubonic plague epidemic in Bombay, India in 1906. This epidemic has been modeled many times, beginning with one of the most famous works in mathematical epidemiology, due to Kermack and McKendrick (1927). These reported case data consist of weekly reported mortality cases over a period of 30 weeks beginning in January, 1906. These data of reported mortality cases are available at https://www.math.psu.edu/treluga/misc.html, and are graphed Fig. 7a (black dots). The population of Bombay in 1906 was approximately 1,000,000, but not all residents had equal likelihood of infection. Infection did not spread from person to person, but was strongly associated with shared dwellings, workplaces, or other localities (Thompson 1906). Additionally, many residents fled the city during the seasonal plague epidemics of that time period, as for example, 850,000 in 1896 (http://en.wikipedia.org/wiki/Mumbai). The case-fatality rate was as high as 90%, and the typical duration of a fatal infection was \(\approx 5\) days with an incubation period of \(\approx 3\) days (Bacaër 2012).

For the model (2.1), (2.2) we take \(S(0) = 100,000\) and \(I(0) = 8\). The cumulative reported mortality data are illustrated by the black dots in Fig. 7b. From these data we estimate \(CR_{\infty } \approx 8840\), \(tp \approx 13.5\) weeks, \(CR(tp) \approx 4330\), and \(\nu _1 I(tp) = 770\). Notice that \(2 \, CR(tp) < CR_{\infty }\). The two solutions of \(F(X) = I_0/S_0\), with F(X) given in (3.3), are \(X = 0.0000159\) and \(X = 0.0000204\). The solution that gives agreement with the data is \(X = 0.0000159\).

Fig. 8
figure 8

Weekly reported cases of seasonal influenza epidemics in Puerto Rico in the 2015–2016 and 2016–2017 seasons. The graph of reported cases in 2015–2016 (yellow) has two peaks. The graph of reported cases in 2016–2017 (black) has only one peak and satisfies \(CR(tp) / CR_{\infty } < 1/2\) (color figure online)

Full size image

For \(X = 0.0000159\) the numerical solutions of (2.9), (2.10), (2.11) are \(\tau \approx 5.2 \times 10^{-5}\), \(\nu _1 \approx 3.3\), \(\nu _2 \approx 1.6\), which correspond to an average infectious period of \(\approx 1/4.9\) weeks \(\approx 1.4\) days, and a ratio of unreported to reported cases of \(\approx 0.48\) to 1. From (2.8) we estimate \(S_{\infty } \approx 87,000\) with corresponding total cases \(\approx 13{,}000\), an attack ratio \(\approx 13\%\), and a reported removal rate of \(\approx 70\%\) (the total mortality rate would be increased by unreported removal cases). The epidemic peak size is \(I(tp) \approx 235\) at \(tp \approx 13.5\) weeks. The basic reproduction number \(\mathbb {R}_0 \approx 1.07\). The graph of the model output of the weekly reported cases \(\nu _1 I(t)\) is given in Fig. 7a. The graph of model output of cumulative weekly reported cases is given in Fig. 7b.

Example 3

The seasonal influenza epidemic in Puerto Rico in 2016–2017.

We apply the parameter identification algorithm to the seasonal influenza epidemic in Puerto Rico in 2016–2017. The reported case data consist of weekly reported cases from Departamento de Salud, Puerto RicoFootnote 1 from week 36 in 2016 to week 23 in 2017 (Fig. 8). The cumulative reported cases are obtained by adding these weekly values over the 37 weeks duration of the epidemic. We take \(S(0) = 2{,}500{,}000\) and \(I(0) = 57\). The 2010 census of Puerto Rico was \(\approx 3{,}500{,}000\), so we assume \(\approx 1 / 3\) of the population has acquired immunity. The weekly reported data and the cumulative reported data are illustrated by the black dots in Fig. 9. From these data we estimate \(CR_{\infty } \approx 45{,}300\), \(tp \approx 14\) weeks, \(CR(tp) \approx 20{,}400\), and \(\nu _1 I(tp) \approx 6200\). Notice that \(2 \, CR(tp) < CR_{\infty }\).

Fig. 9
figure 9

Seasonal influenza epidemic in Puerto Rico in 2016–2017. The weekly reported mortality case data and cumulative reported case data (black dots), and the model output graphs of \(\nu _1 I(t)\) and CR(t) (blue). The vertical lines are at the epidemic turning point \(tp \approx 14\) weeks (color figure online)

Full size image

The two solutions of \(F(X) = I_0/S_0\), with F(X) given in (3.3), are \(X =4.78 \times 10^{-7}\) and \(X = 2.67 \times 10^{-5}\). The solution that gives agreement with the data is \(X = 2.67 \times 10^{-5}\), and the numerical solutions of Eqs. (2.9), (2.10) are \(\tau \approx 6.36 \times 10^{-7}\), \(\nu _1 \approx 0.024\), \(\nu _2 \approx 0.90\), which correspond to an average infectious period in non-fatal cases of \(\approx 1.1\) weeks, and a ratio of unreported to reported cases of \(\approx 37.7\) to 1. The graphs of the reported data and model output for reported cases are given in Fig. 9. From the model output we obtain the epidemic final size \(S_0 + I_0 - S_{\infty } \approx 1{,}756{,}000\). The attack ratio of the susceptible population is \((S_0 + I_0 - S_{\infty }) / (S_0 + I_0) \approx 70\%\). The attack ratio as a fraction of the total population of Puerto Rico is \((S_0 + I_0 - S_{\infty }) / 3,500,000 \approx 48.8\%\). The reported case data include year-round background cases, which may over estimate the reported cases and the attack ratio. The basic reproduction number \(\mathbb {R}_0 \approx 1.73\). The epidemic peak size is \(I(tp) \approx 237{,}000\) at \(tp \approx 14\) weeks. The graphs of S(t), I(t), R(t) are given in Fig. 10.

Fig. 10
figure 10

Seasonal influenza epidemic in Puerto Rico in 2016–2017. The model output graphs of S(t) (blue), I(t) (black), and R(t) (dashed)

Full size image

5 Discussion

For the SIR model (2.1), (2.2), (2.3) we have constructed an algorithm to identify unreported cases using reported case data \(\nu _1 I(t)\). This SIR model yields a simple rise and fall of the infected cases over the course of the epidemic, with a turning point at time tp. A necessary condition for the algorithm is that the data satisfy \(CR(tp) / CR_{\infty } < 1 / 2\). The algorithm uses the cumulative reported case data CR(tp) at the turning point and cumulative reported case data \(CR_{\infty }\) at the end of the epidemic to obtain model parameters \(\nu _1\), \(\nu _2\), and \(\tau \), that fit the reported case data and data turning point tp, assuming \(S_0, I_0\) are known. \(S_0\), however, may not be known, since the number of initially susceptible individuals \(S_0\) may be less than the demographic population of the region due to some individuals having acquired immunity from previous infections. \(I_0\) may not be known, since the number of initially infected individuals \(I_0\) may be much higher than the number of initially reported infected individuals. If one of \(S_0\) and \(I_0\) is known, then the algorithm allows identification of the other using the turning point of the data. If neither \(S_0\) nor \(I_0\) is known, then the ratio \(I_0 / S_0\) in the algorithm can be adjusted to identify parameters \(\tau , \nu _1, \nu _2\) that agree with the reported case data and its turning point tp. In this case any scaling \(c I_0 / c S_0\) will yield parameters that also fit the reported case data and turning point. The ratio \(S_0 / I_0\) is central in our algorithm for identifying the parameters and initial conditions of the model.

Other methods provide alternative approaches for the parameter identification problem for epidemic models, including formal least squares methods and stochastic methods (Andersson and Britton 2000; Becker 1989; Gamado et al. 2014; Li et al. 2008). Stochastic SIR models account for probabilistic individual behavior, but typically have much greater computational requirements for large population sets. Deterministic SIR models provide only mean distribution outputs, but are typically much more computationally efficient. Our deterministic modeling approach emphasizes the role of the initial conditions \(I_0\) and \(S_0\), which for many epidemic diseases such as seasonal influenza, are largely unknown. Our approach also emphasizes the role of reported cases, which for diseases such as seasonal influenza, are a very small fraction of the total number of cases.

5.1 The SEIR epidemic model

A generalization of the SIR model (2.9), (2.10), (2.11) is the SEIR model, which allows for an incubation period of newly infected individuals before they become infectious. The equations of the SEIR model are

$$\begin{aligned} \frac{d}{d t} S(t)= & {} -\tau I(t) S(t) \end{aligned}$$
(5.1)
$$\begin{aligned} \frac{d}{d t} E(t)= & {} \tau I(t) S(t) - \sigma E(t) \end{aligned}$$
(5.2)
$$\begin{aligned} \frac{d}{d t} I(t)= & {} \sigma E(t) - (\nu _1+\nu _2) I(t) \end{aligned}$$
(5.3)
$$\begin{aligned} \frac{d}{d t} R(t)= & {} (\nu _1+\nu _2) I(t) \end{aligned}$$
(5.4)

with initial conditions \( S(0)=S_0 > 0, \quad E(0)=E_0 \ge 0\ \quad I(0)=I_0\ge 0, \quad R(0)=R_0 \ge 0.\) Here E(t) is the number of pre-infectious infected individuals at time t, the parameters \(\tau \), \(\nu _1\), and \(\nu _2\) are as before, and \(\sigma \) is the rate at which incubating infected individuals become infectious. The basic reproduction number \(\mathbb {R}_0\) is \(\tau S_0 / (\nu _1 + \nu _2)\), the same as the SIR model, and \(S_{\infty } = \lim _{t \rightarrow \infty } S(t)\) satisfies

$$\begin{aligned} S_{\infty } = S_0 + E_0 + I_0 + \frac{\nu _1 + \nu _2}{\tau } \log \bigg (\frac{S_{\infty }}{S_0} \bigg ) \end{aligned}$$
(5.5)

(\(\mathbb {R}_0\) and \(S_{\infty }\) are independent of \(\sigma \)). If \(\mathbb {R}_0 > 1\), I(t) may first decrease before increasing and then decreasing at a turning point tp (depending on \(S_0\), \(E_0\), \(I_0\)). The SIR analysis carries over to a similar analysis for the SEIR model, with Eqs. (2.9), (2.10), (2.11) the same, except that \(I_0\) is replaced by \(I_0 + E_0\), and \(F(X) = (I_0 + E_0) / S_0\) in (3.3). With the values of \(\tau , \nu _1, \nu _2\) obtained from the modified Eqs. (2.9), (2.10), (2.11), and (3.3), the Eqs. (5.1), (5.2), (5.3), are solved numerically with values of \(\sigma \) chosen so that the output matches the reported case data and the turning point tp. For seasonal influenza in Puerto Rico in 2016–2017, \(\tau =4.54 \times 10^{-7}\), \(\nu _1 = 0.017\), and \(\nu _2 = 0.90\) as in the SIR model (Example 3) and \(E_0 = 70\), \(I_0 = 70\), and \(\sigma = 21\). Thus, the incubation period is \(\approx 1 / 21\) weeks \(= 0.33\) days, which is consistent with incubation times for influenza in volunteer studies measured by viral shedding in Carrat et al. (2008).

5.2 The general incidence SIR epidemic model

The SIR model (2.9), (2.10), (2.11) does not take into account changes in social behavior or public health policies as the epidemic unfolds, which may reduce transmission. A model that incorporates such change is the general incidence SIR model:

$$\begin{aligned} \frac{d}{d t} S(t)= & {} - \frac{\tau I(t)^p}{1 + \kappa I(t)^q} S(t) \end{aligned}$$
(5.6)
$$\begin{aligned} \frac{d}{d t} I(t)= & {} \frac{\tau I(t)^p}{1 + \kappa I(t)^q} S(t) - \nu I(t) \end{aligned}$$
(5.7)
$$\begin{aligned} \frac{d}{d t} R(t)= & {} \nu I(t) \end{aligned}$$
(5.8)

with initial conditions \( S(0)=S_0> 0, \quad I(0)=I_0 > 0, \quad R(0)=R_0 \ge 0.\) The parameter \(\kappa \) accounts for reduced transmission as I(t) increases, and the positive exponents p and q satisfy \(1 \le p \le q + 1\). The basic reproduction number is \(\mathbb {R}_0 = \tau I_0^{p-1} S_0 / (1 + \kappa I_0^q)\). If \(R_0 < 1\), then S(t) decreases to a limiting value \(S_{\infty } > 0\) and I(t) decreases to 0. If \(R_0 > 1\), then S(t) decreases to a limiting value \(S_{\infty } > 0\) and I(t) first increases, then decreases to 0 (Magal et al. Submitted).

The condition \(CR(tp) / CR_{\infty } < 1 / 2\) may be violated for the general incidence model. If the epidemic data do not satisfy this condition, then the SIR model (2.9), (2.10), (2.11) may not be valid for epidemic analysis. A numerical example of a general incidence model not satisfying \(CR(tp) / CR_{\infty }< 1 / 2\) is given in Fig. 11. Examples of epidemic reported case data not satisfying \(CR(tp) / CR_{\infty } < 1 / 2\) are the 2013–2014 and 2014–2015 seasonal influenza epidemics in Puerto RicoFootnote 2 illustrated in Fig. 12.

Fig. 11
figure 11

The general incidence model (5.6), (5.7) with \(\tau = 0.14 \times 10^{-6}\), \(\nu = 3.0\), \(p = 1.09\), \(q = 0.1\), \(S_0 = 3{,}500{,}000\), \(I_0 = 1000\). \(\nu \) corresponds to all cases being reported. \(\mathbb {R}_0 = 1.105\), \(tp = 22.74\), \(\nu I(tp) = 6829\), \(CR(tp) = 114{,}984\), \(CR_{\infty } = 210{,}828\). The cumulative reported cases do not satisfy \(CR(tp) / CR_{\infty } < 1/2\)

Full size image
Fig. 12
figure 12

Weekly reported cases of seasonal influenza epidemics in Puerto Rico in the 2013–2014 and 2014–2015 seasons. The cumulative reported cases (corresponding to the areas below the graphs) do not satisfy \(CR(tp) / CR_{\infty } < 1/2\) for either season

Full size image

5.3 More general epidemic models

In many cases epidemic data do not show a simple rise and fall of the reported cases. One example is found in models with multi-group infected populations. In Magal et al. (2016) a two-group model, with super-spreaders and ordinary spreaders, yielded a two-peak output of the total infected population for the SARS influenza outbreak in Singapore in 2003. Spatial heterogeneity is also important in modeling the spread of epidemics. One example is given in Fitzgibbon et al. (2017) of the 2015–2016 Zika epidemic in Rio de Janeiro, Brazil. Another example is given in Magal et al. (Submitted) of the geographical spread of the 2015–2016 seasonal influenza epidemic in Puerto Rico, in which the reported infected cases show an early high peak and a later low peak (Fig. 8). The multiple peaks in this case can be attributed to spatial variation in the course of epidemic. Models incorporating the disease phase of infected individuals with a continuum disease age variable can track the rise of infectiousness, and in particular the pre-symptomatic periods of infectiousness (Blaser et al. 2010). Models involving time dependent transmission and removal rates, corresponding to public behavioral changes are also important, and considerations of time dependent parameters were treated by Hadeler (2011a, b) and Stadler et al. (2013). These examples illustrate the need for extension of the work here to identify parameters in epidemic models that incorporate additional features of outbreak epidemic diseases.