1 Introduction

Since December 2019, a total of 499748065 many infections and 6181560 many deaths due to the ongoing COVID-19 pandemic were confirmed until April 12, 2022 [1]. Therefore, biological modelling of such disease outbreaks plays an essential role in containing these and possible other epidemics or pandemics. Such models allow the study of infection dynamics and offer the possibility of assessing the potential consequences of various options for population or government reactions and thus developing a strategy that is optimized as well as feasible.

Studying epidemics by use of compartmental models, which are deterministic mean-field models, is a well-established academic discipline [2,3,4,5]. The Susceptible-Infectious-Removed (SIR) model is a basic model [6] that assumes that the total population can be divided into three classes: susceptible, infected and removed population parts. Note that within this approach demographic effects are neglected since infection dynamics are often much faster than demographic dynamics. Infections occur due to contact between infected and susceptible individuals. The number of infected individuals decreases due to treatment, quarantine measures and death cases.

In addition, variations of the SIR model like SIRD and SIQR models and also their modified versions exist [7,8,9,10,11,12,13,14,15,16,17,18,19,20]. On the one hand, the SIRD model considers the number of susceptible, infected, recovered, and dead people. Correspondingly, a susceptible individual in contact with an infected person is likely to get infected. This individual can either recover from the disease or die. Hence, the sum of susceptible, infected, recovered, and dead remains constant. Furthermore, it is conjectured that all individuals who are exposed to the disease get infected instantly. Thus, no latent time between exposure and infection exists within this model. In addition, confinement or quarantine measures are neglected. On the other hand, the SIQR model is a compartmental model which describes a population by four compartments, assuming the existence of an additional compartment of quarantined individuals in contrast to the SIR model. In common epidemics, only symptomatic patients are considered as infectious and the quarantine measure is launched to remove symptomatic individuals from the population. Nonetheless, infectious diseases like COVID-19 also yield pre-symptomatic and asymptomatic patients who are infectious and thus are also considered as part of the additional compartment.

Remarkably, due to the COVID-19 pandemic modelling and simulation is of great prominence. Many important results were summarized in [6, 21]. The SIR model, introduced in [5], was used for the purpose of studying the COVID-19 outbreak [22, 23], while the SIRD model was also studied for the analysis of the pandemic [24, 25].

Therefore, analytical results are derived for the SIRD and SIQR cases in Sects. 2.12.4. These formulae (describing the epidemic peak, analytical solutions for all times t and the small time limit) are obtained trivially by considering well-known expressions for the SIR model. In addition, a novel analysis of the daily new cases is presented in Sect. 2.5.

Moreover, the loss due to reactions of government and population is investigated by use of novel measure and loss functions. Two different strategies are presented. On the one hand, the epidemic peak is kept constant. On the other hand, the loss is minimized with regard to a given measure. This analysis is crucial since drastic measures were taken for the purpose of containing the spread of COVID-19. However, a major burden of the COVID-19 outbreak and a worldwide recession is observable [26, 27].

2 Results obtained by use of the connection between compartmental models

2.1 SIR, SIRD and SIQR models

Within the framework of compartmental models, SIR [28,29,30,31,32,33], SIRD [34,35,36] and SIQR [37,38,39,40] approaches are well-known methods for the purpose of studying epidemic disease outbreaks. These models are given by sets of ordinary differential equations that describe the dynamics of the outbreak by rate equations. The SIR model is given by

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}s_{1}\left( t\right)&=-\beta s_{1}\left( t\right) i_{1}\left( t\right) , \end{aligned}$$
(1a)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}i_{1}\left( t\right)&=\beta s_{1}\left( t\right) i_{1}\left( t\right) -\gamma i_{1}\left( t\right) ,\end{aligned}$$
(1b)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}r_{1}\left( t\right)&=\gamma i_{1}\left( t\right) , \end{aligned}$$
(1c)

while the SIRD model is characterized by

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}s_{2}\left( t\right)&=-\beta s_{2}\left( t\right) i_{2}\left( t\right) ,\end{aligned}$$
(2a)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}i_{2}\left( t\right)&=\beta s_{2}\left( t\right) i_{2}\left( t\right) -\left( \gamma +\sigma \right) i_{2}\left( t\right) ,\end{aligned}$$
(2b)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}r_{2}\left( t\right)&=\gamma i_{2}\left( t\right) ,\end{aligned}$$
(2c)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}d_{2}\left( t\right)&=\sigma i_{2}\left( t\right) \end{aligned}$$
(2d)

and the SIQR model is described by

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}s_{3}\left( t\right)&=-\beta s_{3}\left( t\right) i_{3}\left( t\right) ,\end{aligned}$$
(3a)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}i_{3}\left( t\right)&=\beta s_{3}\left( t\right) i_{3}\left( t\right) -\left( \gamma +\delta \right) i_{3}\left( t\right) , \end{aligned}$$
(3b)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}q_{3}\left( t\right)&=\delta i_{3}\left( t\right) -\mu q_{3}\left( t\right) ,\end{aligned}$$
(3c)
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}r_{3}\left( t\right)&=\gamma i_{3}\left( t\right) +\mu q_{3}\left( t\right) , \end{aligned}$$
(3d)

where t is the time, \(\beta \) the transmission rate, \(\gamma \) the recovery rate of infectious individuals, \(\delta \) the isolation rate, \(\sigma \) the mortality rate, and \(\mu \) the recovery rate of isolated individuals. Remarkably, these parameters are measurable quantities during an outbreak. Isolation rate \(\delta \) and transmission rate \(\beta \) depend on government reactions to an outbreak, while \(\beta \) also depends on population reactions. The functions \(s_{j}\left( t\right) \), \(i_{j}\left( t\right) \), \(q_{j}\left( t\right) \), \(r_{j}\left( t\right) \) and \(d_{j}\left( t\right) \), \(j=1,2,3\) with accordingly chosen j, describe the fractions of susceptible, infectious, isolated, removed (deceased or recovered for SIR and SIQR while only recovered for SIRD) and deceased individuals within the total population N at time t. Thus, \(s_{1}\left( t\right) +i_{1}\left( t\right) +r_{1}\left( t\right) =1\), \(s_{2}\left( t\right) +i_{2}\left( t\right) +r_{2}\left( t\right) +d_{2}\left( t\right) =1\) and \(s_{3}\left( t\right) +i_{3}\left( t\right) +q_{3}\left( t\right) +r_{3}\left( t\right) =1\) hold. Note that the initial data of the functions \(s_j\left( t\right) \) is represented by \(s_j\left( 0\right) \) for \(j=1,2,3\). Typically, \(s_j\left( 0\right) =1-\frac{1}{N}\) is used since it can be assumed that one infected individual exists in a community at the start of an epidemic outbreak.

2.2 Epidemic peak

Dividing Eq. (3b) with Eq. (3a) in the SIQR model and integrating over \(s_{3}\) leads to

$$\begin{aligned} i_{3}\left( s_{3}\left( t\right) \right) =1-s_{3}\left( t\right) +\frac{\gamma +\delta }{\beta }\ln s_{3}\left( t\right) , \end{aligned}$$
(4)

which describes the fraction of infected individuals in the \(\left( s_{3},i_{3}\right) \) plane for the SIQR model. In analogy, setting the isolation rate \(\delta =0\) (\(\delta =\sigma \)) and changing the index \(3\rightarrow 1\) (\(3\rightarrow 2\)) yields the equation describing the fraction of infected individuals in the \(\left( s_{1},i_{1}\right) \) (\(\left( s_{2},i_{2}\right) \)) plane for the SIR (SIRD) model. The \(\left( s_{3},i_{3}\right) \) plane is depicted in Fig. 1 for varying \(\frac{\gamma +\delta }{\beta }\).

Fig. 1
figure 1

Fraction of infected individuals \(i_{3}\) depending on the fraction of susceptible individuals \(s_{3}\) for \(\frac{\gamma +\delta }{\beta }=0.1,0.2,0.3,0.4\)

The peak of this plane increases for increasing transmission rate \(\beta \) or decreasing \(\gamma +\delta \) and lies at position \(\left( s_3^{*},i_3^{*}\right) \),

$$\begin{aligned} s_3^{*} =\frac{\gamma +\delta }{\beta }, \qquad i_3^{*} =1+\frac{\gamma +\delta }{\beta }\left( \ln \frac{\gamma +\delta }{\beta }-1\right) . \end{aligned}$$
(5)

The epidemic peak \(i_3^{*}\) is depicted in Fig. 2. It shows a minimum at \(\beta =\gamma +\delta \) while increasing transmission rate \(\beta \) or decreasing \(\gamma +\delta \) yield increasing \(i_3^{*}\) for \(\beta >\gamma +\delta \). Contrarily, for \(\beta <\gamma +\delta \) decreasing transmission rate \(\beta \) or increasing \(\gamma +\delta \) increases \(i_3^{*}\). For \(\frac{\gamma +\delta }{\beta } > \text {e} \) (with Euler’s number \(\text {e}\)) the peak \(i_3^{*}\) is larger than 1, which is prohibited in the framework of the SIQR model.

Fig. 2
figure 2

Epidemic peak \(i_3^{*}\) depending on \(\frac{\gamma +\delta }{\beta }\)

2.3 Exact results

Exact results for the SIR model are well-known [41,42,43] and given by

$$\begin{aligned} s_{1}\left( t\right)&=s_{1}\left( 0\right) u_{1}\left( t\right) , \end{aligned}$$
(6a)
$$\begin{aligned} i_{1}\left( t\right)&=1+\frac{\gamma }{\beta }\ln u_{1}\left( t\right) -s_{1}\left( 0\right) u_{1}\left( t\right) , \end{aligned}$$
(6b)
$$\begin{aligned} r_{1}\left( t\right)&=-\frac{\gamma }{\beta }\ln u_{1}\left( t\right) , \end{aligned}$$
(6c)

where the function \(u_{1}\left( t\right) \) is defined by

$$\begin{aligned} t =\intop _{1}^{u_{1}\left( t\right) }\frac{\mathrm {d}\xi }{\xi \left( \beta \xi s_{1}\left( 0\right) -\beta -\gamma \ln \xi \right) }. \end{aligned}$$
(6d)

These solutions are depicted in Fig. 3.

Fig. 3
figure 3

Time t dependence of functions \(s_{1}\left( t\right) \), \(i_{1}\left( t\right) \) and \(r_{1}\left( t\right) \) for \(\beta =0.56\) and \(\gamma =0.2\). The parameters are chosen for the COVID-19 outbreak before any population or government reactions came into effect [13]. Thus, they represent the natural COVID-19 outbreak without any control measures within the SIR approach

The SIRD model reduces to the SIR model by use of \(\frac{\mathrm {d}}{\mathrm {d}t}\left( r_{2}\left( t\right) +d_{2}\left( t\right) \right) =\frac{\mathrm {d}}{\mathrm {d}t}r_{1}\left( t\right) \), changing the index \(2\rightarrow 1\) and setting the mortality rate \(\sigma =0\). Hence, Eqs. (6a), (6b) and (6d) also hold for the SIRD case if the replacements in the indices \(1\rightarrow 2\) are performed while Eq. (6c) is a solution for \(r_{1}\left( t\right) =r_{2}\left( t\right) +d_{2}\left( t\right) \). Hence, exact solutions are given by

$$\begin{aligned} s_{2}\left( t\right)&=s_{2}\left( 0\right) u_{2}\left( t\right) , \end{aligned}$$
(7a)
$$\begin{aligned} i_{2}\left( t\right)&=1+\frac{\gamma +\sigma }{\beta }\ln u_{2}\left( t\right) -s_{2}\left( 0\right) u_{2}\left( t\right) , \end{aligned}$$
(7b)
$$\begin{aligned} r_{2}\left( t\right)&=-\frac{\gamma +\sigma }{\beta }\ln u_{2}\left( t\right) \nonumber \\&\quad -\sigma \intop _{0}^{t}\mathrm {d}t^{\prime }\!\left( \!1+\frac{\gamma +\sigma }{\beta }\ln u_{2}\left( t^{\prime }\right) -s_{2}\left( 0\right) u_{2}\left( t^{\prime }\right) \!\right) ,\end{aligned}$$
(7c)
$$\begin{aligned} d_{2}\left( t\right)&=\sigma \intop _{0}^{t}\mathrm {d}t^{\prime }\,\left( 1+\frac{\gamma +\sigma }{\beta }\ln u_{2}\left( t^{\prime }\right) -s_{2}\left( 0\right) u_{2}\left( t^{\prime }\right) \right) , \end{aligned}$$
(7d)
$$\begin{aligned} t&=\intop _{1}^{u_{2}\left( t\right) }\frac{\mathrm {d}\xi }{\xi \left( \beta \xi s_{2}\left( 0\right) -\beta -\left( \gamma +\sigma \right) \ln \xi \right) }. \end{aligned}$$
(7e)

Solutions for \(r_{2}\left( t\right) \) and \(d_{2}\left( t\right) \) are obtained by use of Eqs. (2d) and (6c), respectively. Figure 4 depicts exact SIRD solutions for \(s_{2}\left( t\right) \), \(i_{2}\left( t\right) \), \(r_{2}\left( t\right) \) and \(d_{2}\left( t\right) \).

Fig. 4
figure 4

Time t dependence of functions \(s_{2}\left( t\right) \), \(i_{2}\left( t\right) \), \(r_{2}\left( t\right) \) and \(d_{2}\left( t\right) \) for \(\beta =0.56\), \(\gamma =0.2\) and \(\sigma =0.01\). The parameters are chosen for the COVID-19 outbreak before any population or government reactions came into effect [13]. Thus, they represent the natural COVID-19 outbreak without any control measures within the SIRD approach

In addition, the SIQR model reduces to the SIR model by use of

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\left( q_{3}\left( t\right) +r_{3} \left( t\right) \right) =\frac{\mathrm {d}}{\mathrm {d}t}r_{1}\left( t\right) , \end{aligned}$$
(8)

changing the index \(3\rightarrow 1\) and setting the isolation rate \(\delta =0\). Hence, Eqs. (6a), (6b) and (6d) also hold for the SIQR case if the replacements \(\gamma \rightarrow \gamma +\delta \), \(1\rightarrow 3\) are performed, while Eq. (6c) is a solution for \(r_{1}\left( t\right) =q_{3}\left( t\right) +r_{3}\left( t\right) \). Hence, exact solutions are given by

$$\begin{aligned} s_{3}\left( t\right)&=s_{3}\left( 0\right) u_{3}\left( t\right) , \end{aligned}$$
(9a)
$$\begin{aligned} i_{3}\left( t\right)&=1+\frac{\gamma +\delta }{\beta }\ln u_{3}\left( t\right) -s_{3}\left( 0\right) u_{3}\left( t\right) , \end{aligned}$$
(9b)
$$\begin{aligned} q_{3}\left( t\right)&=\delta \intop _{0}^{t}\mathrm {d}t^{\prime }\,\text {e}^{\mu \left( t^{\prime }-t\right) }\,\nonumber \\&\quad \times \left( 1+\frac{\gamma +\delta }{\beta }\ln u_{3}\left( t^{\prime }\right) -s_{3}\left( 0\right) u_{3}\left( t^{\prime }\right) \right) ,\end{aligned}$$
(9c)
$$\begin{aligned} r_{3}\left( t\right)&=-\frac{\gamma +\delta }{\beta }\ln u_{3}\left( t\right) -\delta \intop _{0}^{t}\mathrm {d}t^{\prime }\,\text {e}^{\mu \left( t^{\prime }-t\right) }\,\nonumber \\&\quad \times \left( 1+\frac{\gamma +\delta }{\beta }\ln u_{3}\left( t^{\prime }\right) -s_{3}\left( 0\right) u_{3}\left( t^{\prime }\right) \right) , \end{aligned}$$
(9d)
$$\begin{aligned} t&=\intop _{1}^{u_{3}\left( t\right) }\frac{\mathrm {d}\xi }{\xi \left( \beta \xi s_{3}\left( 0\right) -\beta -\left( \gamma +\delta \right) \ln \xi \right) }. \end{aligned}$$
(9e)

Solutions for \(q_{3}\left( t\right) \) and \(r_{3}\left( t\right) \) are obtained by use of Eqs. (3c) and (6c), respectively. Figure 5 depicts exact SIQR solutions for \(s_{3}\left( t\right) \), \(i_{3}\left( t\right) \), \(q_{3}\left( t\right) \) and \(r_{3}\left( t\right) \).

Fig. 5
figure 5

Time t dependence of functions \(s_{3}\left( t\right) \), \(i_{3}\left( t\right) \), \(q_{3}\left( t\right) \) and \(r_{3}\left( t\right) \) for \(\beta =0.56\), \(\gamma =0.2\), \(\delta =0.1\) and \(\mu =0.02\). The parameters are chosen for the COVID-19 outbreak before any population or government reactions came into effect [13]. Thus, they represent the natural COVID-19 outbreak without any control measures within the SIQR approach. The number of isolated individuals reaches its maximum 14 days later in comparison to the peak of infected people

Note that the fraction of isolated individuals reaches its maximum later than the fraction of infected individuals. The difference is typically in the order of 10 days.

2.4 Small time limit

Equation (9e) simplifies for small times t since \(1-u_{3}\left( t\right) \) is small in this case. Hence, this evaluation describes the beginning of an outbreak and yields exact short-term solutions for the SIQR model given by

$$\begin{aligned} s_{3}\left( t\right)&=s_{3}\left( 0\right) +\frac{\beta \left( 1-s_{3}\left( 0\right) \right) }{\gamma +\delta -\beta }\left( \text {e}^{\left( \beta -\gamma -\delta \right) t}-1\right) , \end{aligned}$$
(10a)
$$\begin{aligned} i_{3}\left( t\right)&=\left( 1-s_{3}\left( 0\right) \right) \text {e}^{\left( \beta -\gamma -\delta \right) t}, \end{aligned}$$
(10b)
$$\begin{aligned} q_{3}\left( t\right)&=\frac{\left( 1-s_{3}\left( 0\right) \right) \delta }{\beta -\gamma -\delta +\mu }\left( \text {e}^{\left( \beta -\gamma -\delta \right) t}-\text {e}^{-\mu t}\right) , \end{aligned}$$
(10c)
$$\begin{aligned} r_{3}\left( t\right)&=\frac{\left( 1-s_{3}\left( 0\right) \right) \left( \gamma +\delta \right) }{\beta -\gamma -\delta }\left( \text {e}^{\left( \beta -\gamma -\delta \right) t}-1\right) \nonumber \\&\quad -\frac{\left( 1-s_{3}\left( 0\right) \right) \delta }{\beta -\gamma -\delta +\mu }\left( \text {e}^{\left( \beta -\gamma -\delta \right) t}-\text {e}^{-\mu t}\right) , \end{aligned}$$
(10d)

which also represent short-term solutions for SIR or SIRD models if \(r_{1}\left( t\right) =q_{3}\left( t\right) +r_{3}\left( t\right) \) or \(q_{3}\left( t\right) =d_{2}\left( t\right) \) is used, indices are changed, i.e. \(3\rightarrow 1\) or \(3\rightarrow 2\), and vanishing isolation rate \(\delta =0\) or \(\delta =\sigma \) is assumed, respectively. By considering Eqs. (10a)–(10d), it is obvious that the short-term behaviour of the epidemic outbreak and its growth rate is controlled by \(\beta -\gamma -\delta \) for the three models. Hence, at the start of an epidemic outbreak population and government reactions that lead to negative or at least small but positive \(\beta -\gamma -\delta \) are preferable [44].

Remarkably, this fact can also be derived quite trivially by using the fact that at the early stage of an outbreak the number of infected, quarantined and removed individuals is quite small in comparison with the total population number N, i.e. \(s_3\left( t\right) =1-i_3\left( t\right) -q_3\left( t\right) -r_3\left( t\right) =1+\mathcal {O}\left( i_3\left( t\right) +q_3\left( t\right) +r_3\left( t\right) \right) \) and \(i_3\left( t\right) +q_3\left( t\right) +r_3\left( t\right) \ll 1\) hold. Thus, Eq. (3b) simplifies to

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}i_{3}\left( t\right) =\left( \beta -\gamma -\delta \right) i_{3}\left( t\right) , \end{aligned}$$
(11)

which also leads to the solution given by Eq. (10b). Note that this approach also holds for time-dependent transmission rate \(\beta \left( t\right) \) and isolation rate \(\delta \left( t\right) \), respectively. Hence, Eq. (11) yields

$$\begin{aligned} i_{3}\left( t\right) =i_{3}\left( 0\right) \,\text {e}^{\intop _{0}^{t}\mathrm {d}t^{\prime }\,\left( \beta \left( t^{\prime }\right) -\gamma -\delta \left( t^{\prime }\right) \right) } \end{aligned}$$
(12)

in this case, which is a generalization of the small time limit given in Eq. (10b).

As an example of the observation that \(\beta -\gamma -\delta \) is dominating the early stage of an outbreak, we consider real COVID-19 data from Germany until 28 March 2020 [45]. The first COVID-19 case in Germany was reported on 27 January 2020. However, the number of daily cases was published from 04 March 2020 onwards, i.e. 36 days later. Furthermore, note that data analyses with confirmed cases need to be done carefully due to the delay of 1 to 2 weeks between the calculated and officially recorded number of confirmed cases. Moreover, official reports do not distinguish between infected and isolated individuals. For these reasons, we assume that the number of cases has been recorded from the 45th day of the outbreak and the short-term function \(i\left( t\right) + q\left( t\right) \) given by Eqs. (10b) and (10c) is fitted to real data, where \(\beta \) and \(\delta \) are determined by the fitting process. The result is depicted in Fig. 6. \(\beta =0.5109\) and \(\delta =0.2007\) are obtained which shows that isolation measures were taken and the transmission rate was reduced in comparison with \(\beta =0.56\) [13]. However, \(\beta -\gamma -\delta >0\) still held which explains the occurrence of the first COVID-19 wave in Germany. Thus, it is clear that a further decreasing transmission rate in combination with a higher isolation rate was needed for stamping out the outbreak. Note that within such an approach the continuous determination [13, 46,47,48,49] of this rates is a requirement since they show dynamic behaviour while \(\beta -\gamma -\delta <0\) is aimed for.

Remarkably, within this approach, it is not possible to propose a specific intervention during an epidemic outbreak based on this model. However, there are two possible advantages. First, as demonstrated in Fig. 6, the fitting procedure for the early phase of an outbreak yields the transmission and isolation rates \(\beta \) and \(\delta \), respectively. Thus, quantitative effects of any interventions on the rates can also be determined. Naturally, interventions only have to take effect individually so that the respective influence can be determined directly. However, since ultimately only the sign of \(\beta -\gamma -\delta \) plays a role, this method can be used for measuring the total effect of a number of interventions. Second, in addition to the weighting of the interventions \(\beta -\gamma -\delta <0\) can obviously be aimed for.

Fig. 6
figure 6

Short-time behaviour of \(i\left( t\right) + q\left( t\right) \) in comparison with the real data from Germany

2.5 Analysis of daily new cases

The assumption that infected individuals are quarantined leads to a relationship between the number of daily new cases \(C\left( t\right) \) and infected individuals since considering the incubation period distribution function \(\rho \left( t\right) \) for quarantining infected people and using the fact that a convolution of this function \(\rho \left( t\right) \) with \(i_3\left( t\right) \) describes individuals with starting symptoms leads to

$$\begin{aligned} c\left( t\right) :=\frac{C\left( t\right) }{N}=\intop _{-\infty }^{t}\mathrm {d}t^{\prime }\,\rho \left( t-t^{\prime }\right) i_{3}\left( t^{\prime }\right) . \end{aligned}$$
(13)

Since the incubation period distribution function \(\rho \left( t\right) \) should be a well-behaved function with just a single peak [50, 51], the method of steepest descent [52, 53] yields

$$\begin{aligned} c\left( t\right) \simeq i_{3}\left( t-\tau \right) \rho ^{\frac{3}{2}}\left( \tau \right) \sqrt{\frac{2\pi }{\left| \rho ^{\prime \prime }\left( \tau \right) \right| }},\qquad \rho ^{\prime }\left( \tau \right) =0, \end{aligned}$$
(14)

where \(\tau \) characterizes the position of the peak of the incubation period distribution function \(\rho \left( t\right) \) and \(\rho ^{\prime \prime }\left( t\right) \) is a shorthand notation, i.e. \(\rho ^{\prime \prime }\left( t\right) =\frac{\mathrm {d}^{2}\rho \left( t\right) }{\mathrm {d}t^{2}}\). Thus, we find

$$\begin{aligned} \frac{c\left( t\right) }{c\left( 0\right) }=\frac{i_{3}\left( t-\tau \right) }{i_{3}\left( -\tau \right) }, \end{aligned}$$
(15)

which yields

$$\begin{aligned} c\left( t\right) =\frac{c\left( 0\right) i_{3}\left( t-\tau \right) }{i_{3}\left( -\tau \right) } \end{aligned}$$
(16)

for the daily new cases \(c\left( t\right) \), where \(c\left( 0\right) \) is the initial number of daily cases. Hence, \(c\left( t\right) \) can be evaluated by use of the SIQR approach presented in Eq. (9b) for constant transmission rate \(\beta \) and isolation rate \(\delta \). In addition, for small times t and time-dependent rates \(\beta \left( t\right) \) and \(\delta \left( t\right) \) Eq. (12) yields

$$\begin{aligned} c\left( t\right) =c\left( 0\right) \,\text {e}^{\intop _{0}^{t}\mathrm {d}t^{\prime }\,\left( \beta \left( t^{\prime }-\tau \right) -\gamma -\delta \left( t^{\prime }-\tau \right) \right) }. \end{aligned}$$
(17)

Corresponding to this result, daily new cases \(c\left( t\right) \) were approximated by piecewise exponential functions for the analysis of the first wave of the COVID-19 outbreak [1, 54]. In analogy, Eq. (17) can be used for studying the effects of time-dependent transmission (isolation) rate \(\beta \left( t\right) \) (\(\delta \left( t\right) \)) with regard to the waves of the outbreak.

3 Measure

Lockdowns are widely used as mitigation or suppression strategies during epidemic disease outbreaks. In the analysis presented in the previous Sect. 2.4 it was observed that \(\beta -\gamma -\delta \) is the dominant factor at the beginning of an epidemic outbreak. Hence, for the purpose of controlling the spread of an infectious disease decreasing transmission rate \(\beta \) [13] or increasing isolation rate \(\delta \) is essential since the recovery rate \(\gamma \) is fixed for any given disease. However, the cost of such methodologies has to be taken into account. Replacing \(\beta \) with \(\left( 1-\alpha \right) \beta \) allows the definition of a measure function \(m\left( \alpha ,\delta \right) \) that is used for measuring the lockdown strength. \(\alpha =0\) describes the natural outbreak without any control reactions, while \(\alpha =1\) leads to a vanishing transmission rate and thus describes a total lockdown.

The epidemic peak \(i_3^{*}\) in Eq. (5) is taken as measure function, i.e.

$$\begin{aligned} m\left( \alpha ,\delta \right)= & {} 1+\frac{\gamma +\delta }{\left( 1-\alpha \right) \beta }\nonumber \\&\times \left( \ln \frac{\gamma +\delta }{\left( 1-\alpha \right) \beta }-1\right) ,~ m\left( \alpha ,\delta \right) \ge 0.\qquad \end{aligned}$$
(18)

Within this framework, Fig. 7 illustrates trajectories for which the measure function \(m\left( \alpha ,\delta \right) \) is constant.

Fig. 7
figure 7

\(\alpha \) in dependence of \(\delta \) for constant measure \(m\left( \alpha ,\delta \right) =m=0.1,0.2,0.3,0.4,0.5\), \(\beta =0.56\) and \(\gamma =0.2\) [13]. The trajectories are described by the linear relationship \(\alpha =1-\frac{\gamma +\delta }{\beta c}\) between \(\alpha \) and \(\delta \), where the constant c is connected to the constant value of \(m\left( \alpha ,\delta \right) =m\) by \(\text {e}^{m-1}=\left( \frac{c}{\text {e}}\right) ^{c}\)

Furthermore, a loss function

$$\begin{aligned} l\left( \alpha ,\delta \right) =k_{1}\alpha +k_{2}\alpha ^{2}+k_{3}\frac{\delta ^{2}}{\beta ^{2}} \end{aligned}$$
(19)

of quadratic form is used for the purpose of measuring the cost of any action the government or the population implements. This specification involves \(\alpha \) which describes the strength of measures and the isolation rate \(\delta \). Within this approach, the fraction \(\frac{\delta }{\beta }\) is considered since it leads to a loss function with arbitrary unit. Remarkably, the total loss includes not only the isolation for every individual but also the cost of organization, management, treatment, cooperation, loss of productivity, etc. Hence, the loss function should be nonlinear. Thus, in this study, a quadratic function Eq. (19) is implemented for estimating quarantine measures as suggested by previous studies about epidemic control [55,56,57]. Another quite obvious argument for assuming a nonlinear loss function is that there exists no linear relationship between the effects of medical treatment and disease. Note that the loss function \(l\left( \alpha ,\delta \right) \) increases with increasing \(\alpha \) or isolation rate \(\delta \) and \(l\left( \alpha ,\delta \right) \ge 0\) holds as expected. The coefficients \(k_1\), \(k_2\) and \(k_3\) are balancing cost factors due to scales and importance of the three parts in the loss function \(l\left( \alpha ,\delta \right) \). For instance, \(k_3>k_1,k_2\) (\(k_1,k_2>k_3\)) suggests that the loss caused by medical treatment (social distancing) is higher than the loss arising from social distancing (medical treatment). Moreover, the relationship between \(k_1\) and \(k_2\) determines whether social distancing contributes more linearly or nonlinearly. If any coefficient is equal to another, this leads to a balanced weighting of the related terms in the loss function in Eq. (19). Within this context, constant values for the loss \(l\left( \alpha ,\delta \right) =l\) are obtained for

$$\begin{aligned} \alpha&=-\frac{k_{1}}{2k_{2}}+\frac{\sqrt{\beta ^{2}\left( k_{1}^{2}+4k_{2}l\right) -4k_{2}k_{3}\delta ^{2}}}{2k_{2}\beta }\nonumber \\&=-\frac{k_{1}}{2k_{2}}+\frac{\sqrt{k_{1}^{2}+4k_{2}l}}{2k_{2}}-\frac{k_{3}\delta ^{2}}{\beta ^{2}\sqrt{k_{1}^{2}+4k_{2}l}}+\mathcal {O}\left( \delta ^{4}\right) . \end{aligned}$$
(20)

The minimum of the loss function \(l\left( \alpha ,\delta \right) \) is determined by considering

$$\begin{aligned} \mathcal {L}\left( \alpha ,\delta ,\lambda \right) =l\left( \alpha ,\delta \right) -\lambda \left( m\left( \alpha ,\delta \right) - m \right) , \end{aligned}$$
(21)

where \(\lambda \) is a Lagrange multiplier and \(m\left( \alpha ,\delta \right) =m\) a constraint at some level. Thus, considering \(\varvec{\partial }_{\alpha }\mathcal {L}\left( \alpha ,\delta ,\lambda \right) =\varvec{\partial }_{\delta }\mathcal {L}\left( \alpha ,\delta ,\lambda \right) =\varvec{\partial }_{\lambda }\mathcal {L}\left( \alpha ,\delta ,\lambda \right) =0\) yields the optimum trajectory \(\left( \alpha ^{*},\delta ^{*}\right) \) minimizing the loss function \(l\left( \alpha ,\delta \right) \) for a given measure function \(m\left( \alpha ,\delta \right) \), i.e.

$$\begin{aligned} \beta ^{2}\left( k_{1}+2\alpha ^{*} k_{2}\right) \left( 1-\alpha ^{*}\right) =2k_{3}\delta ^{*}\left( \gamma +\delta ^{*}\right) . \end{aligned}$$
(22)

The trajectory is depicted in Fig. 8 for varying ratios of \(k_1\), \(k_2\) and \(k_3\) and described by

$$\begin{aligned} \alpha ^{*}&=\frac{1}{2}-\frac{k_{1}}{4k_{2}}+\frac{\sqrt{\beta ^{2}\left( k_{1}+2k_{2}\right) ^{2}-16k_{2}k_{3}\delta ^{*}\left( \gamma +\delta ^{*}\right) }}{4k_{2}\beta } \nonumber \\&=1-\frac{2k_{3}\gamma \delta ^{*}}{\beta ^{2}\left( k_{1}+2k_{2}\right) }+\mathcal {O}\left( \delta ^{*2}\right) . \end{aligned}$$
(23)

Note that within this framework, only solutions \(\alpha ^{*}\in \left[ 0,1\right] \) have to be taken into account.

Fig. 8
figure 8

\(\alpha ^{*}\) in dependence of \(\delta ^{*}\) for \(\beta =0.56\), \(\gamma =0.2\) [13] and varying ratios of \(k_1\), \(k_2\) and \(k_3\)

Due to this analysis, it is observable that lockdown strategies depend on the intent. The epidemic peak is constant for certain values of \(\alpha \) and isolation rate \(\delta \), as observable in Fig. 7 while other values for the parameters depicted in Fig. 8 have to be chosen if the aim is governed by fears of loss. Remarkably, a total lockdown with \(\alpha =1\) is obviously not the best strategy. A constant epidemic peak is not achievable with this ansatz while the loss is only minimized with this approach if no quarantine measures are taken at all, i.e. isolation rate \(\delta ^{*}=0\). In contrast, increasing \(\delta ^{*}\) leads to decreasing \(\alpha ^{*}\). The minimum loss on the trajectory (23) lies at position

$$\begin{aligned} \alpha ^{*}=\frac{1}{2}-\frac{k_{1}}{4k_{2}},\qquad \delta ^{*}=-\frac{\gamma }{2}+\sqrt{\frac{\gamma ^{2}}{4}+\frac{\beta ^{2}\left( k_{1}+2k_{2}\right) ^{2}}{16k_{2}k_{3}}} \end{aligned}$$
(24)

and is described by

$$\begin{aligned} l\left( \alpha ^{*},\delta ^{*}\right)= & {} \frac{k_{1}}{2}-\frac{k_{1}^{2}}{8k_{2}}+\frac{k_{2}}{2}+\frac{k_{3}\gamma }{\beta ^{2}}\nonumber \\&\times \left( \frac{\gamma }{2}-\sqrt{\frac{\gamma ^{2}}{4}+\frac{\beta ^{2}\left( k_{1}+2k_{2}\right) ^{2}}{16k_{2}k_{3}}}\right) .\qquad \end{aligned}$$
(25)

This analysis shows that a lockdown, which is described by \(\alpha ^{*}\le \frac{1}{2}\), is quite sufficient for an optimized epidemic containment strategy that minimizes social costs if a corresponding isolation rate \(\delta ^{*}\) depending on \(\beta \), \(\gamma \), \(k_1\), \(k_2\) and \(k_3\) is provided.

This procedure can be combined with the stamping out strategy presented in Sect. 2.4. It was shown that an optimized strategy can be obtained by lowering the transmission rate \(\beta \) to \(\frac{\beta }{2}\) at maximum. The transmission rate should not be lowered further (with regard to the assumptions and models presented in this study). In contrast the isolation rate \(\delta \) needs to be increased in such a case such that the sign of \(\beta -\gamma -\delta \) is negative.

4 Conclusions

Exact solutions of SIRD and SIQR models were presented. In particular, short-term solutions were derived. Results show that an epidemic outbreak is dominated by \(\beta -\gamma -\delta \) at its early stage. This is of major interest since \(\beta \) and \(\delta \) can be modified by government or population reactions. Thus, a stamp out strategy for the epidemic can be used by generating negative \(\beta -\gamma -\delta \). The comparison of the analytical approach with real data from Germany indicates that this strategy was not pursued in Germany at the early stage of the COVID-19 pandemic.

In addition, optimizing strategies for choosing \(\alpha \) and isolation rate \(\delta \) were developed. One strategy leads to a constant epidemic peak \(i_3^{*}\), while another methodology minimizes the social loss function \(l\left( \alpha ,\delta \right) \) for particular values of \(\alpha \) and isolation rate \(\delta \). Accordingly, choosing the correct strategy depends on the intended purpose. One can try to stamp out the epidemic, reduce the epidemic peak or mitigate the social consequences of measures. In any case, it turns out that a complete lockdown with \(\alpha =1\) is not a reasonable alternative. Contrarily, regardless of the parameters \(\alpha \le \frac{1}{2}\) is obtained if the loss is minimized which indicates that well-chosen containment strategies far away from a complete lockdown are achievable.

Furthermore, the methodology with regard to minimizing the loss function presented in this study should be considered as a first step. The implementation of time t dependent transmission rate \(\beta \left( t\right) \) and isolation rate \(\delta \left( t\right) \) is an important possible extension which might lead to additional further insights for epidemic containment strategies.

Note that future research might thematise the incorporation of an input variable describing the proportion of vaccinated residents in a population which may improve the models further. Moreover, such an extension might be of interest for autoregressive models which might also use machine learning techniques [58]. Another open question is the uncertainty in the predictions. Markov chain Monte Carlo (MCMC) methods can be utilized for describing the stochastic behaviour of transitions between the compartmental populations. Moreover, compartmental models can be combined with deep learning models to predict several epidemic peaks based on the date of previous peaks and uncover hidden interconnections [59, 60]. The incorporation of spatial and temporal aspects to the dynamics of epidemic outbreaks is also of great interest. Self-organizing neural networks and fuzzy fractal approaches were proposed as a possible methodology for this purpose [61]. Furthermore, construction of models that can describe both government actions and individual reactions in combination with the influence of weather is of major relevance.