How Re-Infections and Newborns Can Impact Visible and Hidden Epidemic Dynamics?

Abstract

Mathematical modeling allows taking into account registered and hidden infections to make correct predictions of epidemic dynamics and develop recommendations that can reduce the negative impact on public health and the economy. A model for visible and hidden epidemic dynamics (published by the author in February 2025) has been generalized to account for the effects of re-infection and newborns. An analysis of the equilibrium points, examples of numerical solutions, and comparisons with the dynamics of real epidemics are provided. A stable quasi-equilibrium for the particular case of almost completely hidden epidemics was also revealed. Numerical results and comparisons with the COVID-19 epidemic dynamics in Austria and South Korea showed that re-infections, newborns, and hidden cases make epidemics endless. Newborns can cause repeated epidemic waves even without re-infections. In particular, the next epidemic peak of pertussis in England is expected to occur in 2031. With the use of effective algorithms for parameter identification, the proposed approach can ensure effective predictions of visible and hidden numbers of cases and infectious and removed patients.

1. Introduction

Mathematical modeling allows us to understand the dynamics of infectious disease epidemics, make predictions, and develop recommendations that can reduce the negative impact on public health and the economy. Some models of epidemic dynamics can be found in [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. In particular, different approaches are summarized in [13,14]. In this study, we will focus on the impact of newborns, re-infections, and invisible (unregistered) cases. The corresponding differential equations, initial conditions, parameter identification procedure, analysis of equilibrium points, and examples of numerical solutions will be presented.

Children born during an epidemic can significantly increase the number of susceptible patients, since many of them do not have immunity. To take into account this fact, a generalization of the classical SIR (susceptible-infectious-removed) model [1,2,3,4,5,6,7] was proposed in [4]. It was shown that oscillations in the number of new daily cases could occur. In this study, we propose a new approach for taking into account the influence of newborns and estimating the repeating epidemic waves.

Re-infections, typical for many infections including SARS-CoV-2 [15,16,17], were simulated in [18] with the use of the assumption that all infectious persons were registered. The equilibrium points and their stability were also discussed. The global endemic characteristics of SARS-CoV-2 disease were estimated. In this study, we will estimate the influence of re-infections on the visible and hidden epidemic dynamics.

Almost all infections have many asymptomatic and unregistered cases [19,20,21,22,23,24,25] that influence epidemic dynamics. A corresponding novel generalization of the classical SIR model, examples of analytical solutions, and number of registered and hidden cases for the COVID-19 pandemic and pertussis epidemic in England in 2023 and 2024 are presented in [26,27]. In this paper, we continue investigations of these infections, taking into account newborns and asymptomatic patients.

In this study, we propose the most general mathematical model that takes into account all the above factors. We will analyze the equilibrium points of the corresponding set of ordinary differential equations. The stability of these equilibria is investigated. Some examples of numerical solutions reflecting the dynamics of the COVID-19 pandemic in South Korea and Austria and the pertussis epidemic in England in 2023 and 2024 [28] are presented.

2. Materials and Methods

2.1. Differential Equations, Initial Conditions, and Parameter Identification Procedure

For every epidemic wave i, let us suppose the constant rate of increase in the number of susceptible persons ${\mu }_i$ (due to newborns, see Equation (1)) and divide the compartment of infectious persons I(t) (t is time) into visible (registered) $I^{(v)}$ and hidden $I^{(h)}$ (invisible/asymptomatic and unregistered) parts ($I=I^{(v)}+I^{(h)}$, see Equations (1)–(5)). As in the classical SIR model [1,2,3,4,5,6,7], we suppose that new infections appear at a rate ${\alpha }_{i}SI$ and reduce the number of susceptible persons (see Equation (1)). Only a part of these infectious persons are visible (symptomatic). That is why the increase in $I^{(v)}$ is divided by the visibility coefficient ${\beta }_i\ge 1$ (the ratio of total infections to the visible ones; case ${\beta }_i=1$ corresponds to the fully visible epidemic) in Equation (2) and multiplied by $({\beta }_i-1)/{\beta }_i$ in Equation (3) (then Equation $I=I^{(v)}+I^{(h)}$ holds, since $1/{\beta }_i+({\beta }_i-1)/{\beta }_i=1$).

As in [26,27], we divide the compartment of removed persons R(t) into visible (registered) and hidden parts $R=R^{(v)}+R^{(h)}$ and suppose that infectious persons are removed at rates ${\rho }_i^{(v)}{I}^{(v)}$ and ${\rho }_i^{(h)}{I}^{(h)}$, respectively (see Equations (4) and (5)). Individuals who have been infected may lose immunity and become infected again. To take into account these re-infections, let us suppose the waning immunity rates to be proportional to the number of removed persons ${\delta }_i^{(v)}{R}^{(v)}$ and ${\delta }_i^{(h)}{R}^{(h)}$ [18]. The corresponding terms appear in Equations (4) and (5) with a minus sign, and in Equation (1) with a plus sign.

Thus, the model takes the following form:

$${\displaystyle \frac{dS}{dt}}={\mu }_i-{\alpha }_{i}S(I^{(v)}+I^{(h)})+{\delta }_i^{(v)}{R}^{(v)}+{\delta }_i^{(h)}{R}^{(h)},$$

(1)

$${\displaystyle \frac{d{I}^{(v)}}{dt}}={\displaystyle \frac{{\alpha }_i}{{\beta }_i}}S(I^{(v)}+I^{(h)})-{\rho }_i^{(v)}{I}^{(v)},$$

(2)

$${\displaystyle \frac{d{I}^{(h)}}{dt}}={\displaystyle \frac{({\beta }_i-1){\alpha }_i}{{\beta }_i}}S(I^{(v)}+I^{(h)})-{\rho }_i^{(h)}{I}^{(h)},$$

(3)

$${\displaystyle \frac{d{R}^{(v)}}{dt}}={\rho }_i^{(v)}{I}^{(v)}-{\delta }_i^{(v)}{R}^{(v)},$$

(4)

$${\displaystyle \frac{d{R}^{(h)}}{dt}}={\rho }_i^{(h)}{I}^{(h)}-{\delta }_i^{(h)}{R}^{(h)}.$$

(5)

Infection, removal, and waning immunity rates (${\alpha }_i$, ${\rho }_i^{(v)}$, ${\rho }_i^{(h)}$, ${\delta }_i^{(v)}$, ${\delta }_i^{(h)}$), the visibility coefficient ${\beta }_i$ and the increasing rate of susceptible persons ${\mu }_i$ are supposed to be constant for every epidemic wave, i.e., for the time periods: $t_i^{*}\le t\le t_{i+1}^{*},\hspace{0.33em}i=1,2,3,\dots$.

Summarizing Equations (1)–(5) yields a non-zero value of the derivative:

$${\displaystyle \frac{d(S+I^{(v)}+I^{(h)}+R^{(v)}+R^{(h)})}{dt}}={\mu }_i$$

(6)

and the solution of the differential Equation (6)

$$S+I^{(v)}+I^{(h)}+R^{(v)}+R^{(h)}=N_i+{\mu }_i(t-t_i^{*})$$

(7)

As in [7,26,27], we will consider the value N_i to be an unknown parameter of the model corresponding to the i-th wave, which is not equal to the known volume of the population and must be estimated by observations. There is no need to assume that before the outbreak, all people are susceptible (see, e.g., [4]), since many of them are protected by their immunity, distance, lockdowns, etc. Thus, we do not reduce the problem to a 4-dimensional one. This means that the solution can be obtained by the numerical integration of the set of five differential Equations (1)–(5) using the initial conditions:

$$\begin{array}{c}{I}^{(v)}(t_i^{*})=I_{vi}, I^{(h)}(t_i^{*})=I_{hi}, R^{(v)}(t_i^{*})=R_{vi}, R^{(h)}(t_i^{*})=R_{hi},\\ S(t_i^{*})=N_i-I_{vi}-I_{hi}-R_{vi}-R_{hi}\end{array}$$

(8)

If at the moment $t_i^{*}$ all previously infected persons are removed, we can take into account only cases starting to appear during the i-th wave and use the following values of parameters:

$$I_{vi}=1, I_{hi}={\beta }_i-1, R_{vi}=0, R_{hi}=0$$

(9)

In comparison with the zero re-infections case [26], the accumulated number of registered cases $V^{(v)}$ is no longer equal to $I^{(v)}+R^{(v)}$. We can state only that the derivative of this function (daily or monthly number of new cases) equals

$${\displaystyle \frac{d{V}^{(v)}}{dt}}={\displaystyle \frac{{\alpha }_i}{{\beta }_i}}S(I^{(v)}+I^{(h)})$$

(10)

To determine $V^{(v)}$ values, we have to integrate (10) as follows:

$$V^{(v)}={\displaystyle \frac{{\alpha }_i}{{\beta }_i}}{\displaystyle \underset{t_i^{*}}{\overset{t}{\int }}S(I^{(v)}+I^{(h)})dt}$$

(11)

(the accumulation of cases has started at the moment $t_i^{*}$).

The values of 13 parameters ${\alpha }_i$, ${\rho }_i^{(v)}$, ${\rho }_i^{(h)}$, ${\delta }_i^{(v)}$, ${\delta }_i^{(h)}$, ${\beta }_i$, ${\mu }_i$, $N_i$, $t_i^{*}$, $I_{vi}$, $I_{hi}$, $R_{vi}$, $R_{hi}$ are unknown and have to be determined based on the results of observations (e.g., accumulated number of visible cases $V_j^{(v)}$ registered at moments of time $t_j\hspace{0.33em},j=1,2,\dots ,n$). In particular, the method of least squares [29] can be used:

$${\displaystyle \sum _{j=1}^n{\left[V^{(v)}(t_j)-V_j^{(v)}\right]}^2}\to \min$$

The values $V^{(v)}(t_j)$ can be calculated using (11). To estimate the values of all parameters, we need at least 13 observations. Since the results of the observations are random, the accuracy of the parameter identification increases with increasing n. However, a larger number of observations requires a longer time during which the parameters can change and cannot be considered constant. The experience of determining the optimal values of the four parameters of the classical SIR model for the first waves of the COVID-19 pandemic showed that 14 observations are enough for fairly accurate and long-term forecasts [7]. Since the set of three differential equations corresponding to the classical SIR model has an exact solution [7], the calculations of the optimal values of the four parameters do not require a lot of time. The set of differential Equations (1)–(5) can be solved only numerically, and the identification of 13 parameters requires the use of high-performance computing, parallel codes, and/or AI methods.

2.2. Equilibrium Points

Let us find the values $S^{*}$, $I^{*(v)}$, $I^{*(h)}$, $R^{*(v)}$, $R^{*(h)}$ corresponding to zero derivatives on the left-hand side of Equations (1)–(5). These equilibrium points provide the endemic characteristics of a disease. It follows from (2) and (3) that,

$$I^{*(h)}={\displaystyle \frac{{\rho }_i^{(v)}{\beta }_i-{\alpha }_i{S}^{*}}{{\alpha }_i{S}^{*}}}{I}^{*(v)}.$$

(12)

If $I^{*(v)}\ne 0$, then Equations (3) and (12) yield:

$$S^{*}={\displaystyle \frac{{\beta }_i{\rho }_i^{(v)}{\rho }_i^{(h)}}{{\alpha }_i\left[{\rho }_i^{(h)}+{\rho }_i^{(v)}({\beta }_i-1)\right]}}.$$

(13)

By substituting dS/dt = 0 into Equation (1), we obtain

$${\alpha }_{i}S(I^{*(v)}+I^{*(h)})={\mu }_i+{\delta }_i^{(v)}{R}^{*(v)}+{\delta }_i^{(h)}{R}^{*(h)}.$$

After taking into account (12), the following relationship is valid:

$$I^{*(v)}{\rho }_i^{(v)}{\beta }_i={\mu }_i+{\delta }_i^{(v)}{R}^{*(v)}+{\delta }_i^{(h)}{R}^{*(h)}.$$

(14)

Equations (4), (5), (12), and (14) yield

$$I^{*(v)}\left[{\rho }_i^{(v)}({\beta }_i-1)-{\rho }_i^{(h)}{\displaystyle \frac{{\rho }_i^{(v)}{\beta }_i-{\alpha }_i{S}^{*}}{{\alpha }_i{S}^{*}}}\right]={\mu }_i.$$

(15)

Taking into account (13), it can be shown that the expression in the square brackets equals zero. Then non-trivial equilibrium ($I^{*(v)}\ne 0$) occurs only at ${\mu }_i$ = 0 with arbitrary values of $I^{*(v)}$. Corresponding characteristics $I^{*(h)}$ and $S^{*}$ can be calculated from (12), (13), and the equilibrium number of removed persons with the use of (4), (5), and (12):

$$R^{*(v)}={\displaystyle \frac{{\rho }_i^{(v)}{I}^{*(v)}}{{\delta }_i^{(v)}}},$$

(16)

$$R^{*(h)}={\displaystyle \frac{{\rho }_i^{(h)}({\rho }_i^{(v)}{\beta }_i-{\alpha }_i{S}^{*})}{{\delta }_i^{(h)}{\alpha }_i{S}^{*}}}{I}^{*(v)}$$

(17)

The endemic characteristics of the fully visible epidemic (${\beta }_i$ = 1) and the stability of the equilibrium were considered in [18].

When re-infections are neglected (e.g., for pertussis), ${\delta }_i^{(v)}$ = ${\delta }_i^{(h)}$ = 0 and Equations (1)–(3) do not depend on Equations (4) and (5). Equation (12) holds for the equilibrium points of set (1)–(3), and after substituting into (1), we obtain

$$I^{*(v)}={\displaystyle \frac{{\mu }_i}{{\rho }_i^{(v)}{\beta }_i}}$$

(18)

The stability of this equilibrium can be analyzed numerically after parameter identification. Here, we consider some particular cases in which this analysis can be performed.

If the removal rates for visible and hidden patients are equal (${\rho }_i^{(v)}$ = ${\rho }_i^{(h)}$), summarizing (2) and (3) yields the following equation:

$${\displaystyle \frac{dI}{dt}}={\displaystyle \frac{{\alpha }_i}{{\beta }_i}}SI-{\rho }_i^{(v)}I,$$

(19)

where $I=I^{(v)}+I^{(h)}$ is the total number of infectious persons. Without re-infection, Equations (1) and (19) do not depend on (3), and the equilibrium values are as follows:

$$S^{*}={\displaystyle \frac{{\beta }_i{\rho }_i^{(v)}}{{\alpha }_i}}, I^{*}={\displaystyle \frac{{\mu }_i}{{\alpha }_i{S}^{*}}}={\displaystyle \frac{{\mu }_i}{{\beta }_i{\rho }_i^{(v)}}}$$

(20)

Jacobian matrix [30,31,32] for the set of differential Equations (1) and (19) is equal to:

$$J=‖\begin{array}{l}-{\alpha }_i{I}^{*}\hspace{1em}-{\alpha }_i{S}^{*}\hspace{1em}\\ {\alpha }_i{I}^{*}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\alpha }_i{S}^{*}-{\rho }_i^{(v)}\end{array}‖$$

(21)

Taking into account (20), Jacobian (21) can be written as follows:

$$J=‖\begin{array}{l}-{\displaystyle \frac{{\alpha }_i{\mu }_i}{{\beta }_i{\rho }_i^{(v)}}}\hspace{1em}{\beta }_i{\rho }_i^{(v)}\\ {\displaystyle \frac{{\alpha }_i{\mu }_i}{{\beta }_i{\rho }_i^{(v)}}}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}0\end{array}‖$$

(22)

The eigenvalues of (22)

$${\lambda }_{1,2}={\displaystyle \frac{1}{2}}\left[-{\displaystyle \frac{{\alpha }_i{\mu }_i}{{\beta }_i{\rho }_i^{(v)}}}\pm \sqrt{{\left({\displaystyle \frac{{\alpha }_i{\mu }_i}{{\beta }_i{\rho }_i^{(v)}}}\right)}^2-4{\alpha }_i{\mu }_i}\right]$$

(23)

have a negative real part; thus, the equilibrium is stable [30,31,32]. The case of the fully visible epidemic ($I=I^{(v)}$) can be easily obtained from (20)–(23) by putting ${\beta }_i$ = 1.

It follows from (10) and (20) that the number of new cases increases at a constant rate:

$${\displaystyle \frac{d{V}^{(v)}}{dt}}={\displaystyle \frac{{\alpha }_i}{{\beta }_i}}{S}^{*}{I}^{*}={\displaystyle \frac{{\mu }_i}{{\beta }_i}},$$

(24)

when this equilibrium is reached. According to (24), the total number of cases and susceptible persons increases at an equal rate ${\mu }_i$. Pertussis in England demonstrated an endemic state in 2018 and 2019, with average monthly numbers of new cases of 246 (2948/12), and 307 (3680/12), respectively (see [28] and Supplementary Figure S1). By multiplying these values by the visibility coefficient, we can estimate the increase rate of susceptible newborns (see (24)). At moderate values of ${\beta }_i$ this rate is much lower than the total birth rate (563,561 live births occurred in England in 2023, [33]). To decrease the endemic level, vaccination of children and pregnant women is necessary, since at a fixed birth rate, only these interventions can decrease the value of the parameter ${\mu }_i$.

2.3. Quasi-Equilibrium Point

Effective isolation of symptomatic patients can lead to a situation where $I^{(v)}\ll I^{(h)}$. Then it follows from (1) and (3) that

$${\displaystyle \frac{dS}{dt}}\approx {\mu }_i-{\alpha }_{i}S{I}^{(h)}+{\delta }_i^{(v)}{R}^{(v)}+{\delta }_i^{(h)}{R}^{(h)},$$

(25)

$${\displaystyle \frac{d{I}^{(h)}}{dt}}\approx {\displaystyle \frac{({\beta }_i-1){\alpha }_i}{{\beta }_i}}S{I}^{(h)}-{\rho }_i^{(h)}{I}^{(h)},$$

(26)

and a state with almost constant number of invisible infectious persons can occur at

$$S^{*}\approx {\displaystyle \frac{{\rho }_i^{(h)}{\beta }_i}{{\alpha }_i({\beta }_i-1)}}$$

(27)

(it follows from (26) at $d{I}^{(h)}/dt=0$). If the number of removed persons is constant, Equations (4), (5), and (25) yield:

$${\displaystyle \frac{dS}{dt}}\approx \left\{\begin{array}{l}{\mu }_i-{\alpha }_{i}S{I}^{(h)}+{\rho }_i^{(h)}{I}^{(h)},\hspace{1em}{\delta }_i^{(h)}>0\\ \hspace{1em}{\mu }_i-{\alpha }_{i}S{I}^{(h)},\hspace{1em}{\delta }_i^{(h)}=0\end{array}\right.$$

(28)

and approximately constant values of susceptible persons at

$$I^{*(h)}\approx \left\{\begin{array}{l}{\displaystyle \frac{{\mu }_i}{{\alpha }_i{S}^{*}-{\rho }_i^{(h)}}}\approx {\displaystyle \frac{{\mu }_i({\beta }_i-1)}{{\rho }_i^{(h)}}},\hspace{1em}{\delta }_i^{(h)}>0\\ \hspace{1em}\hspace{1em}{\displaystyle \frac{{\mu }_i}{{\alpha }_i{S}^{*}}}\approx {\displaystyle \frac{{\mu }_i({\beta }_i-1)}{{\beta }_i{\rho }_i^{(h)}}},\hspace{1em}{\delta }_i^{(h)}=0.\end{array}\right.$$

(29)

For the set of approximate differential Equations (26) and (28), the Jacobian matrix [30,31,32] is equal to:

$$J=\left\{\begin{array}{l}‖\begin{array}{l}-{\alpha }_i{I}^{*(h)}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{1em}\hspace{1em}-{\alpha }_i{S}^{*}+{\rho }_i^{(h)}\\ {\displaystyle \frac{{\alpha }_i{I}^{*(h)}({\beta }_i-1)}{{\beta }_i}}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}{\displaystyle \frac{{\alpha }_i{S}^{*}({\beta }_i-1)}{{\beta }_i}}-{\rho }_i^{(h)}\end{array}‖,\hspace{1em}{\delta }_i^{(h)}>0\\ ‖\begin{array}{l}-{\alpha }_i{I}^{*(h)}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{1em}\hspace{1em}-{\alpha }_i{S}^{*}\\ {\displaystyle \frac{{\alpha }_i{I}^{*(h)}({\beta }_i-1)}{{\beta }_i}}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}{\displaystyle \frac{{\alpha }_i{S}^{*}({\beta }_i-1)}{{\beta }_i}}-{\rho }_i^{(h)}\end{array}‖,\hspace{1em}{\delta }_i^{(h)}=0\end{array}\right.$$

(30)

Taking into account (27) and (29), Jacobian (30) can be written as follows:

$$J\approx \left\{\begin{array}{l}‖\begin{array}{l}-{\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\rho }_i^{(h)}}}\hspace{1em}\hspace{1em}-{\displaystyle \frac{{\rho }_i^{(h)}}{{\beta }_i-1}}\\ {\displaystyle \frac{{\alpha }_i{\mu }_i{({\beta }_i-1)}^2}{{\beta }_i{\rho }_i^{(h)}}}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}0\end{array}‖,\hspace{1em}{\delta }_i^{(h)}>0\\ ‖\begin{array}{l}-{\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i{\rho }_i^{(h)}}}\hspace{1em}\hspace{1em}-{\displaystyle \frac{{\rho }_i^{(h)}}{{\beta }_i-1}}\\ {\displaystyle \frac{{\alpha }_i{\mu }_i{({\beta }_i-1)}^2}{{\beta }_i^2{\rho }_i^{(h)}}}\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{0.33em}0\end{array}‖,\hspace{1em}{\delta }_i^{(h)}=0\end{array}\right.$$

(31)

The eigenvalues of Jacobian (31)

$${\lambda }_{1,2}\approx \left\{\begin{array}{l}{\displaystyle \frac{1}{2}}\left[-{\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\rho }_i^{(h)}}}\pm \sqrt{{\left({\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\rho }_i^{(h)}}}\right)}^2-{\displaystyle \frac{4{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i}}}\right],\hspace{1em}{\delta }_i^{(h)}>0\\ {\displaystyle \frac{1}{2}}\left[-{\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i{\rho }_i^{(h)}}}\pm \sqrt{{\left({\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i{\rho }_i^{(h)}}}\right)}^2-{\displaystyle \frac{4{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i^2}}}\right],\hspace{1em}{\delta }_i^{(h)}=0\end{array}\right.$$

(32)

have a negative real part at ${\beta }_i$ > 1, thus the quasi-equilibrium is stable [30,31,32] and it follows from (10), (27), and (29) that the number of new visible cases increases at approximately a constant rate:

$${\displaystyle \frac{d{V}^{(v)}}{dt}}\approx {\displaystyle \frac{{\alpha }_i}{{\beta }_i}}{S}^{*}{I}^{*(h)}\approx \left\{\begin{array}{l}\hspace{1em}{\mu }_i,\hspace{1em}{\delta }_i^{(h)}>0\\ {\mu }_i/{\beta }_i,\hspace{1em}{\delta }_i^{(h)}=0\end{array}\right.$$

(33)

When re-infections occur (${\delta }_i^{(h)}$ > 0), the rate of the number of visible cases is approximately equal to ${\mu }_i$ and does not depend on infection, removal, re-infection, and visibility coefficients. Without re-infections, the number of visible cases increases at ${\beta }_i$ times smaller rate (see (33)). In both cases, the epidemic is driven by invisible (asymptomatic) infectious patients.

Such a quasi-equilibrium state probably occurred in South Korea, Austria, Spain, and France in June 2020 between the first and second waves of the COVID-19 pandemic (see average daily numbers of visible cases calculated in Chapter 8 of the book [7]). For example, in South Korea in June 2020, the smoothed daily numbers of new cases DC varied from 38 to 48.1 (see black “crosses” in Figure 1 representing dataset from [34], version uploaded on 23 December 2023) with the average value 44.4 = (12,799 – 11,468)/30, which according to (33) could be equal to ${\mu }_i$. The relative variation is approximately 23% (10.1/44.4). In 2020, 272,337 births were registered in South Korea [35] (the average value is 744 per day). Thus, only 6% of newborns become susceptible.

In Austria, the smoothed daily numbers of new cases varied from 24.6 to 35.4 in the period from 27 May to 24 June 2020 (see black “crosses” in Figure 2 representing dataset from [34], version of 23 December 2023) with an average value of 31.1, and relative variation of 36%. The average daily number of newborns was 243 in 2020 (the figure was calculated with the use of [36,37]), 12.8% of which became susceptible. A comparison of the percentages of susceptible newborns allows us to conclude that in June 2020, tracing and isolation of infections in South Korea were more effective. It must be noted that many countries (e.g., Ukraine, the UK, the US, Italy, Germany, Sweden, and the Republic of Moldova) have not achieved the quasi-steady state, and in some of them, the second COVID-19 wave started before finishing the first one [7]. Probably, in these countries, the efficacy of isolation infectious patients was not enough to have $I^{(v)}\ll I^{(h)}$.

The existence of a non-trivial stable quasi-equilibrium makes epidemics endless since newborns are in every country, and it is very difficult to isolate all asymptomatic infectious persons. Moreover, in the case of complex eigenvalues (32), the epidemic waves can repeat with the period.

$$T={\displaystyle \frac{2\pi }{w}};\hspace{1em}w\approx \left\{\begin{array}{l}{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{4{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i}}-{\left({\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\rho }_i^{(h)}}}\right)}^2},\hspace{1em}{\mu }_i<{\displaystyle \frac{4{({\rho }_i^{(h)})}^2}{{\alpha }_i{\beta }_i({\beta }_i-1)}},\hspace{1em}{\delta }_i^{(h)}>0\\ {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{4{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i^2}}-{\left({\displaystyle \frac{{\alpha }_i{\mu }_i({\beta }_i-1)}{{\beta }_i{\rho }_i^{(h)}}}\right)}^2},\hspace{1em}{\mu }_i<{\displaystyle \frac{4{({\rho }_i^{(h)})}^2}{{\alpha }_i({\beta }_i-1)}},\hspace{1em}{\delta }_i^{(h)}=0\end{array}\right.$$

(34)

This is because, in the vicinity of the equilibrium point, the linearized set of differential equations has oscillatory solutions [4,32]. For example, in 2020 and 2021, Zero-COVID countries [38] attempted to stop community transmission of SARS-CoV-2 infection using contact tracing, mass testing, border quarantine, lockdowns, and mitigation software. In particular, in Hong Kong, the smoothed daily numbers of new COVID-19 cases were less than 20 per million in 2020 and 2021 [34], i.e., the epidemic was controlled but not removed completely, since in 2022 a huge epidemic wave occurred [34]. Other zero-COVID countries have also experienced severe pandemic waves after reducing quarantine limitations and decreasing the test-per-case ratio [39].

A similar oscillatory solution can exist for the stable equilibrium discussed in the previous Section (${\delta }_i^{(v)}$ = ${\delta }_i^{(h)}$ = 0, ${\rho }_i^{(v)}$ = ${\rho }_i^{(h)}$). It follows from (23) that the corresponding frequency is

$$w={\displaystyle \frac{1}{2}}\left[\sqrt{4{\alpha }_i{\mu }_i-{\left({\displaystyle \frac{{\alpha }_i{\mu }_i}{{\beta }_i{\rho }_i^{(v)}}}\right)}^2}\right],\hspace{1em}{\mu }_i<{\displaystyle \frac{4{\left[{\beta }_i{\rho }_i^{(v)}\right]}^2}{{\alpha }_i}}$$

(35)

3. Results and Discussion

3.1. The COVID-19 Pandemic Dynamics in South Korea and Austria

The set of differential Equations (1)–(5) was integrated numerically using the initial conditions (8), (9), and the fourth-order Runge-Kutta method [32]. The results for the COVID-19 pandemic in South Korea and Austria were tested using exact solutions for the classical SIR model [7] and are shown in Figure 1, Figure 2, Figure 3 and Figure 4 for different parameter values. Blue curves represent the accumulated numbers of visible cases $V^{(v)}$; black ones—the theoretical estimations of the daily numbers of visible cases $d{V}^{(v)}/dt$. The daily numbers of hidden cases have the same trends, since according to (3) and (10), $d{V}^{(h)}/dt=({\beta }_i-1)d{V}^{(v)}/dt$. The method of least squares was not used to detect the optimal parameter values. We used only parameter sets that provide some similarity to the observed numbers of accumulated (AC) and daily cases (DC) registered after outbreaks, as shown by blue and black “crosses”, respectively.

Figure 2 and Figure 3 illustrate the dynamics of the COVID-19 pandemic in Austria. The solid lines correspond to the values of the parameters:

${\alpha }_i$ = 1.924971 × 10⁻⁵ day⁻¹; ${\rho }_i^{(v)}$ = 0.92365 day⁻¹; ${\rho }_i^{(h)}$ = 0.97226 day⁻¹; ${\delta }_i^{(v)}$ = ${\delta }_i^{(h)}$ = 0.0202 day⁻¹; ${\mu }_i$ = 31.1 day⁻¹; ${\beta }_i$ = 1.5; $N_i$ = 56,382; $t_i^{*}$ = 17.19408 days (zero value of time corresponds to 15 January 2020, day⁻¹ means persons per day); $I_{vi}$ = 1; $I_{hi}$ = 0.5; $R_{vi}$ = $R_{hi}$ = 0. The dashed curves correspond to the same values of parameters, but zero birth rate ${\mu }_i$ = 0. Dotted lines correspond to the fully visible epidemic with ${\beta }_i$ = 1; ${\delta }_i^{(v)}$ = ${\delta }_i^{(h)}$ = 0; ${\rho }_i^{(v)}$ = 0.92365 day⁻¹, which can be simulated with the use of the classical SIR model [7]. The optimal values of the parameters calculated in [7] based on previous versions of the AC dataset [34] differ from the listed set of classical SIR model parameters.

Figure 1 and Figure 4 illustrate the dynamics of the COVID-19 pandemic in South Korea. Solid lines correspond to the values of the parameters:

${\alpha }_i$ = 8.711987 × 10⁻⁵ day⁻¹; ${\rho }_i^{(v)}$ = 5.6226988 day⁻¹; ${\rho }_i^{(h)}$ = 5.5614615 day⁻¹; ${\delta }_i^{(v)}$ = ${\delta }_i^{(h)}$ = 0.0202 day⁻¹; ${\mu }_i$ = 44.4 day⁻¹; ${\beta }_i$ = 1.1; $N_i$ = 68,038; $t_i^{*}$ = 29.805 days (zero value of time corresponds to 15 January 2020); $I_{vi}$ = 1; $I_{hi}$ = 0.5; $R_{vi}$ = $R_{hi}$ = 0. The dashed curves correspond to the same values of parameters, but zero birth rate ${\mu }_i$ = 0. Dotted lines correspond to the fully visible epidemic with ${\beta }_i$ = 1; ${\delta }_i^{(v)}$ = ${\delta }_i^{(h)}$ = 0; ${\rho }_i^{(v)}$ = 5.6226988 day⁻¹, which can be simulated with the use of the classical SIR model [7]. The optimal values of parameters calculated in [7] based on previous versions of the AC dataset [34] differ from the set of classical SIR model parameters.

The classical SIR model yields very optimistic predictions of the epidemic duration and final numbers of visible cases (see dotted curves in Figure 1, Figure 2, Figure 3 and Figure 4). Re-infections yield infinite epidemic durations even at zero birth rate ${\mu }_i$ = 0 (see dashed curves in Figure 1, Figure 2, Figure 3 and Figure 4, which tend to a non-trivial equilibrium $I^{*(v)}\ne 0$). Corresponding characteristics $I^{*(h)}$, $S^{*}$, $R^{*(v)}$ and $R^{*(h)}$ can be calculated with the use of (12), (13), (16), and (17). Then, the constant equilibrium number of new daily visible cases, DC^*, can be determined from Equation (10) as follows:

$$D{C}^{*}={\rho }_i^{(v)}{I}^{*(v)}$$

(36)

South Korea has stopped reporting new COVID-19 cases at a rather high level of DC (see black “crosses” in Figure 4). Equation (36) demonstrates that the number of infectious persons in South Korea remained high in August 2023. In Austria, there was a decreasing DC trend before stopping with notifications of new cases in June 2023 (see black “crosses” in Figure 3).

Solid curves in Figure 1, Figure 2, Figure 3 and Figure 4 represent the case ${\mu }_i>0$ and show no equilibrium trends (as mentioned in Section 3). Both $V^{(v)}$ (blue) and $d{V}^{(v)}/dt$ (black) values increased in 2023 (see Figure 3 and Figure 4). The accumulated number of cases $V^{(v)}$ was quite consistent with AC values registered in South Korea before October 2021 (compare solid blue curves and “crosses” in Figure 1 and Figure 4). AC and DC numbers registered in Austria from mid-March to mid-August 2020 are rather close to the values $V^{(v)}$ and $d{V}^{(v)}/dt$ corresponding to the case ${\mu }_i$ = 0 (compare “crosses” and dashed curves in Figure 2).

Re-infections cause oscillations in the daily numbers of new cases (see solid and dashed black curves in Figure 1, Figure 2, Figure 3 and Figure 4) and create the illusion of different pandemic waves, although the model parameters were considered fixed throughout the time period shown in Figure 1, Figure 2, Figure 3 and Figure 4. To achieve better coincidence with the real dynamics (influenced by changes in quarantine restrictions, social behavior, new virus strains, vaccinations, etc.), simulations must be repeated using observations for different periods (for example, for February–March 2022 in South Korea, see Figure 4).

3.2. The Pertussis Epidemic in England

To simulate the pertussis outbreak in England in 2023 and 2024, the set of differential Equations (1)–(5) was integrated numerically using initial conditions (8), (9), and the fourth-order Runge-Kutta method [32]. The results were tested using the exact solutions for the SIR model [26,40] and are shown in Figure 5 and Figure 6 for different parameter values. Blue curves represent the accumulated numbers of visible cases $V^{(v)}$; black ones—the theoretical estimations of the daily numbers of visible cases $d{V}^{(v)}/dt$. The red lines represent the number of infectious persons.

The accumulated registered number of cases is listed in Table S1 according to the dataset [28] (version available on 13 February 2025) and is shown by blue ”crosses”. The average daily number of new cases DC was calculated using the formula presented in [40], and is listed in Table S1 and shown by black “crosses”. Solid lines correspond to the values of the parameters:

${\alpha }_i$ = 2.112667 × 10⁻⁶ day⁻¹; ${\rho }_i^{(v)}$ = 7.717292 day⁻¹; ${\rho }_i^{(h)}$ = 0; ${\delta }_i^{(v)}$ = ${\delta }_i^{(h)}$ = 0; ${\mu }_i$ = 7.48 day⁻¹; ${\beta }_i$ = 1; $N_i$ = 3,657,890; $t_i^{*}$ = 0 (zero value of time corresponds to 31 December 2022); $I_{vi}$ = 0.029115; $I_{hi}$ = $R_{vi}$ = $R_{hi}$ = 0. The dotted and dashed curves represent SIR simulations of the first and second pertussis waves, respectively (${\mu }_i$ = 0, ${\beta }_i$ = 1, the values of other parameters are listed in [26]). All simulations suppose that the epidemic is completely visible, i.e., $I=I^{(v)}$; $V=V^{(v)}$; ${\beta }_i$ = 1.

In comparison with SIR simulations at ${\mu }_i$ = 0 (dotted and dashed curves), taking into account the birth rate (${\mu }_i$ = 7.48 day⁻¹, solid curves) allowed us to obtain rather good agreement between the theory and results of all observations (compare blue and black solid curves with blue and black “crosses” in Figure 5). Moreover, long epidemic waves were revealed, repeating after around 2500 days (6.85 years; see Figure 6). Formula (35) yields the period of 1678 days (4.6 years) for the solution of linearized differential equations in the vicinity of the asymptotically stable spiral point [32] corresponding to eigenvalues (23). The difference can be explained by the large deviation of the numerical solution (red and black curves in Figure 6) from the equilibrium values $I^{*(v)}={\mu }_i/({\rho }_i^{(v)}{\beta }_i)\approx 0.97$ (see (18)) and $d{V}^{(v)}/dt={\mu }_i/{\beta }_i\approx 7.48$ day⁻¹ (see (24)). To decrease the size of new waves (maximum total daily numbers of new cases), the parameter ${\mu }_i$ has to be diminished (due to vaccinations of children and pregnant women).

According to [28], “pertussis is a cyclical disease that peaks every 3 to 5 years, with the last cyclical increase occurring in 2016 and the last major outbreak occurring in 2012”. The number of laboratory-confirmed pertussis cases in England by quarter demonstrates that the previous peak was approximately 7.75 years before the peak registered in the second quarter of 2024 (Figure S2, taken from [28]). Thus, the results of the observations correlate with the theoretical findings. The accuracy of the predictions can be improved using the optimal parameter values. Since these values cannot be constant for a long time, the procedure of the parameter identification should be repeated as often as possible. The application of the least-squares method could be a topic for further investigation.

4. Conclusions

A recently proposed model for visible and hidden epidemic dynamics was generalized to take into account the impact of re-infection and newborns. The set of five differential equations and initial conditions contained 13 unknown parameters, which could be identified using the method of least squares. The analysis of the equilibrium points and the examples of numerical solutions provided allowed us to draw important conclusions regarding the influence of various factors on the epidemic dynamics.

It was shown that in the general case, no equilibrium exists, but stable equilibria are possible when the influence of re-infections or newborns can be neglected. If the number of visible infectious patients is much lower than the number of hidden cases, a stable quasi-equilibrium exists with constant daily numbers of new visible cases. Re-infections, newborns, and hidden cases make epidemics endless. These facts were illustrated by numerical results and comparisons with COVID-19 epidemic dynamics in Austria and South Korea. When re-infections can be neglected (e.g., for pertussis), newborns can cause repeated epidemic waves. In particular, numerical simulations of the pertussis epidemic in England in 2023 and 2024 demonstrated that the next epidemic peak is expected to occur in 2031.

The proposed model can be recommended for the calculations and predictions of visible and hidden numbers of cases and infectious and removed patients. With the use of effective algorithms for parameter identification, the accuracy of the method can be rather high.