Dynamics of Infectious Diseases Incorporating a Testing Compartment

Abstract

In this paper, we construct an infectious disease model with a testing compartment and analyze the existence and stability of its endemic states. We obtain the basic reproduction number, ${\mathcal{R}}_0$, and demonstrate the existence of one endemic equilibrium without testing and one endemic equilibrium with testing and prove their local and global stabilities based on the value of the basic reproduction number, ${\mathcal{R}}_0$. We then apply our model to the US COVID-19 pandemic and find that, for a large parameter set, including those relevant to the SARS-CoV-2 virus, our analytic and numerical results suggest that the trajectories will be trapped to the testing-free state when the testing number is small enough. This indicates that the pandemic may end with a testing-free endemic state through a novel and surprising mechanism called stochastic trapping.

1. Introduction

An infectious disease, also known as a transmissible disease or communicable disease, is an illness resulting from an infection. The total population can be simply classified into three different classes: susceptible, infected, and recovered [1]. The susceptible class represents individuals who are healthy but can contract the disease, the infected class represents individuals who have contracted the disease and are also infectious, and the recovered class represents individuals who have recovered. For real-world applications such as the collected or reported data, a positive test is actually required for an infected individual to be identified or confirmed. Testing for infectious diseases for this paper is defined to be any diagnosis that results in case numbers on record. A test-positive patient may either recover or die later. For example, the coronavirus Disease 2019 (COVID-19) is an infectious disease caused by the novel coronavirus that requires testing to be confirmed because the symptoms of COVID-19 are very similar to influenza, like fever, cough, breathing difficulties, etc. [2]. Due to its high transmission rate and morbidity, the disease quickly spread worldwide, resulting in a global pandemic in 2020 and causing massive social and economic losses.

Many model studies [3,4,5,6,7,8,9,10,11,12] have been done on the COVID-19 outbreak to investigate its transmission dynamics. The modeling studies are mostly based on the classic Susceptible–Infected–Recovered (SIR) compartmental model [1] in epidemiology. If the total population is fixed as 1, then the model can be written as follows:

$$S^{\prime }=-cSI,I^{\prime }=cSI-rI,R^{\prime }=rI,S+I+R=1,$$

where S is the proportion of the susceptible in the total population at time t, I is the infected, R is the recovered, and $S^{\prime }=S^{\prime }\left(t\right)$ is the rate of change (derivative) of variable $S\left(t\right)$. Parameter c is the disease transmission rate and r is the recovery rate. In [3], the authors added an asymptomatic (A) compartment in a diffusion model to study the dynamics of COVID-19 and determined the existence and local asymptotic stability of the endemic and disease-free equilibrium states. Other research works incorporated at least one exposed (E) compartment to study COVID-19 transmission [4,5,6,7,8,9], where [4] also took into account the hospitalized compartment to emphasize the vulnerable people in the population group, ref. [7] further considered a quarantined compartment and conducted the stability analysis, ref. [5] divided the infected class (I) into detected and undetected and found backward bifurcation for the model, and refs. [6,8,9] incorporated the virus concentration in the environment into the model and studied the impact of the indirect transmission route on COVID-19 dynamics. Another interesting work was presented in [10]. The author modeled COVID-19 via the fractional order mathematical model with diagnosed, ailing, and threatened compartments. There are more mathematical models of COVID-19 discussed in [11]. However, as far as we know, there is little research that has focused on the testing compartment for COVID-19 transmission or did not study the dynamics behaviors with a testing compartment. If the testing time cannot be neglected, like COVID-19, then the testing process may have an impact on disease transmission. The epidemiological dynamics of an outbreak is a known-unknown without testing. It is through testing that we gain a window on the spread of the disease in terms of speed and scope. Because of such an essential role testing plays in epidemiological understanding, theoretical models should consider testing as an important compartment.

In this paper, we first introduce a model modified from the SICM model with testing from [13], where S represents the susceptible class, I represents the infected class, C represents the confirmed class after a positive test, and M represents the monitored class to which the test-positive individuals will go. For the purpose of understanding the long-term dynamic behavior of infectious diseases in large population sizes, we further incorporate both the natural birth rate and the natural death rate into the new model, which is referred to as the SICMR model for distinction, where S, I, C, and M represent the same meanings as above and R represents the recovered class. We then obtain the existence as well as the local and global stabilities of endemic equilibrium states, which are of two types: endemic equilibrium without testing and endemic equilibrium with testing. As an application, we apply our theory to the U.S. COVID-19 pandemic by first best-fitting the model to the case and death numbers, and then analyzing the long-term behaviors of the best-fitted model. To our surprise, we find that our model is capable of deducing that the endemic state without large-scale testing is the outcome. This happens either because the endemic state has insignificant numbers of testing or due to the fact that the model has a testing-free invariant manifold and the SARS-CoV-2’s outbreak trajectories tend to fall towards an exponentially attracting region of the manifold, and as a result, when a trajectory stays long enough near the trapping region, stochastic fluctuations will eventually push the trajectory into the testing-free zone for good.

2. Model Formulation

There are three major modifications proposed in [13]. The first is the openness hypothesis that the total population is not fixed at the total population of a geographic region or state, say the U.S., but instead, it is a parameter, referred to as the effective susceptible population for a period of time. This is because, for example, when SARS-CoV-2 first appeared in Seattle, WA, it did not make every person in Nebraska susceptible. Also, if a person isolates themselves in terms of mitigation from the population, they cannot be susceptible to the disease at such times. The second modification is the inclusion of testing because it is the case numbers and death numbers from the disease that we can see, not the S or I, which are not directly observable and can only be triangulated by the testing numbers. The inclusion of testing results in two more compartments: the class C for confirmed by testing, and the class M for monitored after confirmation that requires at least one more test before going into the recovered class. The third modification is the incorporation of the demography (the natural birth and death rates) and the intrinsic recovery rate for the purpose of long-term disease dynamics study. We first present our dimensional model as follows:

$$\left\{\begin{array}{cc}{\overline{S}}^{\prime }\hfill & =\overline{\alpha }-\overline{c}\overline{S}\overline{I}-\mu \overline{S},\hfill \\ {\overline{I}}^{\prime }\hfill & =\overline{c}\overline{S}\overline{I}-\frac{a_1\overline{I}}{1+a_{1}h\overline{I}+a_{2}h\overline{M}}\overline{C}-\gamma \overline{I}-\mu \overline{I},\hfill \\ {\overline{C}}^{\prime }\hfill & =\frac{a_1\overline{I}}{1+a_{1}h\overline{I}+a_{2}h\overline{M}}\overline{C}-m\overline{C}-d\overline{C}-\mu \overline{C},\hfill \\ {\overline{M}}^{\prime }\hfill & =m\overline{C}-q\overline{M}-\mu \overline{M},\hfill \\ {\overline{R}}^{\prime }\hfill & =q\overline{M}+\gamma \overline{I}-\mu \overline{R},\hfill \end{array}\right.$$

(1)

where $\overline{S},\overline{I},\overline{C},\overline{M}$, and $\overline{R}$ are the dimensional state variables, $\overline{\alpha }$ is the influx rate, approximately the natural per-capita daily birth and immigration rate, $\overline{c}$ is the disease transmission rate, $\mu$ is the efflux rate, i.e., the natural per-capita daily death other than the disease caused, which is assumed to the same for all compartments, $\gamma$ is the product of the recovery rate, r, and the proportion of those infected but not tested. Note that $\overline{M}$ is the class of test-positive individuals who will eventually recover from the infection and who will receive at least one more test before being put into the recovered category at the monitored recovery rate, q. This class of individuals is taken out from the infected class, $\overline{I}$, by themselves or institutionalized isolation. Parameter m is the monitoring rate with which test-positive individuals are put into the monitored class, $\overline{M}$. Parameter d is the death rate of those who are tested positive and eventually die from the disease.

Parameters $a_1,a_2$, and h are all related to the daily test-positive rate $\frac{a_1\overline{I}}{1+a_{1}h\overline{I}+a_{2}h\overline{M}}\overline{C}$. It is Holling’s Type II functional form [13] from theoretical ecology. Holling’s theory, derived from predation, is universal to all processes involving two entities, one of which must take time to change the encountering of both into something else. In our setting, disease testing is an agent or infrastructure that is to find infected individuals by diagnostic interaction before putting them into the confirmed class, $\overline{C}$. Testing is also the means to find out if an infected individual under monitoring is no longer infectious and thus can be released to the recovered class, $\overline{R}$. For the first class, there is a discovery probability rate, $a_1$, of the infected class, $\overline{I}$, that will be tested and confirmed. For the second class, there is a repeating test rate, $a_2$, which is the average number of tests an individual will receive over an average period of days under monitoring. For both cases, there is an average time, h, needed to complete a test. We rewrite the number of daily cases confirmed in the following Holling Type II function,

$$\frac{a_1\overline{I}}{1+a_{1}h\overline{I}+a_{2}h\overline{M}}\overline{C}=\frac{(1/h)\overline{C}/N_0}{1/\left(a_{1}h{N}_0\right)+\overline{I}/N_0+(a_2/a_1)\overline{M}/N_0}\overline{I}=\frac{pC}{\epsilon +I+aM}\overline{I},$$

where $p=1/h$ is the rate of testing and h is test processing time, $a=a_2/a_1$ is the ratio of testing rates for monitored and infected, and $\epsilon =1/\left(a_{1}h{N}_0\right)$, with $N_0$ being the effective susceptible population. $I=\overline{I}/N_0,C=\overline{C}/N_0$, etc. are dimensionless variables. Because $a_{1}h$ is moderate and $N_0$ is large, we will keep $\epsilon$ as a small parameter. Alternatively, one can start with the assumption that the daily confirmed number is proportional to the product of the infected and the confirmed because one class has positive feedback on the other class has the so-called Holling Type I functional form. Therefore, because testing takes time, the daily rate must be constrained by the time allowed and the constraining factor is exactly in the form of the denominator by Holling’s theory. See [13] for more explanations on the functional form.

For simplicity, we convert the dimensional model (1) into its dimensionless one by dividing each equation of (1) by the parameter $N_0$ and replace $S=\frac{\overline{S}}{N_0},I=\frac{\overline{I}}{N_0}$ etc., and all the parameters remain the same, except for the fact that $\frac{\overline{\alpha }}{N_0}$ is replaced by $\alpha$ and $\overline{c}{N}_0$ is replaced by c. For the dimensionless model, we assume the initial values sum up approximately equal to 1: $S\left(0\right)+I\left(0\right)+C\left(0\right)+M\left(0\right)+R\left(0\right)\simeq 1$. If we rewrite the daily testing rate as $P=\frac{C}{\epsilon +I+aM}pI$, then the factor $\frac{C}{\epsilon +I+aM}$ is the ratio that infected are tested and the complement $1-\frac{C}{\epsilon +I+aM}$ is the fraction of the infected class going directly into the recovered class, R, with the natural recover rate, r. To keep the model simple, we will use a parameter, namely $\gamma$, for the product as the overall recovery rate.

To summarize the above discussion result and better illustrate our model, we present the disease transmission flow chart, the dimensionless version of the model, and the parameter definition in

Table 1. Definition of model parameters.

Parameter	Definition	Units
$\alpha$	The influx rate	${\mathrm{day}}^{-1}$
c	The disease transmission rate	${\mathrm{day}}^{-1}$
$\mu$	The natural death rate of the population	${\mathrm{day}}^{-1}$
p	The rate of testing	${\mathrm{day}}^{-1}$
$\epsilon$	The parameter related to Holling Type II functional response	-
a	The ratio of testing rates for monitored and infected	-
$\gamma$	The overall recovery rate for the infected individuals	${\mathrm{day}}^{-1}$
m	The monitoring rate	${\mathrm{day}}^{-1}$
d	Disease-induced death rate	${\mathrm{day}}^{-1}$
q	The recovery rate for the monitored individuals	${\mathrm{day}}^{-1}$

below.

Disease transmission flow chart:

Dimensionless mathematical model:

$$\left\{\begin{array}{cc}{S}^{\prime }\hfill & =\alpha -cSI-\mu S,\hfill \\ I^{\prime }\hfill & =cSI-\frac{pC}{\epsilon +I+aM}I-\gamma I-\mu I,\hfill \\ C^{\prime }\hfill & =\frac{pC}{\epsilon +I+aM}I-mC-dC-\mu C,\hfill \\ M^{\prime }\hfill & =mC-qM-\mu M,\hfill \\ R^{\prime }\hfill & =qM+\gamma I-\mu R.\hfill \end{array}\right.$$

(2)

Last, we note that the SICM model of [13] is system (2) without all the terms with parameters $\alpha ,\mu ,\epsilon$, and $\gamma$. Also note that the equation R is decoupled from the rest, which makes analysis and computation easier.

3. Model Analysis

We first present the existence, uniqueness, positivity, and boundedness of solutions for model (2) in the following. The proof is provided in Appendix A.

Theorem 1. 

For any non-negative initial values $(S_0,I_0,C_0,M_0,R_0)$, the solution of the model (2) exists and it is unique, boundness, and non-negative for all $t>0$ on the invariant domain:

$$\Omega =\{(S,I,C,M,R)\in {\mathbb{R}}_{+}^5:S+I+C+M+R\le \frac{\alpha }{\mu }\}.$$

3.1. Basic Reproduction Number

In model (2), we consider $I,C$, and M to be all infectious compartments. For $I=C=M=0$, we can obtain a unique disease-free equilibrium:

$$E_0=(\frac{\alpha }{\mu },0,0,0,0,0).$$

Using the next-generation matrix method [14], we choose $cSI,\frac{pC}{\epsilon +I+aM}I,mC$ to be the new appearance rates and $\frac{pC}{\epsilon +I+aM}I,(m+d+\mu )C,(q+\mu )M$ to be the transfer-out rates for each infectious compartment, respectively. Then, taking the partial derivatives of those rates with respect to variables $I,C$, and M, we obtain the new infection matrix, F, and the transitive matrix, V, at the disease-free equilibrium, $E_0$, as

$$F=\left(\begin{array}{ccc}\frac{c\alpha }{\mu }& 0& 0\\ 0& 0& 0\\ 0& m& 0\end{array}\right)\mathrm{and}V=\left(\begin{array}{ccc}\mu +\gamma & 0& 0\\ 0& m+d+\mu & 0\\ 0& 0& q+\mu \end{array}\right).$$

Hence, the next-generation matrix is given by

$$F{V}^{-1}=\left(\begin{array}{ccc}\frac{c\alpha }{\mu (\mu +\gamma )}& 0& 0\\ 0& 0& 0\\ 0& \frac{m}{m+d+\mu }& 0\end{array}\right).$$

The basic reproduction number can be defined as the spectral radius of $F{V}^{-1}$ [14], that is,

$${\mathcal{R}}_0=\rho \left(F{V}^{-1}\right)={\displaystyle \frac{c\alpha }{\mu (\mu +\gamma )}}.$$

(3)

We now show that the disease can be eliminated if ${\mathcal{R}}_0\le 1$. This can be given in the following theorem.

Theorem 2. 

For system (2), the disease-free equilibrium $E_0$ is globally asymptotically stable in Ω for ${\mathcal{R}}_0\le 1$ and it is unstable for ${\mathcal{R}}_0>1$.

Proof. 

For ${\mathcal{R}}_0\le 1$, consider the following Lyapunov function:

$${\displaystyle \mathcal{V}=I+C+M.}$$

In $\Omega$, it is easy to see that

$$\begin{array}{cc}\hfill {\mathcal{V}}^{\prime }& =cSI-(\gamma +\mu )I-(d+\mu )C-(q+\mu )M\hfill \\ & \le (\frac{c\alpha }{\mu }-\gamma -\mu )I-(d+\mu )C-(q+\mu )M\hfill \\ & \le (\mu +\gamma )({\mathcal{R}}_0-1)I-(d+\mu )C-(q+\mu )M\hfill \\ & \le 0.\hfill \end{array}$$

Note that ${\mathcal{V}}^{\prime }=0$ implies $S=\frac{\alpha }{\mu }$ and $C=M=0$, and the largest positive invariant subset of the set $\{(S,I,C,M,R)\in \Omega :S=\frac{\alpha }{\mu },C=M=0\}$ is the disease-free equilibrium $\{E_0\}$. Thus, $\{E_0\}$ is the largest positive invariant set on $\{(S,I,C,M,R)\in \Omega :{\mathcal{V}}^{\prime }=0\}$. By the LaSalle invariant principle [15], $E_0$ is globally asymptotically stable in $\Omega$.

For ${\mathcal{R}}_0>1$, it follows from the Jacobian of system (2) at $E_0$,

$$J\left(E_0\right)=\left(\begin{array}{ccccc}-\mu & -\frac{c\alpha }{\mu }& 0& 0& 0\\ 0& (\mu +\gamma )({\mathcal{R}}_0-1)& 0& 0& 0\\ 0& 0& -m-d-\mu & 0& 0\\ 0& 0& m& -q-\mu & 0\\ 0& \gamma & 0& q& -\mu \end{array}\right),$$

which has a positive eigenvalue $(\mu +\gamma )({\mathcal{R}}_0-1)$, that $E_0$ is unstable. □

3.2. Endemic Equilibirum

When $I>0$, the endemic equilibrium $(S,I,C,M,R)$ satisfies the following equations:

$$\begin{array}{cc}\hfill \alpha -cSI-\mu S& =0,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill cS-\frac{pC}{\epsilon +I+aM}-\gamma -\mu & =0,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \frac{pC}{\epsilon +I+aM}I-mC-dC-\mu C& =0,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill mC-qM-\mu M& =0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill qM+\gamma I-\mu R& =0.\hfill \end{array}$$

(8)

Equations (7) and (8) give $C=\frac{q+\mu }{m}M$ and $R=\frac{1}{\mu }(qM+\gamma I)$. By substituting $C=\frac{q+\mu }{m}M$ into Equations (5) and (6), we have

$$\begin{array}{cc}\hfill cS-\frac{p(q+\mu )M}{m(\epsilon +I+aM)}-\mu -\gamma & =0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill M\left(\frac{pI}{\epsilon +I+aM}-m-d-\mu \right)& =0.\hfill \end{array}$$

(10)

For $M=0$, system (2) has a boundary equilibrium, or testing-free endemic equilibrium, below if ${\mathcal{R}}_0>1$.

$${\displaystyle E_1=(S_1^{\ast },I_1^{\ast },0,0,R_1^{\ast })=\left(\frac{\mu +\gamma }{c},\frac{\mu }{c}({\mathcal{R}}_0-1),0,0,\frac{\gamma }{c}({\mathcal{R}}_0-1)\right).}$$

For $M\ne 0$, by solving Equations (4)–(10), we can find

$$S_{\ast }=\frac{\alpha }{cb(M_{\ast }+\frac{\epsilon }{a})+\mu },I_{\ast }=b(M_{\ast }+\frac{\epsilon }{a}),C_{\ast }=\frac{q+\mu }{m}{M}_{\ast },R_{\ast }=\frac{1}{\mu }(q{M}_{\ast }+\gamma I_{\ast }),$$

where $b=\frac{a(m+d+\mu )}{p-m-d-\mu }$ and $M_{\ast }$ is the positive root of the equation

$$a_2{M}^2+a_{1}M+a_0=0$$

(11)

from (6), where

$$\begin{array}{cc}\hfill a_2& =\frac{(q+\mu )(m+d+\mu )}{m}+(\mu +\gamma )b,\hfill \\ \hfill a_1& =\frac{(q+\mu )(m+d+\mu )}{m}(\frac{\epsilon }{a}+\frac{\mu }{bc})+\frac{2\epsilon b(\mu +\gamma )}{a}+\frac{\mu (\mu +\gamma )}{c}-\alpha ,\hfill \\ \hfill a_0& =\frac{\epsilon }{a}(\frac{\epsilon b(\mu +\gamma )}{a}+\frac{\mu (\mu +\gamma )}{c}-\alpha )=\frac{\epsilon \mu (\mu +\gamma )}{ac}(\frac{\epsilon bc}{a\mu }+1-{\mathcal{R}}_0).\hfill \end{array}$$

Note that $a_1\ge \frac{2\epsilon b(\mu +\gamma )}{a}+\frac{\mu (\mu +\gamma )}{c}-\alpha \ge \frac{a}{\epsilon }{a}_0$. Hence, if $a_0\ge 0$, that is, ${\mathcal{R}}_0\le 1+\frac{\epsilon bc}{a\mu }$, then there is no positive root for Equation (11) because $f\left(0\right)=a_0>0$ and $f^{\prime }\left(0\right)=a_1\ge \frac{a}{\epsilon }{a}_0>0$ for the quadratic polynomial $f\left(x\right)=a_2{x}^2+a_{1}x+a_0$. If $a_0<0$, that is, ${\mathcal{R}}_0>1+\frac{\epsilon bc}{a\mu }$, then there is a unique positive root $M_{\ast }$ for Equation (11). Thus, if $p>m+d+\mu$ and ${\mathcal{R}}_0>1+\frac{\epsilon bc}{a\mu }$, then $S_{\ast },I_{\ast },C_{\ast },M_{\ast }$, and $R_{\ast }$ are all positive, and hence system (2) admits an interior endemic equilibrium:

$$E_{\ast }=(S_{\ast },I_{\ast },C_{\ast },M_{\ast },R_{\ast }).$$

In addition, when $E_1$ and $E_{\ast }$ both exist, we can show that the number of infected individuals, $I_{\ast }$, at the interior endemic equilibrium, $E_{\ast }$, is less than the number of infected individuals, $I_1^{\ast }$, at the testing-free endemic equilibrium, $E_1$. In fact, let $\Delta ={\mathcal{R}}_0-1-\frac{\epsilon bc}{a\mu }>0$. Note that

$${\displaystyle I_{\ast }<I_1^{\ast }⟺b(M_{\ast }+\frac{\epsilon }{a})<\frac{\mu }{c}({\mathcal{R}}_0-1)⟺M_{\ast }<\frac{\mu }{bc}({\mathcal{R}}_0-1)-\frac{\epsilon }{a}⟺M_{\ast }<\frac{\mu }{bc}\Delta .}$$

Since $a_2{M}_{\ast }^2+a_1{M}_{\ast }+a_0=0$, it suffices to show that $a_2{(\frac{\mu }{bc}\Delta )}^2+a_1(\frac{\mu }{bc}\Delta )+a_0>0$. By direct algebra calculation, one can verify that

$$\begin{array}{cc}& {\displaystyle a_2{(\frac{\mu }{bc}\Delta )}^2+a_1(\frac{\mu }{bc}\Delta )+a_0}\hfill \\ \hfill =& \left(\frac{(q+\mu )(m+d+\mu )}{m}+(\mu +\gamma )b\right){\left(\frac{\mu }{bc}\Delta \right)}^2+\left(\frac{(q+\mu )(m+d+\mu )\mu }{mbc}\left(1+\frac{\epsilon bc}{a\mu }\right)\right.\hfill \\ & +\left.\frac{\epsilon b(\mu +\gamma )}{a}-\frac{\mu (\mu +\gamma )}{c}\Delta \right)\frac{\mu }{bc}\Delta -\frac{\epsilon \mu (\mu +\gamma )}{ac}\Delta \hfill \\ \hfill =& \left[\left(\frac{(q+\mu )(m+d+\mu )}{m}+(\mu +\gamma )b\right)\frac{\mu }{bc}\Delta +\frac{(q+\mu )(m+d+\mu )\mu }{mbc}({\mathcal{R}}_0-\Delta )\right.\hfill \\ & -\left.\frac{\mu (\mu +\gamma )}{c}\Delta \right]\frac{\mu }{bc}\Delta \hfill \\ \hfill =& \left[\left((\frac{(q+\mu )(m+d+\mu )}{m}+(\mu +\gamma )b)\frac{\mu }{bc}-\frac{(q+\mu )(m+d+\mu )\mu }{mbc}-\frac{\mu (\mu +\gamma )}{c}\right)\right.\Delta \hfill \\ & +\left.\frac{(q+\mu )(m+d+\mu )\mu }{mbc}{\mathcal{R}}_0\right]\frac{\mu }{bc}\Delta \hfill \\ \hfill =& \frac{(q+\mu )(m+d+\mu ){\mu }^2}{m{\left(bc\right)}^2}{\mathcal{R}}_0\Delta \hfill \\ \hfill >& 0.\hfill \end{array}$$

We summarize the above results in the following theorem.

Theorem 3. 

For system (2),

1.

there always exists a unique disease-free equilibrium $E_0$;

2.

if ${\mathcal{R}}_0>1$, a unique testing-free endemic equilibrium $E_1$ occurs;

3.

if $p\le m+d+\mu$, there is no interior endemic equilibrium;

4.

if $p>m+d+\mu$ and ${\mathcal{R}}_0>1+\frac{\epsilon bc}{a\mu }$, a unique interior endemic equilibrium $E_{\ast }$ exists. Furthermore, $I_{\ast }<I_1^{\ast }$.

3.3. Stability Analysis

For the stability analysis of the endemic equilibria, we have the following results.

Theorem 4. 

For system (2),

1.

if $p\le m+d+\mu$ and ${\mathcal{R}}_0>1$, the testing-free endemic $E_1$ is locally asymptotically stable; and in fact $E_1$ is also globally asymptotically stable in $\Omega \setminus \{S-\mathit{axis}\}$;

2.

if $p>m+d+\mu$ and $1<{\mathcal{R}}_0\le 1+\frac{\epsilon bc}{a\mu }$, the testing-free endemic equilibrium $E_1$ is globally asymptotically stable in $\Omega \setminus \{S-\mathit{axis}\}$;

3.

if $p>m+d+\mu$ and ${\mathcal{R}}_0>1+\frac{\epsilon bc}{a\mu }$, the testing-free endemic equilibrium $E_1$ is unstable.

Proof. 

1. The Jacobian of system (2) at $E_1$ is

$$J\left(E_1\right)=\left(\begin{array}{ccccc}-c{I}_1^{\ast }-\mu & -c{S}_1^{\ast }& 0& 0& 0\\ c{I}_1^{\ast }& 0& -\frac{p{I}_1^{\ast }}{\epsilon +I_1^{\ast }}& 0& 0\\ 0& 0& \frac{p{I}_1^{\ast }}{\epsilon +I_1^{\ast }}-m-d-\mu & 0& 0\\ 0& 0& m& -q-\mu & 0\\ 0& \gamma & 0& q& -\mu \end{array}\right),$$

and the associated characteristic equation is

$$(\lambda +\mu )(\lambda +q+\mu )(\lambda -\frac{p{I}_1^{\ast }}{\epsilon +I_1^{\ast }}+m+d+\mu )({\lambda }^2+(c{I}_1^{\ast }+\mu )\lambda +c^2{S}_1^{\ast }{I}_1^{\ast })=0,$$

which has four negative eigenvalues and one eigenvalue

$$\frac{p{I}_1^{\ast }}{\epsilon +I_1^{\ast }}-m-d-\mu =\frac{\mu (p-m-d-\mu )}{c(\epsilon +I_1^{\ast })}({\mathcal{R}}_0-1-\frac{\epsilon bc}{a\mu }).$$

Hence, for $p<m+d+\mu$, we have $b<0$ and then $\frac{p{I}_1^{\ast }}{\epsilon +I_1^{\ast }}-m-d-\mu <0$ for ${\mathcal{R}}_0>1$. Thus, all eigenvalues of $J\left(E_1\right)$ are negative, and thereby $E_1$ is locally asymptotically stable for ${\mathcal{R}}_0>1$. If $p=m+d+\mu$, we can confirm this case in the global asymptotic stability. Similar to the global stability analysis in [16], we note that $S^{\prime }\le \alpha -cSI-\mu S,I^{\prime }\le cSI-\gamma I-\mu I$ for system (2) and the following SI system

$$\left\{\begin{array}{c}{S}^{\prime }=\alpha -cSI-\mu S,\hfill \\ I^{\prime }=cSI-\gamma I-\mu I\hfill \end{array}\right.$$

has a global attractor $(S_1^{\ast },I_1^{\ast })$ in ${\mathring{\Omega }}_1$ for ${\mathcal{R}}_0>1$, where ${\Omega }_1=\{(S,I)={\mathbb{R}}_{+}^2:S+I\le \frac{\alpha }{\mu }\}$. By the comparison theorem [17] , we have ${\displaystyle lim\; supS\le S_1^{\ast }}$ and ${\displaystyle lim\; supI\le I_1^{\ast }}$. Hence, we may consider the following attracting set in $\Omega$ for model (2):

$$\Gamma =\{(S,I,C,M,R)\in {\mathbb{R}}_{+}^5:S+I+C+M+R\le \frac{\alpha }{\mu },S\le S_1^{\ast },I\le I_1^{\ast }\}.$$

Let the Lyapunov function

$${\mathcal{V}}_1=C.$$

Then, we have

$$\begin{array}{cc}\hfill {\mathcal{V}}_1^{\prime }=& \frac{pCI}{\epsilon +I+aM}-mC-dC-\mu C\hfill \\ \hfill \le & \frac{pC{I}_1^{\ast }}{\epsilon +I_1^{\ast }}-(m+d+\mu )C\hfill \\ \hfill =& \frac{\mu (p-m-d-\mu )}{c(\epsilon +I_1^{\ast })}({\mathcal{R}}_0-1-\frac{\epsilon bc}{a\mu })C\hfill \\ \hfill \le & 0\hfill \end{array}$$

for ${\mathcal{R}}_0\ge 1$. Since ${\mathcal{V}}_1^{\prime }=0$ implies either $C=0$ or $I=I_1^{\ast }$ and $M=0$, the largest positive invariant subset of the set $\{(S,I,C,M,R)\in \Gamma :{\mathcal{V}}_1^{\prime }=0\}$ is $\{E_0,E_1\}$. By the LaSalle invariant principle, all solution curves in $\Gamma$ will approach either $E_0$ or $E_1$. Note that $E_0$ only attracts the solution curve on the S-axis for ${\mathcal{R}}_0>1$. Hence, all other solution curves in $\Gamma \setminus \{S-\mathrm{axis}\}$ will approach $E_1$, which proves that $E_1$ is globally asymptotically stable in $\Omega \setminus \{S-\mathrm{axis}\}$.

2. Similarly, by using the same Lyapunov function ${\mathcal{V}}_1=C$, one can obtain that $E_1$ is globally asymptotically stable in $\Omega \setminus \{S-\mathrm{axis}\}$ for $1<{\mathcal{R}}_0\le 1+\frac{\epsilon bc}{a\mu }$.

3. If $p>m+d+\mu$, then $E_1$ is unstable for ${\mathcal{R}}_0>1+\frac{\epsilon bc}{a\mu }$ since the eigenvalue $\frac{p{I}_1^{\ast }}{\epsilon +I_1^{\ast }}-m-d-\mu$ of $J\left(E_1\right)$ is positive. □

Theorem 4 shows that under the condition $p\le m+d+\mu$, which indicates that the test processing time $h=\frac{1}{p}$ is long enough, then the infectious disease will either end at a disease-free state, $E_0$, or end at a testing-free endemic state, $E_1$, determined by whether the basic reproduction number ${\mathcal{R}}_0\le 1$ or ${\mathcal{R}}_0>1$; under the condition $p>m+d+\mu$, i.e., the test processing time is short, then the dynamics will become complicated. If the test processing time is fast enough, like $h=0$, to be negligible, then the Holling Type II form can be reduced to the Holling Type I form, which we will investigate in the next section.

4. Simplified SICMR Model

For comparison purposes and to understand the global stability of the interior endemic equilibrium, $E_{\ast }$, better, we consider a simplified model by using Holling’s Type I form for the testing rate in the following:

$$\left\{\begin{array}{cc}{S}^{\prime }\hfill & =\alpha -cSI-\mu S,\hfill \\ I^{\prime }\hfill & =cSI-pCI-\gamma I-\mu I,\hfill \\ C^{\prime }\hfill & =pCI-mC-dC-\mu C,\hfill \\ M^{\prime }\hfill & =mC-qM-\mu M,\hfill \\ R^{\prime }\hfill & =qM+\gamma I-\mu R,\hfill \end{array}\right.$$

(12)

where all parameters have the same meanings as model (2), except that $p=a_1$ is the discovery probability rate. We could analyze model (12) also in the invariant set

$$\Omega =\{(S,I,C,M,R)\in {\mathbb{R}}_{+}^5:S+I+C+M+R\le \frac{\alpha }{\mu }\}.$$

In model (12), there exists a unique disease-free equilibrium, $E_0=(\frac{\alpha }{\mu },0,0,0,0,0)$, and the basic reproduction number is still

$${\mathcal{R}}_0={\displaystyle \frac{c\alpha }{\mu (\mu +\gamma )}}.$$

In addition, we can also obtain a testing-free endemic equilibrium,

$${\displaystyle E_1=(S_1^{\ast },I_1^{\ast },0,0,R_1^{\ast })=\left(\frac{\mu +\gamma }{c},\frac{\mu }{c}({\mathcal{R}}_0-1),0,0,\frac{\gamma }{c}({\mathcal{R}}_0-1)\right),}$$

for ${\mathcal{R}}_0>1$ and an interior endemic equilibrium,

$$E_{\ast }=(S_{\ast },I_{\ast },C_{\ast },M_{\ast },R_{\ast }),$$

for ${\mathcal{R}}_0>1+\frac{c(m+d+\mu )}{\mu p}$, where

$$I_{\ast }=\frac{m+d+\mu }{p},S_{\ast }=\frac{\alpha }{c{I}_{\ast }+\mu },C_{\ast }=\frac{\mu (\mu +\gamma )}{p(c{I}_{\ast }+\mu )}({\mathcal{R}}_0-1-\frac{c(m+d+\mu )}{\mu p}),M_{\ast }=\frac{m{C}_{\ast }}{q+\mu },\mathrm{and}{R}_{\ast }=\frac{q{M}_{\ast }+\gamma I_{\ast }}{\mu }.$$

Clearly, ${\mathcal{R}}_0>1+\frac{c(m+d+\mu )}{\mu p}$ implies that $I_{\ast }<I_1^{\ast }$. Thus, we have the following theorem.

Theorem 5. 

For system (12),

1.

there always exists a unique disease-free equilibrium $E_0$;

2.

there exists a unique testing-free endemic equilibrium $E_1$ for ${\mathcal{R}}_0>1$;

3.

there is a unique interior endemic equilibrium $E_{\ast }$ for ${\mathcal{R}}_0>1+\frac{c(m+d+\mu )}{\mu p}$. Furthermore, $I_{\ast }<I_1^{\ast }$.

Similarly, for stability results, we have

Theorem 6. 

For system (12),

1.

the disease-free equilibrium $E_0$ is globally asymptotically stable in Ω for ${\mathcal{R}}_0\le 1$ and it is unstable for ${\mathcal{R}}_0>1$;

2.

the testing-free endemic equilibrium $E_1$ is globally asymptotically stable in $\Omega \setminus \{S-\mathit{axis}\}$ for $1<{\mathcal{R}}_0\le 1+\frac{c(m+d+\mu )}{\mu p}$, and becomes unstable for ${\mathcal{R}}_0>1+\frac{c(m+d+\mu )}{\mu p}$;

3.

the interior endemic equilibrium $E_{\ast }$ is globally asymptotically stable in $\mathring{\Omega }$ for ${\mathcal{R}}_0>1+\frac{c(m+d+\mu )}{\mu p}$.

Proof. 

By using the same proof in Theorem 4, it is not hard to obtain the stabilities of $E_0$ and $E_1$. We only prove (3) by using the following Lyapunov function (see [6,18,19]) in $\mathring{\Omega }$:

$${\displaystyle {\mathcal{V}}_2=\frac{1}{2}{(S-S_{\ast })}^2+S_{\ast }(I-I_{\ast }-I_{\ast }ln\frac{I}{I_{\ast }}+C-C_{\ast }-C_{\ast }ln\frac{C}{C_{\ast }}).}$$

It follows from $\alpha =c{S}_{\ast }{I}_{\ast }+\mu S_{\ast },\gamma +\mu =c{S}_{\ast }-p{C}_{\ast }$, and $m+d+\mu =p{I}_{\ast }$ that

$$\begin{array}{cc}\hfill {\mathcal{V}}_2^{\prime }=& (S-S_{\ast })S^{\prime }+S_{\ast }(\frac{I-I_{\ast }}{I}{I}^{\prime }+\frac{C-C_{\ast }}{C}{C}^{\prime })\hfill \\ \hfill =& (S-S_{\ast })(c(S_{\ast }{I}_{\ast }-SI)+\mu (S_{\ast }-S))+S_{\ast }((I-I_{\ast })(c(S-S_{\ast })-p(C-C_{\ast }))\hfill \\ & +(C-C_{\ast })p(I-I_{\ast }))\hfill \\ \hfill \le & c(S-S_{\ast })(S_{\ast }{I}_{\ast }-S_{\ast }I+I{S}_{\ast }-IS)+c{S}_{\ast }(I-I_{\ast })(S-S_{\ast })\hfill \\ \hfill \le & c{S}_{\ast }(S-S_{\ast })(I_{\ast }-I)+c{S}_{\ast }(I-I_{\ast })(S-S_{\ast })\hfill \\ \hfill =& 0.\hfill \end{array}$$

Note that ${\mathcal{V}}_2^{\prime }=0$ implies that $S=S_{\ast }$. Any trajectory that starts in the space $S=S_{\ast }$ and then remains in $S=S_{\ast }$ for all $t>0$ must satisfy $S^{\prime }=0$, i.e., $I=I_{\ast }$, and similarly, we have $C=C_{\ast },M=M_{\ast }$, and $R=R_{\ast }$. That is, the largest positive invariant set on $\{(S,I,C,M,R)\in \mathring{\Omega }:{\mathcal{V}}_2^{\prime }=0\}$ is the singleton $\{E_{\ast }\}$. By the LaSalle invariant principle, $E_{\ast }$ is globally asymptotically stable in $\mathring{\Omega }$. □

The last result of Theorem 6 raises the question of whether or not the interior endemic equilibrium, $E_{\ast }$, is also globally asymptotically stable for the original system (2).

5. Application to U.S. COVID-19 Pandemic

5.1. Fit Model to Data

The first date when the COVID-19 case and death numbers were reported for the U.S. from the CDC is 22 January 2020. The end date of the data for this study is 1 September 2021 [20]. To apply our model (2), we do not expect its parameters to remain constant for this period of the U.S. COVID-19 pandemic. We will adopt the same protocol of the best-fitting model to data from [21]. That is, starting from day 50 (12 March 2020) to day 590 (1 September 2021), we fit the model to data from the past 21 days. There are a total of 10 parameters in the model (2). We use the initial parameter values from [21] for $S_0,I_0,C_0,M_0$, and $c,p,a,m,d,q$, respectively, as initial guesses for the same gradient decent algorithm as for [21] for our expanded model (2). As for the additional parameter $\alpha$, we use a U.S. birth rate of 12.012 per 1000 per year which translates to a fixed $\alpha$ value at $\alpha =12.012/1000/365$. For parameter $\mu$, we use a U.S. death rate of 8.4 per 1000 per year, which translates to a fixed $\mu$ value at $\mu =8.4/1000/365$. For parameter $\epsilon$, we fix it at $\epsilon ={10}^{-8}$. For parameter $\gamma$, we use $0.001$ as the initial guess for the best-fit searching algorithm. All these parameter values and initial conditions are provided in

Table 2. Parameter values and initial conditions.

Parameter	Variant 1	Variant 2	Variant 3	Variant 4	Variant 5
c	0.444	0.33	0.163	0.358	0.295
p	23.989	12.873	3.488	14.85	14.261
a	2.643	1.698	0.729	1.003	0.699
m	5.055	1.615	1.581	3.657	5.306
d	0.324	0.028	0.022	0.057	0.035
q	0.082	0.013	0.027	0.023	0.028
$\gamma$	0.012	0.012	0.01	0.011	0.011
$\alpha$	0.0000321	0.0000321	0.0000321	0.0000321	0.0000321
$\mu$	0.00002301	0.00002301	0.00002301	0.00002301	0.00002301
$\epsilon$	${10}^{-8}$	${10}^{-8}$	${10}^{-8}$	${10}^{-8}$	${10}^{-8}$
$t_0$	50	55	85	300	450
$S_0$	0.98993	0.99915	0.99998	0.99997	0.99992
$I_0$	0.0039355	0.00013194	$7.7\times {10}^{-6}$	$8.53\times {10}^{-6}$	$2.176\times {10}^{-5}$
$C_0$	0.00032705	$2.752\times {10}^{-5}$	$2\times {10}^{-8}$	$1.5\times {10}^{-7}$	$2.17\times {10}^{-6}$
$M_0$	0.0058068	0.0006905	$8.13\times {10}^{-6}$	$1.983\times {10}^{-5}$	$5.449\times {10}^{-5}$
$N_0$	$0.16\times {10}^7$	$0.7\times {10}^7$	$3.64\times {10}^7$	$0.5\times {10}^7$	$2.05\times {10}^7$
$R_0$	0	0	0	0	0

, and we use Matlab R2022 for plotting all figures. On each matching day (between day 50 and day 590), a large number of searches are carried out for the remaining 7 parameters; each search results in a ‘local minimum’ of 7 parameter values, and the best 30 results are ranked and archived. These best-fitted initials and parameters, including all the figures generated below, are included in figshare [22].

Figure 1 shows the result of how our SICMR model is fitted to the U.S. case and death data. The graph is assembled by the same protocol as [21]. It uses only the first ranked fit for each day of the 30 best-fits archived. Specifically, for each day’s case number there are 21 best-fits: on the day the datum belongs, on the day after, and up to the 20th day after. Each day’s fitting is treated as equally as every other 20-day fitting. Thus, each day’s plotting point is the average of the 21 best-fits. The same method is applied to the death data and matching curve. The main graphs are for the daily numbers, with the inserted graphs for the seven-day average, and the cumulative total; all are computed from the daily numbers.

For each of the best-fits (from a total of $541\times 30$), the best-fitted model satisfies condition (4) of Theorem 3, which is the same as condition (3) of Theorem 4. Figure 2 shows the I-component of the testing-free equilibrium, $E_1$, and the interior equilibrium $E_{\ast }$. Each day’s datum is the average of 21 best-fitted values for both $I_1^{\ast }$ and $I_{\ast }$, respectively. It shows that $I_{\ast }<I_1^{\ast }$, as predicted by Theorem 3 (4).

5.2. Variant Outbreaks

We also know the world was hit by the appearance of new variants of the COVID-19 virus. For this paper, we will define variants only from the data by the underlining long-term peaks of the data. For the period from day 50 to day 590, we can identify five such peaks. The first is due to the original outbreak. The second peaks around day 177, the third around day 347, the fourth around day 442, and the last continues on day 590. One can argue for only four variant peaks because the fourth can be considered as a part of the third variant.

We used the parameter values from the best-fit of Figure 1 to find good fits for each of the variant outbreaks. The shared data contains 500 fits for each variant. The best-fits are searched only for the shapes and magnitudes of the variants, foregoing the secondary oscillation modes with 7-day and 3-day periodicity, respectively. Figure 3 shows the first ranked fit for each variant. The parameter values and initial conditions are all listed in Table 2. The reason that the dimensionless initials in $S_0$ through $M_0$ need to be accurate to the fifth decimal place is because the daily effective susceptible population, $N_0$, is in the ${10}^6$ to ${10}^7$ range.

5.3. Local Stability of $E_1$ and $E_{\ast }$

In Figure 4a, the parameter values for system (2) are the same as the variant 3 best-fit from Figure 3, rounded to two digits in their decimals. The system satisfies condition (4) of Theorem 3 and condition (3) of Theorem 4. Hence, $E_1$ is unstable and there is a unique $E_{\ast }$. It can be demonstrated numerically that it has, at $E_1$, one negative eigenvector, $-\mu$, with eigenvector $[0,0,0,0,1]$ because the R equation is decoupled from the rest, two complex eigenvalues with a negative real part with eigenvectors in the invariant space $C=M=0$ for the reduced SIR system, in which $E_1$ is globally stable. It also has one negative eigenvalue, $-0.03$, with eigenvector $v=(0,0,0,1,-1)$, one positive eigenvalue, $1.89$, with an eigenvector of all non-vanishing entries. Denote the eigenvector by $v^u$ for the positive eigenvalue that points into the positive side of variable C and has the unit length. As for $E_{\ast }$, it can be checked numerically that it is locally asymptotically stable.

Figure 4a shows three numerical orbits in addition to the equilibrium solutions $E_1$ and $E_{\ast }$. The unstable manifold orbit, denoted by $W^u\left(E_1\right)$, is generated by the initial point $X_0=E_1+{10}^{-5}{v}^u$. The small perturbation orbit of $E_{\ast }$ is generated by an initial $X_0=E_{\ast }+(0.01,0.001,0.001,0,0)$ and a typical outbreak orbit with the same initial values as the variant 3 best-fit from Figure 3. (An outbreak orbit is loosely defined with the property that the initial value of $S_0$ is near 1 while all others are very small.) The parameter values are the same as the variant 3 fit. The unstable manifold orbit $W^u\left(E_1\right)$ returns to $E_1$ and appears to be a homoclinic orbit.

In Figure 4b, the corresponding system (2) satisfies condition (3) of Theorem 3 and condition (1) of Theorem 4. Hence, $E_1$ is globally asymptotically stable and $E_{\ast }$ does not exist. The simulation confirms the theory.

6. Stochastic Trapping and Homoclinic Connection

Let $\Lambda =\{(S,I,C,M,R)\in \Omega :C=0\}$ and ${\Lambda }_0=\{(S,I,C,M,R)\in \Omega :C=0,M=0\}$. Obviously, $\Lambda$ is a smooth invariant manifold for the model. On it, the model is reduced to the basic SIR model with $E_1$ being globally stable with ${\mathcal{R}}_0>1$. Motivated by Fenichel’s theory of hyperbolic invariant manifolds ([23,24,25]), we can partition $\Lambda$ into hyperbolic regions by finding the eigenspace at every point on $\Lambda$ because $C=M=0$ is invariant for the system. To do so, we first evaluate the Jacobian J from the proof of Theorem 4 at $X_0=(S_0,I_0,0,M_0,R_0)\in \Lambda$ to get

$$J_0=\left(\begin{array}{ccccc}-c{I}_0-\mu & -c{S}_0& 0& 0& 0\\ c{I}_0& c{S}_0-\gamma -\mu & -\frac{p{I}_0}{\epsilon +I_0+a{M}_0}& 0& 0\\ 0& 0& \frac{p{I}_0}{\epsilon +I_0+a{M}_0}-m-d-\mu & 0& 0\\ 0& 0& m& -q-\mu & 0\\ 0& \gamma & 0& q& -\mu \end{array}\right).$$

One can check easily that it has eigenvalues of ${\lambda }_1=-\mu ,{\lambda }_2=-q-\mu ,{\lambda }_3=\frac{p{I}_0}{\epsilon +I_0+a{M}_0}-m-d-\mu$ and ${\lambda }_{4,5}$ from the $2\times 2$ top-left block of $J_0$, which corresponds to the eigenvalues for the reduced SIR model with $C=0$. For ${\mathcal{R}}_0>1$, $\mathrm{Re}{\lambda }_{4,5}<0$ because $E_1$ is asymptotically stable for the SIR system. Hence, the manifold $\Lambda$ is partitioned into two open regions

$${\Lambda }_1=\Lambda \cap \{{\lambda }_3<0\}\mathrm{and}{\Lambda }_2=\Lambda \cap \{{\lambda }_3>0\}.$$

Their boundary (${\lambda }_3=0$) can be solved easily to be this hyperplane:

$$I_0=L\left(M_0\right):=\frac{(\epsilon +a{M}_0)(m+d+\mu )}{p-m-d-\mu }.$$

Thus, on any interior compact subset of ${\Lambda }_1$, the full SICMR system is uniformly attracting, and the eigenvector, $v_3=(0,0,1,0,0)$, for ${\lambda }_3$ is perpendicular to ${\Lambda }_1$. Locally around such a compact subset, if the eigenvalue ${\lambda }_3$ is greater than the others in magnitude, then similar to Fenichel’s theory, the system admits a hyperbolic splitting transversal to the invariant manifold, uniformly attracting at each point, having an invariant foliation transversal to the manifold.

Recall that the one-dimensional eigenspace of $E_1$ is transversal to $\Lambda$ with a non-negative C-component. The unstable manifold $W^u\left(E_1\right)$ is an orbit outside $\Lambda$. It is called a pseudo-homoclinic orbit if the unstable manifold is connected to a stable foliation of a point on ${\Lambda }_1$ that admits a transversal hyperbolicity. Dynamics near true homoclinic orbits can be extremely complex [26,27,28,29,30]; we expect nontrivial dynamics near pseudo-homoclinic orbits.

By definition, the attracting manifold ${\Lambda }_1$ is said to stochastically trap an orbit outside if any numerical simulation of the orbit sinks into the manifold with its C-component non-positive, $C\left(t\right)\le 0$, for some future time $t>0$. This can happen when the orbit is attracted to ${\Lambda }_1$ and stays long enough near ${\Lambda }_1$ so that the numerical approximation of its C-component is indistinguishable from zero. When this happens, a typical solver will keep $C=0$ because of the invariance of $\Lambda$ to the SICMR system. Biologically, it means that testing comes to a sudden stop when the number of confirmed C is too small.

A pseudo-homoclinic orbit is called a stochastic homoclinic orbit if the orbit $W^u\left(E_1\right)$ is stochastically trapped by ${\Lambda }_1$. This is what happens to $W^u\left(E_1\right)$ for Figure 4a and Figure 5a. More specifically, we can see that in Figure 5a the orbit first comes out from $E_1$, makes a U-turn, and then heads towards ${\Lambda }_1$. It appears to be trapped by ${\Lambda }_1$ because the orbit makes a right-angle downturn following the dynamics on $\Lambda$ on which M is strictly decreasing with the exponential rate $-q-\mu$, towards the sub-manifold ${\Lambda }_0$, on which the orbit has nowhere to go but is asymptotically attracted to $E_1$ in the $SIR$ subspace. Because of ${\Lambda }_1$’s hyperbolicity, the trapping to the manifold is exponential, with a rate of ${\lambda }_3<0$. Thus, the farther away from the boundary of ${\Lambda }_1$, the greater the attraction becomes and the more likely that trapping takes place. Stochastic trapping was confirmed empirically because all our numerical simulations had their C-components sink below zero, even when the absolute error and relative error tolerances for the Matlab ODE solver, ode15s, were reduced all the way down to ${10}^{-16}$. Stochastic trapping did not happen to the outbreak orbit for higher accuracy of the solver for the outbreak initials of Figure 5a. We also carried out the same analysis for variant 5’s outbreak in Figure 3. Stochastic trapping takes place up to ${10}^{-16}$ solver accuracy for both the unstable manifold of $E_1$ and the outbreak orbit, c.f., Figure 5b. Because there is a much lower accuracy (>${10}^{-16}$) in reality, our result suggests that the phenomenon of stochastic homoclinic orbits (i.e., stochastic trapping) is going to happen in real time.

7. Concluding Remarks

As pointed out in [13], the U.S. daily numbers exhibit a 7-day oscillation, which then changes to a 3-day oscillation. The inclusion of Holling’s Type II functional form for testing can capture this feature of the U.S. pandemic data because we suspect that the handling time constantly introduces a break-and-go effect on the data and we failed to do the same with the simplified model (12). Note also that the SICM model of [13] is the minimal model to capture such oscillations at the daily scale.

Because the $E_{\ast }$ is globally stable for the simplified SICMR system (12) by Theorem 6 (3), it is reasonable to conjecture the same for the original SICMR system (2) with condition (4) of Theorem 4. However, there is a hint that may not be true. If the pseudo-homoclinic orbit of Figure 5 is a real homoclinic orbit, converging to $E_1$ along the principal stable manifold tangent to the $SI$-plane, then it is Shilnikov’s saddle-focus type, because the real part of the stable eigenvalue is $-2.4\times {10}^{-4}$, and the unstable eigenvalue is $1.89>2.4\times {10}^{-4}$ ([27,29,30]). As a result, the dynamic behavior in a small neighborhood of the homoclinic orbit is chaotic, having infinitely many periodic orbits at the minimum. For pseudo-homoclinic orbits of the same saddle-focus type, we should expect the same. Hence, the existence of periodic orbits in a neighborhood of the orbit would prevent the endemic equilibrium state $E_{\ast }$ from being globally stable. However, it remains an open problem to show the existence of chaos near a pseudo-homoclinic orbit of Shilnikov’s type.

As for the long-term prospect of the U.S. pandemic, our results suggest two possibilities. One, the outbreak is stochastically trapped to the testing-free endemic state $E_1$, and two, the outbreak settles into the endemic state $E_{\ast }$ with testing. For the latter scenario, the simulated equilibrium in $C_{\ast }$ and $M_{\ast }$ are approximately $1.1734\times {10}^{-5}$ and $6.1757\times {10}^{-4}$, respectively, which translates to roughly 6000 cases for C and M classes together each day because the effective susceptible population, $N_0$, is in the order of ${10}^7$. This means that even if the endemic ends with testing, the scale is too small to equate it with the large scale of testing we have had throughout the pandemic. For the first scenario, the time needed to be stochastically trapped to the complete testing-free state $C=M=0$ is about a year, after the last outbreak. Altogether, this suggests that testing in the U.S. is to come to an end shortly after the end of the pandemic. This is apparent for everyone to see, but it is nonetheless surprising that the same picture can come from a mathematical model. Modeling and analyzing infectious diseases are complicated and worth exploring. There are still some limitations to our study and application. For example, our model does not consider vaccination, which is common for COVID-19, the immunity waning to the disease, the exposed individuals, the virus concentration in the environment, and the coronavirus viability in different seasons, which may impact the disease transmission behaviors and could be described by using variable transmission rates in a non-autonomous model, etc. However, all those limitations would also be a part of our future research direction, and hopefully we could gain a deeper understanding of infectious disease transmission by considering a combination of multiple factors.