The importance of quarantine: modelling the COVID-19 testing process

Abstract

We incorporate the disease state and testing state into the formulation of a COVID-19 epidemic model. For this model, the basic reproduction number is identified and its dependence on model parameters related to the testing process and isolation efficacy is discussed. The relations between the basic reproduction number, the final epidemic and peak sizes, and the model parameters are further explored numerically. We find that fast test reporting does not always benefit the control of the COVID-19 epidemic if good quarantine while awaiting test results is implemented. Moreover, the final epidemic and peak sizes do not always increase along with the basic reproduction number. Under some circumstances, lowering the basic reproduction number increases the final epidemic and peak sizes. Our findings suggest that properly implementing isolation for individuals who are waiting for their testing results would lower the basic reproduction number as well as the final epidemic and peak sizes.

1 Introduction

The Coronavirus Disease 2019 (COVID-19) was declared a global pandemic by the World Health Organization (WHO) on March 11, 2020, and remains a global threat to public health and economics worldwide (WHO 2022). Since early 2020, tremendous efforts have been made to understand the spread patterns of COVID-19, and various prevention and control strategies from mask-wearing and social distancing to massive city lockdowns have been implemented in affected areas (Fanelli and Piazza 2020; He et al. 2020; Jung et al. 2020; Li et al. 2020; Tang et al. 2020; Wang et al. 2020; Wu et al. 2020). COVID-19 related data such as daily new cases, hospitalizations, and deaths become crucial for policymakers and public health authorities to make informed decisions on appropriate interventions and resource allocations (Farsalinos et al. 2021).

Lab tests such as the polymerase chain reaction (PCR) test and at-home rapid tests based either on molecular or on antigen technology have been used in many countries to detect infection cases and assist clinical diagnosis (Web 2022). Testing processes as well as prevention strategies and social behavior changes based on testing results have greatly affected the dynamics of the COVID-19 epidemic (Lyng et al. 2021), and hence, mathematical models of their effects and implications merit further study (Akman et al. 2022; Gharouni et al. 2022).

In their recent work, Gharouni et al. proposed an SIR type model combining testing and disease states to gain some insights into testing and isolation efficacy on the effectiveness of controlling COVID-19 infection (Gharouni et al. 2022). Their model used three disease states (susceptible, infectious, and recovered) and four testing states (untested, waiting-for-positive, waiting-for-negative, and confirmed positive) to group individuals. One assumption in this model is that the individuals who are waiting for their test results do not change their disease states during the waiting period. This assumption does not hold in reality because some individuals may change their disease states; namely, an individual who was susceptible at the time of testing may get infected and become infectious during the waiting period. Since the infection happens after the testing, this individual may end up with a negative test result. After receiving a negative result, this individual will not self-isolate, even though is infectious. As a result, this individual is more likely to infect other individuals.

Assigning compartment-specific relative testing weights, Gharouni et al. showed that under some circumstances, both increased testing intensity and faster test reporting can reduce the effectiveness of control in the sense that the basic reproduction number would be increased (Gharouni et al. 2022). Note that the basic reproduction number cannot fully characterize the disease dynamics (Cui et al. 2019, 2022; Shaw and Kennedy 2021). For epidemic models, besides the basic reproduction number, the peak size and the final epidemic size are also very important for controlling and assessing the disease (Arino et al. 2007). The final epidemic size and the basic reproduction number are in general positively related in the homogeneous models (Cui et al. 2022), but may be negatively correlated in some heterogeneous models (Cui et al. 2022, 2019).

In what follows, we will extend the model of Gharouni et al. by assuming that individuals may change their disease states during the waiting period and characterize the properties of the basic reproduction number. We will also study how parameters related to disease states and testing states affect the final and peak sizes.

The rest of this paper is organized as follows. In Sect. 2, we formulate our model which is governed by a system of fifteen differential equations. Model analysis, including the well-posedness of the model and studies of the basic reproduction number and the final epidemic size relation, is presented in Sect. 3. In Sect. 4, we carry out some numerical simulations to explore the effects of parameters on the basic reproduction number, the final epidemic size, and the peak size. We summarize our findings in Sect. 5.

2 Model formulation

As in the model of Gharouni et al., we consider three disease states (Susceptible, Infectious and Recovered). However we use a slightly different model of testing: we set the testing states as untested, waiting for results, received negative results, or received positive results. Let $X_{T}$ denote the population of each subcompartment with $X\in \{S,I,R\}$ indicating disease state and $T\in \{u,w_s,w_i,w_r,n,p\}$ indicating testing state. Testing states are u, untested; $w_x$, waiting with disease state x at time of test and $x\in \{s,i,r\}$; n, recently received negative results; and p, recently received positive results. Specifically, individuals waiting for test results are subdivided into six compartments:

$S_{w_s},$:

susceptible individuals who were tested and are waiting for testing results;

$I_{w_i},$:

infectious individuals who were tested and are waiting for testing results;

$I_{w_s},$:

infectious individuals who were infected during their waiting period (the period after the tests and before receiving testing results);

$R_{w_r},$:

individuals who were recovered at their time of testing and are waiting for testing results;

$R_{w_i},$:

individuals who were infectious at their time of testing and recovered during their waiting period;

$R_{w_s},$:

individuals who were susceptible at their time of testing, became infectious and recovered during their waiting period.

Fig. 1

Flow diagram for the transmission and testing of COVID-19

We further make the following assumptions:

• (i) Infectivity is multiplied by factors $\eta _w\in [0,1]$, $\eta _p\in [0,1]$, respectively, in the classes $I_{w_i},$ $I_{w_s},$ $I_p$ due to quarantine and isolation, that is, the force of infection $\Lambda$ is given by

  $$\begin{aligned} \Lambda =\frac{\beta \left[ I_u+\eta _w(I_{w_i}+I_{w_s})+I_n+\eta _p I_p\right] }{N}, \end{aligned}$$

  (2.1)

  where $\eta _w$ measures quarantine efficacy during the waiting period and $\eta _p$ measures isolation efficacy for testing positive individuals, and N is the total population;

• (ii) Infection-induced death is negligible;

• (iii) Random testing (Web 2022) is conducted across the untested population, namely, each untested individual is equally likely to get tested.

Note that $\eta _w{=0}$ or $\eta _p=0$ means the quarantine or isolation is perfect, $\eta _w{=1}$ or $\eta _p=1$ means no quarantine or isolation at all; $\rho _s{=1}$ or $\rho _r=1$ means a susceptible (recovered) individual receives a negative testing result; $\rho _i=1$ means an infectious individual receives a positive testing result (The test is $100\%$ accurate).

Our model consists of the following system of differential equations. See Fig. 1 for an illustration of the progression of individuals through testing and disease states.

$$\begin{aligned} \frac{dS_u}{dt}{} & {} = -\Lambda S_u-\theta S_u+\omega _1 S_n+\omega _2S_p,\nonumber \\ \frac{dS_{w_s}}{dt}{} & {} =-\eta _w\Lambda S_{w_s}+\theta S_u-\alpha S_{w_s}{,}\nonumber \\ \frac{dS_n}{dt}{} & {} =-\Lambda S_n+\rho _s\alpha S_{w_s}-\omega _1 S_n,\nonumber \\ \frac{dS_p}{dt}{} & {} =-\eta _p\Lambda S_p+(1-\rho _s)\alpha S_{w_s}-\omega _2 S_p,\nonumber \\ \frac{dI_u}{dt}{} & {} = \Lambda S_u-\theta I_u+\omega _1 I_n+\omega _2 I_p-\gamma I_u,\nonumber \\ \frac{dI_{w_i}}{dt}{} & {} =\theta I_u-\alpha I_{w_i}-\gamma I_{w_i},\nonumber \\ \frac{dI_{w_s}}{dt}{} & {} =\eta _w\Lambda S_{w_s}-\alpha I_{w_s}-\gamma I_{w_s},\nonumber \\ \frac{dI_n}{dt}{} & {} =\Lambda S_n+(1-\rho _i)\alpha I_{w_i}+\rho _s\alpha I_{w_s}-\omega _1 I_n-\gamma I_n,\nonumber \\ \frac{dI_p}{dt}{} & {} =\eta _p\Lambda S_p+\rho _i \alpha I_{w_i}+(1-\rho _s)\alpha I_{w_s} -\omega _2 I_p-\gamma I_p,\nonumber \\ \frac{dR_u}{dt}{} & {} = \gamma I_u-\theta R_u+\omega _1 R_n+\omega _2R_p,\nonumber \\ \frac{dR_{w_r}}{dt}{} & {} =\theta R_u-\alpha R_{w_r},\nonumber \\ \frac{dR_{w_i}}{dt}{} & {} =\gamma I_{w_i}-\alpha R_{w_i},\nonumber \\ \frac{dR_{w_s}}{dt}{} & {} =\gamma I_{w_s}-\alpha R_{w_s},\nonumber \\ \frac{dR_n}{dt}{} & {} =\gamma I_n+\rho _r\alpha R_{w_r}+(1-\rho _i)\alpha R_{w_i}+\rho _s \alpha R_{w_s}-\omega _1 R_n,\nonumber \\ \frac{dR_p}{dt}{} & {} =\gamma I_p+(1-\rho _r)\alpha R_{w_r}+\rho _i\alpha R_{w_i}+(1-\rho _s)\alpha R_{w_s}-\omega _2 R_p, \end{aligned}$$

(2.2)

with the nonnegative initial conditions

$$\begin{aligned} \begin{aligned}&S_u(0)=N-I_{u0},S_{w_s}(0)=0,S_n(0)=0,S_p(0)=0,\\&I_u(0)=I_{u0},I_{w_i}(0)=0,I_{w_s}(0)=0,I_n(0)=0,I_p(0)=0,\\&R_u(0)=0,R_{w_r}(0)=0,R_{w_i}(0)=0,R_{w_s}(0)=0,R_n(0)=0,R_p(0)=0. \end{aligned} \end{aligned}$$

Since the total population is conserved, we can assume without loss of generality $N=1$. The parameters appearing in System (2.2) are presented in Table 1.

Table 1 Model parameters

3 Model analysis

For convenience, we set

$$\begin{aligned}Q_1=\{u,w_s,n,p\}, Q_2=\{u,w_i,w_s,n,p\}\ \text {and}\ Q_3=\{u,w_r,w_i,w_s,n,p\}. \end{aligned}$$

First, we show that our model (2.2) is well posed and the disease eventually dies out.

Theorem 3.1

For any nonnegative initial condition, System (2.2) possesses a unique solution, which remains nonnegative and is bounded for $t\ge 0$. Moreover, $\lim \limits _{t\rightarrow +\infty } I_j(t)= 0, j\in Q_2$, $\lim \limits _{t\rightarrow +\infty }R_{w_i}(t)=0$ and $\lim \limits _{t\rightarrow +\infty }R_{w_s}(t)=0$.

Proof

Using the proof similar to (Ji et al. 2022, Theorem 3.1), one can show that System (2.2) possesses a unique solution, which remains nonnegative for any given nonnegative initial condition. In addition, this solution is bounded since the total population is conserved. Adding all S-component and I-component equations of (2.2), we obtain

$$\begin{aligned} \frac{d\left( \sum _{i\in Q_1} S_i(t)+\sum _{j\in Q_2} I_j(t)\right) }{dt}=-\gamma (\sum _{j\in Q_2} I_j(t))\le 0. \end{aligned}$$

(3.1)

Therefore, $\sum _{i\in Q_1} S_i(t)+\sum _{j\in Q_2} I_j(t)$ is decreasing. This, together with the nonnegativity, shows that the limit of $\sum _{i\in Q_1} S_i(t)+\sum _{j\in Q_2} I_j(t)$ as $t\rightarrow +\infty$ exists. According to the Fluctuations Lemma (Hirsch et al. 1985), we conclude that

$$\begin{aligned} \lim _{t\rightarrow +\infty } \frac{d(\sum _{i\in Q_1} S_i(t)+\sum _{j\in Q_2} I_j(t))}{dt}=0, \end{aligned}$$

which implies $\lim \limits _{t\rightarrow +\infty } I_j(t)=0$, for any $j\in Q_2$ via (3.1). That is to say for any $\epsilon >0$, there exists a $T>0$ such that $I_j(t)<\epsilon /\gamma$ holds for $t>T$. In addition, from the equations of $R_{w_i}$ and $R_{w_s}$ in System (2.2), we have

$$\begin{aligned}\frac{dR_{w_i}(t)}{dt}<\epsilon -\alpha R_{w_i}(t),\ \text {and}\ \frac{dR_{w_s}(t)}{dt}<\epsilon -\alpha R_{w_s}(t)\end{aligned}$$

for $t>T$. Consequently, the comparison principle gives $\limsup \nolimits _{t\rightarrow +\infty } R_{w_i}(t)\le \epsilon /\alpha$ and $\limsup \nolimits _{t\rightarrow +\infty } R_{w_s}(t)\le \epsilon /\alpha$. Since $\epsilon >0$ can be arbitrarily small, we obtain $R_{w_i}(t),R_{w_s}(t)\rightarrow 0$, as $t\rightarrow +\infty$. $\square$

In the following theorem, we investigate the dependence of the basic reproduction number $\mathcal {R}_0$ on the model parameters. The calculation of $\mathcal {R}_0$ and proof of this theorem are given in the appendix.

Theorem 3.2

The partial derivatives of the basic reproduction number $\mathcal {R}_0$ given in (A.1) with respect to model parameters have the following properties.

• (1) $D_{\eta _w}\mathcal {R}_0\ge 0$ and $D_{\eta _p} \mathcal {R}_0\ge 0$;

• (2) If $\eta _w\ge \eta _p$, then $D_{\rho _i}\mathcal {R}_0\le 0$ and $D_{\rho _s}\mathcal {R}_0\ge 0$;

• (3) Assume $\rho _i=\rho _s=1$ and $\eta _p=0$. Then $D_\theta \mathcal {R}_0 \le 0$, $D_{\omega _1} \mathcal {R}_0\le 0$ and $D_{\omega _2} \mathcal {R}_0\ge 0$. Further, if $\eta _w=0$, then $D_{\alpha }\mathcal {R}_0\ge 0$ and if $\eta _w=1$, then $D_{\alpha }\mathcal {R}_0\le 0$.

Remark 3.3

Theorem 3.2 implies that improving isolation efficacy of individuals receiving positive test results or quarantine efficacy of individuals awaiting for their test results, by decreasing $\eta _p$ and/or $\eta _w$, respectively, always decreases the basic reproduction number. If individuals with positive test results have better isolation than those in the waiting period ($\eta _p\le \eta _w$), then increasing accuracy of tests for infectious individuals (increasing test sensitivity) or reducing accuracy of tests for susceptible ones (reducing test specificity) lowers $\mathcal {R}_0$. If isolation of individuals with positive test results is perfect and testing is $100\%$ accurate, increasing the testing rate, $\theta$, or lengthening the isolation period (decreasing $\omega _2$) decreases $\mathcal {R}_0$. Meanwhile, increasing $\omega _1$, the rate back to untested group for individuals with negative results, will also help reducing the value of $\mathcal {R}_0$. Further, the relation between $\mathcal {R}_0$ and test turnaround time, $\alpha$, depends on the quarantine efficacy, $\eta _w$: if individuals are perfectly quarantined while waiting for their test results ($\eta _w=0$), then a slower test turnaround (a longer wait for test results, or equivalently, a lower value for $\alpha$) leads to a lower $\mathcal {R}_0$; while if individuals do not self-quarantine at all while waiting for their test results, then a faster return of test results leads to a lower $\mathcal {R}_0$.

Theorem 3.1 implies the disease dies out eventually. After the epidemic passes, it is of great importance to find the number of individuals who were infected during the epidemic period, that is, the final epidemic size, $\mathcal {F}$, defined as

$$\begin{aligned} \mathcal {F}=\sum _{i\in Q_1}S_i(0)-\lim \limits _{t\rightarrow +\infty }\sum _{i\in Q_1} S_i(t). \end{aligned}$$

Let $S(t)=\sum \nolimits _{i\in Q_1} S_i(t)$, $I(t)=\sum \nolimits _{j\in Q_2}I_j(t)$, and $R(t)=\sum \nolimits _{k\in Q_3}R_k(t)$. From System (2.2), the dynamics of S, I and R are then governed by

$$\begin{aligned} \begin{aligned} \frac{dS(t)}{dt}&=-\Lambda \left( S_u(t)+\eta _w S_{w_s}(t)+S_n(t)+\eta _p S_p(t)\right) ,\\ \frac{dI(t)}{dt}&=\Lambda \left( S_u(t)+\eta _w S_{w_s}(t)+S_n(t)+\eta _p S_p(t)\right) -\gamma I(t),\\ \frac{dR(t)}{dt}&=\gamma I(t), \end{aligned} \end{aligned}$$

(3.2)

together with the nonnegative initial conditions $S(0)=S_u(0), I(0)=I_{u}(0), R(0)=0$. Theorem 3.1 states that the solution of (2.2) is nonnegative, which implies that $dS(t)/dt\le 0$. This, together with $0\le S(t)\le 1$, shows that $\lim \nolimits _{t\rightarrow +\infty } S(t):=S(\infty )$ exists.

Theorem 3.4

Suppose $\eta _w>0$ and $\eta _p>0$ and set $\eta =\min \{\eta _w,\eta _p\}$. Then the final epidemic size $\mathcal {F}$ of System (2.2) satisfies

$$\begin{aligned}S_u(0)-S_2(\infty )\le \mathcal {F}\le S_u(0)-S_1(\infty ),\end{aligned}$$

where $S_1(\infty )$ and $S_2(\infty )$ are, respectively, the roots of the two closely related final epidemic size relations given below

$$\begin{aligned} \ln \frac{S_u(0)}{S_1(\infty )}&=\frac{\beta }{\gamma } \left[ \eta ^2 (S_u(0)-S_1(\infty ))+I_{u}(0) \right] , \\ \ln \frac{S_u(0)}{S_2(\infty )}&=\frac{\beta }{\gamma } \left[ S_u(0)-S_2(\infty )+\eta ^2 I_{u}(0)\right] . \end{aligned}$$

Proof

Assume $\eta _w\ne 0$ and $\eta _p\ne 0$ and set $\eta =\min \{\eta _w,\eta _p\}$. We next estimate the lower and upper bounds of $S(\infty )$. Let $(S_1(t),I_1(t),R_1(t))$ be the solution to the following system

$$\begin{aligned} \begin{aligned} S'_1&=-\beta S_1 I_1,\\ I'_1&=\beta \eta ^2S_1 I_1-\gamma I_1,\\ R'_1&=\gamma I_1, \end{aligned} \end{aligned}$$

(3.3)

with the nonnegative initial conditions

$$\begin{aligned} S_1(0)=S_{u}(0), I_1(0)=I_{u}(0), R_1(0)=0. \end{aligned}$$

(3.4)

By a similar approach as in the proof of Theorem 3.1, we obtain

$$\begin{aligned}\lim _{t\rightarrow +\infty }I_1(t)=0.\end{aligned}$$

Adding the first two equations of (3.3) leads to

$$\begin{aligned} \eta ^2 S'_1+I'_1=-\gamma I_1. \end{aligned}$$

(3.5)

It follows from (3.5) that

$$\begin{aligned}\int \limits _0^\infty (\eta ^2 S'_1(t)+I'_1(t))dt=-\gamma \int \limits _0^\infty I_1(t)dt.\end{aligned}$$

By (3.4), we get

$$\begin{aligned}\int \limits _0^\infty I_1(t)dt=\frac{\eta ^2 S_1(0)+I_1(0)-\eta ^2 S_1(\infty )-I_1(\infty )}{\gamma }=\frac{\eta ^2 S_u(0)+I_{u}(0)-\eta ^2 S_1(\infty )}{\gamma }.\end{aligned}$$

Integrating the first equation of System (3.3) then yields

$$\begin{aligned} \ln \frac{S_u(0)}{S_1(\infty )}=\beta \int \limits _0^\infty I_1(t)dt=\frac{\beta }{\gamma }\left[ \eta ^2 (S_u(0)-S_1(\infty ))+I_{u}(0) \right] {.} \end{aligned}$$

(3.6)

A comparison implies S(t), the solution of (3.2), satisfies $S(t)\ge S_1(t)$ for $t\ge 0$, which yields $\lim \nolimits _{t\rightarrow +\infty }S(t)\ge \lim \nolimits _{t\rightarrow +\infty }S_1(t):=S_1(\infty )$. In a similar manner, we can prove $S(t)\le S_2(t)$, that is, $S(\infty )\le \lim \nolimits _{t\rightarrow +\infty }S_2(t):=S_2(\infty )$, where $S_2(t)$ is the solution to the following system

$$\begin{aligned} \begin{aligned} S'_2&=-\beta \eta ^2 S_2 I_2,\\ I'_2&= \beta S_2 I_2-\gamma I_2,\\ R'_2&= \gamma I_2,\\ \end{aligned} \end{aligned}$$

with the nonnegative initial conditions

$$\begin{aligned} S_2(0)=S_{u}(0), I_2(0)=I_{u}(0), R_2(0)=0; \end{aligned}$$

and $S_2(\infty )$ satisfies

$$\begin{aligned} \ln \frac{S_{u}(0)}{S_2(\infty )}=\frac{\beta }{\gamma }[S_{u}(0)-S_2(\infty )+\eta ^2 I_{u}(0)]. \end{aligned}$$

(3.7)

This completes the proof. $\square$

4 Numerical simulations

In this section, we numerically explore the dynamics of System (2.2). The parameter values used for simulations are listed in Table 1, of which, four parameter values are assumed for illustration purposes as no related references on these parameters are available. In practice, the individuals with positive results will be isolated and the individuals with negative results will not be isolated. Thus, we are more interested in the effects of parameters related to the testing states of waiting, $(\alpha ,\eta _w,\theta )$, on the dynamics of System (2.2).

We first take $\beta =0.5,\omega _1=0.25,\omega _2=0.071,\gamma =0.3,\eta _p=0.4$ and examine how $\mathcal {R}_0$ depends on $\eta _w\in [0,1]$ and $\alpha$. Figure 2a plots $\mathcal {R}_0$ as a function of $\alpha$ and $\eta _w$ in three-dimensional space, where $\theta =0.5,\rho _i=0.8,\rho _s=0.8$ from Table 1 are used. If we fix $\eta _w=0.75$, then, as seen in Fig. 2b, $\mathcal {R}_0$ increases in $\alpha$ for $\alpha \in [0,0.28]$ and decreases in $\alpha$ for $\alpha \in [0.28,1]$.

Set $\alpha =0.5, \eta _w=0.7$. $\mathcal {R}_0$ is a decreasing function of $\theta$ under the following three cases:

• (i) For any $\rho _s\in [0,1]$, fix $\rho _i=0.8$ and $\eta _p=0.4$, see Fig. 3a;

• (ii) For any $\rho _i\in [0,1]$, fix $\rho _s=0.8$ and $\eta _p=0.4$, see Fig. 3b;

• (iii) For any $\eta _p\in [0,\eta _w]$, fix $\rho _i=0,8$ and $\rho _s=0.8$, see Fig. 3c.

Fig. 3d depicts the values of $\mathcal {R}_0$ for different values of $\rho _i,\rho _s,\eta _p$ selected from the above three cases.

Fig. 2

The dependency of $\mathcal {R}_0$ on $\alpha$. Parameter values are $\beta =0.5,\omega _1=0.25,\omega _2=0.071,\gamma =0.3$, $\eta _p=0.4$, $\rho _i=0.8$, $\rho _s=0.8$ and $\theta =0.5$

Fig. 3

The effect of $\theta$ on the basic reproduction number. The parameter values are $\beta =0.5,\alpha =0.5,\omega _1=0.25,\omega _2=0.071,\gamma =0.3,\eta _w=0.7$; for a, $\rho _i=0.8$, $\eta _p=0.4$ and $\rho _s\in [0,1]$; for b, $\rho _s=0.8$, $\eta _p=0.4$ and $\rho _i\in [0,1]$ for c, $\rho _i=\rho _s=0.8$ and $\eta _p\in [0,\eta _w]$

We denote the peak size by $\mathcal {P}=\max \nolimits _{t\ge 0}(\sum \nolimits _{j\in Q_2}I_j(t))$ and numerically demonstrate how $\alpha$ affects the final epidemic size, $\mathcal {F}$, and the peak size, $\mathcal {P}$. We choose the initial condition $S_u(0)=1-10^{-4}$, $S_{w_s}(0)=S_n(0)=S_p(0)=0$, $I_u(0)=10^{-4}$, $I_{w_i}(0)=I_{w_s}(0)=I_n(0)=I_p(0)=0$, $R_{u}(0)=R_{w_r}(0)=R_{w_i}(0)=R_{w_s}(0)=R_n(0)=R_p(0)=0$, and take $\theta =0.2,\beta =0.5,\gamma =0.2,\omega _1=0.25,\omega _2=0.071,\rho _i=0.8,\rho _s=0.8,\rho _r=0.8,\eta _p=0.4$. Figure 4a–d indicates that as $\alpha$ increases in [0.33, 1],

• (i) $\mathcal {F}$, $\mathcal {P}$ and $\mathcal {R}_0$ decrease for $\eta _w=0.82$; see Fig. 4a–b.

• (ii) $\mathcal {F}$, $\mathcal {P}$ and $\mathcal {R}_0$ increase for $\eta _w=0.6$; see Fig. 4c–d.

• (iii) $\mathcal {F}$ and $\mathcal {P}$ have a positive correlation with $\mathcal {R}_0$ for $\eta _w=0.82$ or $\eta _w=0.6$; see Fig. 4a–d.

As shown in Fig. 4e–f, the final epidemic size $\mathcal {F}$ and the peak size $\mathcal {P}$ are non-monotonic with respect to $\alpha$ for $\eta _w=0.75$. For $\alpha \in [0.33,1]$, we observe that

• (i) $\mathcal {F}$ is increasing for $\alpha \in [0,33,0.464]$ and decreasing for $\alpha \in (0.464,1]$; $\mathcal {F}$ has a positive correlation with $\mathcal {R}_0$ for $\alpha \in [0.33,0.464]\cup [0.6047,1]$ and a negative correlation for $\alpha \in (0.464,0.6047)$; see Fig. 4e.

• (ii) $\mathcal {P}$ increases for $\alpha \in [0.33,0.7923]$ and decreases for $\alpha \in (0.7923,1]$; $\mathcal {P}$ and $\mathcal {R}_0$ are positively correlated for $\alpha \in [0.33,0.6047]\cup [0.7923,1]$ and negatively correlated for $\alpha \in (0.6047,0.7923)$; see Fig. 4f.

This observation indicates that $\mathcal {F}$ and $\mathcal {P}$ are not always positively related to $\mathcal {R}_0$ and the value of $\eta _w$ plays an important role to determine the relation. Under some circumstances (better quarantine while awaiting test results), faster test reporting (shorter waiting period) may reduce the effectiveness of control (See Fig. 4c–d). The basic reproduction number, $\mathcal {R}_0$, alone does not completely characterize the disease dynamics and lowering the basic reproduction number may increase the final epidemic size and the peak size (see Fig. 4e–f).

Fig. 4

The effect of $\alpha$ on final epidemic size and peak size for different $\eta _w$. The parameter values are $\alpha \in [0.33,1],\theta =0.2, \beta =0.5, \gamma =0.2, \omega _1=0.25, \omega _2=0.071, \rho _i=0.8, \rho _s=0.8, \rho _r=0.8$ and $\eta _p=0.4$

Finally, we explore how $\eta _w,\theta$ affect $\mathcal {F}$, $\mathcal {P}$ and peak time $\mathcal {T}$ at which $\sum \nolimits _{j\in Q_2}I_j(t)$ achieves its peak value. We observe that

• (i) $\mathcal {F}$, $\mathcal {P}$ and $\mathcal {R}_0$ increase, but $\mathcal {T}$ decreases as $\eta _w$ increases; see Fig. 5a–c;

• (ii) $\mathcal {F}$, $\mathcal {P}$ and $\mathcal {R}_0$ decrease, but $\mathcal {T}$ increases as $\theta$ increases; see Fig. 5d–f;

• (iii) $\mathcal {F}$ and $\mathcal {P}$ are positively correlated with $\mathcal {R}_0$, and $\mathcal {T}$ has a negative correlation with $\mathcal {R}_0$ (see Fig. 5a–f).

Fig. 5

Effect of $\eta _w$ and $\theta$ on the final epidemic size, peak size, and peak time. Parameter values are $\alpha =0.5,\beta =0.5,\gamma =0.2,\omega _1=0.25,\omega _2=0.071,\rho _i=0.8,\rho _r=0.8,\rho _s=0.8,\eta _p=0.4$; for a–c, $\eta _w\in [0.4,1]$ and $\theta =0.2$; for d–f, $\theta \in [0.2,0.5]$ and $\eta _w=0.7$

5 Summary and discussion

In this work, we incorporated testing processes into an SIR model to explore the dynamics of the COVID-19 epidemic. Focusing on some parameters related to the testing process, several results concerning the basic reproduction number, $\mathcal {R}_0$, the final epidemic size, $\mathcal {F}$, and the peak size, $\mathcal {P}$, were obtained.

We find that if good quarantine while awaiting test results is implemented (small $\eta _w$), then slower but more accurate test reporting increases the effectiveness of control (lowers the basic reproduction number). One interesting finding is that under some circumstances, the faster test reporting is (higher $\alpha$), the smaller the basic reproduction number, $\mathcal {R}_0$, but the larger the final epidemic size, $\mathcal {F}$ (see Fig. 4e–f). This suggests that lowering the basic reproduction number may increase the final epidemic size if isolation and test reporting are not properly managed.

Our findings suggest that the effectiveness of control will be improved by increasing the testing rate and increasing test sensitivity. This is not surprising as both lead to faster and more reliable identification of infected individuals. On the other hand, it is important that implement good self-quarantine by individuals who were tested but haven’t received their testing results. Perhaps perversely, our model shows a benefit of delaying tests returns and reduced test specificity if strict self-quarantine is adhered to while awaiting results. In practice, our expectation is that this would reduce public confidence in testing and lead to reduced adherence to self-quarantine.

We point out that our model can speculate on effect of including more disease states such as exposed, infectious but asymptomatic, and infectious with symptoms (Ge et al. 2020). Our analysis extends the results of Arino et al. (2007) to allow for transitions between susceptible states, but does not incorporate their generality of disease states. Our future work will close this gap.

Appendices

Appendix

A Calculation of basic reproduction number

We calculate the basic reproduction number by the next generation matrix method (van den Driessche and Watmough 2002). It is readily seen that System (2.2) has a unique disease-free equilibrium (DFE), which is the solution of

$$\begin{aligned} \begin{aligned} \theta S_u^*&=\omega _1S_n^*+\omega _2 S_p^*,\\ \theta S_u^*&=\alpha {S_{w_s}^*},\\ \rho _s\alpha {S_{w_s}^*}&=\omega _1S_n^*,\\ (1-\rho _s)\alpha {S_{w_s}^*}&=\omega _2S_p^*, \end{aligned} \end{aligned}$$

with $S_u^*+{S_{w_s}^*}+S_n^*+S_p^*=1$. Specifically, the DFE is given by

$$\begin{aligned} E_0=(S_u^*,{S_{w_s}^*},S_n^*,S_p^*,0,0,0,0,0,0,0,0,0,0,0), \end{aligned}$$

where

$$\begin{aligned}{S_{w_s}^*}=\frac{\theta }{\alpha }S_u^*,\ S_n^*=\frac{\rho _s\theta }{\omega _1}S_u^*,\ S_p^*=\frac{(1-\rho _s)\theta }{\omega _2}S_u^*,\ \text {and}\ S_u^*=\frac{1}{1+\theta \left( \frac{1}{\alpha }+\frac{\rho _s}{\omega _1}+\frac{1-\rho _s}{\omega _2}\right) }.\end{aligned}$$

The Jacobian matrix associated with the infectious compartments at $E_0$ is given by

$$\begin{aligned} J=\begin{pmatrix} \beta S^*_u-\theta -\gamma &{}\beta \eta _w S^*_u&{}\beta \eta _w S^*_u&{}\beta S^*_u+\omega _1&{}\beta \eta _p S^*_u+\omega _2\\ \theta &{}-\alpha -\gamma &{}0&{}0&{}0\\ \beta \eta _w {S_{w_s}^*}&{}\beta \eta ^2_w {S_{w_s}^*}&{}\beta \eta ^2_w {S_{w_s}^*}-\alpha -\gamma &{}\beta \eta _w {S_{w_s}^*}&{}\beta \eta _w\eta _p {S_{w_s}^*}\\ \beta S^*_n&{}\beta \eta _w S^*_n+(1-\rho _i)\alpha &{}\beta \eta _w S^*_n+\rho _s\alpha &{}\beta S^*_n-\omega _1-\gamma &{}\beta \eta _p S^*_n\\ \beta \eta _p S^*_p&{}\beta \eta _p\eta _w S^*_p+\rho _i\alpha &{}\beta \eta _p\eta _w S^*_p+(1-\rho _s)\alpha &{}\beta \eta _p S^*_p&{}\beta \eta ^2_p S^*_p-\omega _2-\gamma \end{pmatrix}. \end{aligned}$$

We decompose $J=F-V$, where

$$\begin{aligned} F=\beta \begin{pmatrix} S^*_u&{}\eta _w S^*_u&{}\eta _w S^*_u&{} S^*_u&{}\eta _p S^*_u\\ 0&{}0&{}0&{}0&{}0\\ \eta _w {S_{w_s}^*}&{}\eta _w\eta _w {S_{w_s}^*}&{}\eta _w\eta _w {S_{w_s}^*}&{}\eta _w {S_{w_s}^*}&{}\eta _w\eta _p {S_{w_s}^*}\\ S^*_n&{}\eta _w S^*_n&{}\eta _w S^*_n&{} S^*_n&{}\eta _p S^*_n\\ \eta _p S^*_p&{}\eta _p\eta _w S^*_p&{}\eta _p\eta _w S^*_p&{}\eta _p S^*_p&{}\eta _p\eta _p S^*_p \end{pmatrix}, \end{aligned}$$

and

$$\begin{aligned} V=\begin{pmatrix} \theta +\gamma &{}0&{}0&{}-\omega _1&{}-\omega _2\\ -\theta &{}\alpha +\gamma &{}0&{}0&{}0\\ 0&{}0&{}\alpha +\gamma &{}0&{}0\\ 0&{}-(1-\rho _i)\alpha &{}-\rho _s\alpha &{}\omega _1+\gamma &{}0\\ 0&{}-\rho _i\alpha &{}-(1-\rho _s)\alpha &{}0&{}\omega _2+\gamma \end{pmatrix}. \end{aligned}$$

$\mathcal {R}_0$ is the spectral radius of $F V^{-1}$ (van den Driessche and Watmough 2002), defined by $\mathcal {R}_0=r(F V^{-1})$. Note that $F=\beta F_1^T F_2$, where $F_1=(S^*_u,0,\eta _w {S_{w_s}^*},S^*_n,\eta _p S^*_p)$ and $F_2=(1,\eta _w,\eta _w,1,\eta _p)$ are two row vectors. It follows from (Osnaga 2005, Proposition 1) that $R_0=r(\beta F_1^T F_2V^{-1})=\beta F_2V^{-1}F_1^T$. Denote ${\bar{\alpha }}=\alpha /(\alpha +\gamma )$, ${\bar{\theta }}=\theta /(\theta +\gamma )$, ${{\bar{\omega }}}_1=\omega _1/(\omega _1+\gamma )$, and ${{\bar{\omega }}}_2=\omega _2/(\omega _2+\gamma )$. We also set $D = diag\left( (1-{\bar{\theta }})^{-1}, (1-{\bar{\alpha }})^{-1}, (1-{\bar{\alpha }})^{-1}, (1-{\bar{w}}_1)^{-1}, (1-{\bar{w}}_2)^{-1} \right)$, $W_i={{\bar{\omega }}}_1{{\bar{\alpha }}}(1-\rho _i)+{{\bar{\omega }}}_2{{\bar{\alpha }}}\rho _i$, $W_s={{\bar{\omega }}}_2{{\bar{\alpha }}}(1-\rho _s)+{{\bar{\omega }}}_1{{\bar{\alpha }}}\rho _s$, $W_1={{\bar{\alpha }}}(1-\rho _s)+{{\bar{\omega }}}_1{{\bar{\alpha }}}^2{{\bar{\theta }}}(\rho _i+\rho _s-1)$, $W_2={{\bar{\omega }}}_2{\bar{a}}^2{{\bar{\theta }}}(1-\rho _i-\rho _s)+{{\bar{\alpha }}}\rho _s$, and

$$\begin{aligned} {\bar{V}} = \begin{pmatrix} 1 &{} 0 &{} 0 &{} -{\bar{\omega }}_1 &{} -{\bar{\omega }}_2 \\ -{\bar{\theta }} &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} -(1-\rho _i){\bar{a}} &{} -\rho _s{\bar{a}} &{} 1 &{} 0 \\ 0 &{} -\rho _i{\bar{a}} &{} -(1-\rho _s){\bar{a}} &{} 0 &{} 1 \end{pmatrix}. \end{aligned}$$

We observe $V = \gamma {\bar{V}}D$ and $V^{-1} = (1/\gamma )D^{-1} {{\bar{V}}}^{-1}$, where

$$\begin{aligned} {\bar{V}}^{-1}&= \frac{1}{1-W_i{{\bar{\theta }}}}\times \\ {}&\begin{pmatrix} 1&{} W_i &{} W_s &{} {{\bar{\omega }}}_1 &{} {{\bar{\omega }}}_2\\ {\bar{\theta }}&{} 1 &{} W_s {{\bar{\theta }}} &{} {{\bar{\omega }}}_1{{\bar{\theta }}} &{} {{\bar{\omega }}}_2{{\bar{\theta }}} \\ 0 &{}0&{}1-W_i{{\bar{\theta }}} &{} 0 &{} 0 \\ {{\bar{\alpha }}}{{\bar{\theta }}}(1-\rho _i) &{} {\bar{a}}(1-\rho _i) &{} W_2&{} 1-{{\bar{\omega }}}_2{{\bar{\alpha }}}{{\bar{\theta }}}\rho _i&{} {{\bar{\omega }}}_2{{\bar{\alpha }}}{{\bar{\theta }}}(1-\rho _i)\\ {{\bar{\alpha }}}{{\bar{\theta }}}\rho _i&{}\rho _i{\bar{a}}&{} W_1 &{}{{\bar{\omega }}}_1{{\bar{\alpha }}}{{\bar{\theta }}}\rho _i &{}1-{{\bar{\omega }}}_1{{\bar{\alpha }}}{{\bar{\theta }}}(1-\rho _i) \end{pmatrix}. \end{aligned}$$

Consequently,

$$\begin{aligned} \mathcal {R}_0=\frac{\beta }{\gamma } F_2 D^{-1} {{\bar{V}}}^{-1}F_1^T. \end{aligned}$$

(A.1)

B Proof of Theorem 3.2 (1)

We observe that in (A.1), $D^{-1}{\bar{V}}^{-1}$ is nonnegative and independent of the $\eta _w$ and $\eta _p$. Then $\mathcal {R}_0$ is a quadratic polynomial in $\eta _w$ or $\eta _p$. Hence, the derivative of $\mathcal {R}_0$ with respect to $\eta _w$ or $\eta _p$ is nonnegative.

C Proof of Theorem 3.2 (2)

By simple calculations, we obtain

$$\begin{aligned} D_{\rho _i}\mathcal {R}_0=\frac{-\omega _1\omega _2\alpha ^2\beta \theta c_3^{\rho _i}c_4^{\rho _i}}{\gamma (\alpha +\gamma ) (c_1^{\rho _i})^2c_2^{\rho _i}}, \end{aligned}$$

(C.1)

where

$$\begin{aligned}{} & {} c_1^{\rho _i}=(\omega _2+\gamma )\left[ (\alpha +\gamma )(\theta +\gamma +\omega _1)+\theta \omega _1\right] +\alpha \rho _i\theta (\omega _1-\omega _2), \\{} & {} c_2^{\rho _i}=\omega _1\left[ \omega _2(\alpha +\theta )+\alpha \theta (1-\rho _s)+\omega _2\alpha \theta \rho _s\right] , \\{} & {} c_3^{\rho _i}=(1-\eta _p)(\alpha +\gamma )(\omega _1+\theta +\gamma )+\omega _2\theta (1-\eta _w)+\omega _1\theta (\eta _w-\eta _p){,} \end{aligned}$$

and

$$\begin{aligned} c_4^{\rho _i}=\omega _1\left[ \omega _2(\alpha +\gamma +\eta _w\theta )+c_{41}^{\rho _i}\right] +\omega _2 c_{42}^{\rho _i}+c_{40}^{\rho _i}; \end{aligned}$$

see expressions of $c_{4j}^{\rho _i},j=0,1,2$ in Table 2. It is readily seen that for $j=0,1,2$, $c_{4j}^{\rho _i}\ge 0$ since $\rho _s\in [0,1]$, which implies $c_4^{\rho _i}\ge 0$. If further $\eta _w\ge \eta _p$ and $\rho _i,\eta _p,\eta _w\in [0,1]$, then $c_k^{\rho _i}\ge 0$, $k=1,2,3$ holds. Hence, by (C.1), we derive $D_{\rho _i}\mathcal {R}_0\le 0$ if $\rho _i,\rho _s,\eta _w,\eta _p\in [0,1]$ and $\eta _w\ge \eta _p$.

Table 2 The expressions of $b_{ij}^\theta$ and $c_{4j}^{\rho _i}$

Table 3 The expressions of $c_{ij}^{\rho _s}$

We next take the derivative of $\mathcal {R}_0$ with respect to $\rho _s$, and get

$$\begin{aligned} D_{\rho _s}\mathcal {R}_0= & {} \frac{\omega _1\omega _2\alpha \beta \theta }{\gamma (\alpha +\gamma ) c_1^{\rho _i}(c_2^{\rho _i})^2} \left\{ \omega _1^2\left[ \omega _2c_1^{\rho _s}+c_2^{\rho _s}\right] +\omega _1\left[ \omega _2^2c_3^{\rho _s}+\omega _2c_4^{\rho _s}+c_5^{\rho _s}\right] \right. \nonumber \\{} & {} \left. +\theta \left[ \alpha \gamma (\alpha +\gamma )^2(1-\eta _p^2)(\theta +\gamma )+\omega _2^2 c_6^{\rho _s}+\omega _2 c_7^{\rho _s}\right] \right\} , \end{aligned}$$

(C.2)

where

$$\begin{aligned}{} & {} c_1^{\rho _s}=\alpha ^3(1-\eta _p)+\alpha ^2c_{12}^{\rho _s}+\alpha c_{11}^{\rho _s}+c_{10}^{\rho _s}, \\{} & {} c_2^{\rho _s}=\gamma ^2(\eta _w^2-\eta _p^2)\theta (\theta +\gamma )+\alpha ^3(1-\eta _p)(\gamma +\gamma \eta _p+\eta _p\theta \rho _i)+\alpha ^2 c_{22}^{\rho _s}+\alpha c_{21}^{\rho _s}, \\{} & {} c_3^{\rho _s}=\theta (1-\eta _w)\left\{ \alpha ^2+\alpha \left[ 2\gamma (1+\eta _w)+\eta _w\theta \right] +\gamma (\gamma +\gamma \eta _w+2\eta _w\theta )\right\} , \\{} & {} c_4^{\rho _s}=\alpha ^3(1-\eta _p)\left[ \gamma +\theta (2-\rho _i)\right] +\alpha ^2 \left[ c_{40}^{\rho _s}+\theta c_{41}^{\rho _s}+\theta ^2c_{42}^{\rho _s}\right] +\alpha c_{43}^{\rho _s}+c_{44}^{\rho _s}, \\{} & {} c_5^{\rho _s}=(\alpha +\gamma )\left\{ \gamma ^2(\eta _w^2-\eta _p^2)\theta (\theta +\gamma )+\alpha c_{51}^{\rho _s}+\alpha ^2c_{52}^{\rho _s}\right\} , \\{} & {} c_6^{\rho _s}=(1-\eta _w)\left[ \alpha ^2(2\gamma +\theta -\theta \rho _i)+\alpha c_{61}^{\rho _s}+\gamma ^2(1+\eta _w)(\theta +\gamma )\right] , \end{aligned}$$

and

$$\begin{aligned} c_7^{\rho _s}=(\alpha +\gamma )\left[ \alpha ^2 c_{72}^{\rho _s}+\alpha c_{71}^{\rho _s}+\gamma ^2(1-\eta _w^2)(\theta +\gamma )\right] ; \end{aligned}$$

see the expressions of $c_{ij}^{\rho _s}$ in Table 3. We claim in $c_{43}^{\rho _s}$, $d_1=\eta _p(2\rho _i-1-4\eta _w)+\eta _w(4+\eta _w-2\eta _w\rho _i)\ge 0$. If $(2\rho _i-1-4\eta _w)\ge 0$, it obviously holds; if $2\rho _i-1-4\eta _w<0$, then $\eta _w\ge \eta _p$ and $\eta _w\in [0,1]$ yield $\eta _p\le \eta _w\le \eta _w(4+\eta _w-2\eta _w\rho _i)/(1+4\eta _w-2\rho _i)$, which is equivalent to $d_1>0$. Note that $\eta _w,\eta _p,\rho _i \in [0,1]$ and $\eta _w\ge \eta _p$. Therefore, $c_{ij}^{\rho _s}\ge 0$, $c_i^{\rho _s}\ge 0$ for $1\le i\le 7$, $1\le j\le 3$. It follows from (C.2) that $D_{\rho _s}\mathcal {R}_0\ge 0$.

D Proof of Theorem 3.2 (3)

We first calculate $D_{\theta }\mathcal {R}_0$, $D_{\omega _1}\mathcal {R}_0$ and $D_{\omega _2}\mathcal {R}_0$ when $\rho _i=\rho _s=1$ and $\eta _p=0$. Then,

$$\begin{aligned} D_\theta \mathcal {R}_0=\frac{-\omega _1\alpha \beta \left\{ b_{0}^\theta +\omega _1^2\left[ \omega _2^2b_{1}^\theta +\omega _2 b_{2}^\theta +b_{3}^\theta \right] +\omega _1\left[ \omega _2^2b_{4}^\theta +\omega _2 b_{5}^\theta + b_{6}^\theta \right] \right\} }{\gamma (\omega _1+\gamma )(\alpha +\gamma )\left[ \alpha \theta +\omega _1(\alpha +\theta )\right] ^2\left[ (\alpha +\gamma )(\theta +\gamma )+\omega _2(\alpha +\gamma +\theta )\right] ^2},\nonumber \\ \end{aligned}$$

(D.1)

where

$$\begin{aligned}{} & {} b_{0}^\theta =\omega _2\alpha ^2(\alpha +\gamma )\theta ^2\left[ (\omega _2+\gamma )(1-\eta _w)+\alpha \right] , \\{} & {} b_{1}^\theta =(\alpha +\gamma )(1-\eta _w)(2\alpha ^2+b_{10}^\theta +\alpha b_{11}^\theta ), \\{} & {} b_2^\theta =\alpha ^4+\alpha ^3 \left[ \gamma (6-4\eta _w)+2\theta \right] +\alpha ^2 b_{22}^\theta +\alpha b_{21}^\theta +b_{20}^\theta , \\{} & {} b_3^\theta =\gamma (\alpha +\gamma )\left\{ \alpha ^3+\alpha ^2 \left[ \gamma (3-2\eta _w)+2\theta \right] +\alpha b_{31}^\theta +b_{30}^\theta \right\} , \\{} & {} b_4^\theta =2\alpha ^3(1-\eta _w)(\theta +\gamma )+\alpha ^2b_{42}^\theta +\alpha b_{41}^\theta +\gamma ^2(1-\eta _w^2)(\theta +\gamma )^2, \\{} & {} b_5^\theta =(\alpha +\gamma )\left[ \alpha ^3(\gamma +2\theta )+\alpha ^2 b_{52}^\theta +\alpha b_{51}^\theta +2\gamma ^2(1-\eta _w^2)(\theta +\gamma )^2\right] , \end{aligned}$$

and

$$\begin{aligned} b_6^\theta =\gamma (\alpha +\gamma )\left[ \gamma ^2(1-\eta _w^2)(\theta +\gamma )^2+\alpha ^3(\gamma +2\theta )+\alpha ^2 b_{62}^\theta +\alpha b_{61}^\theta \right] ; \end{aligned}$$

see the expressions of $b_{ij}^\theta$ in Table 2. It follows from $\eta _w\in [0,1]$ that for $1\le i\le 6$ and $0\le j\le 2$ all the above $b_{ij}^\theta \ge 0$, which implies $b_i^\theta \ge 0$. Then, $D_\theta \mathcal {R}_0\le 0$ if $\rho _i=\rho _s=1$, $\eta _p=0$. A simple calculation yields

$$\begin{aligned} D_{\omega _2}\mathcal {R}_0=\frac{\omega _1\alpha ^2\beta \theta (\alpha +\gamma +\eta _w\theta )\left[ (\alpha +\gamma )(\theta +\gamma )+\omega _1(\alpha +\gamma +\eta _w\theta )\right] }{\gamma (\omega _1+\gamma )(\alpha +\gamma )\left[ \alpha \theta +\omega (\alpha +\theta )\right] \left[ (\alpha +\gamma )(\theta +\gamma )+\omega _2(\alpha +\gamma +\theta )\right] ^2}. \end{aligned}$$

We obtain from the above equation that $D_{\omega _2}\mathcal {R}_0\ge 0$. Moreover, we have

$$\begin{aligned} D_{\omega _1}\mathcal {R}_0=\frac{-\alpha \beta \theta ^2\left\{ \gamma \left[ \omega _2b_0^{\omega _1}+b_1^{\omega _1}\right] +\omega _1^2\left[ \omega _2 b_2^{\omega _1}+b_3^{\omega _1}\right] +2\omega _1\gamma \left[ \omega _2b_4^{\omega _1}+b_5^{\omega _1}\right] \right\} }{\gamma (\omega _1+\gamma )^2(\alpha +\gamma )\left[ \alpha \theta +\omega _1(\alpha +\theta )\right] ^2\left[ (\alpha +\gamma )(\theta +\gamma )+\omega _2(\alpha +\gamma +\theta )\right] },\nonumber \\ \end{aligned}$$

(D.2)

where

$$\begin{aligned}{} & {} b_0^{\omega _1}=(1-\eta _w)\left\{ \gamma ^2(1+\eta _w)(\theta +\gamma )+\alpha ^2(2\gamma +\theta )+\alpha \gamma \left[ \gamma (3+\eta _w)+2\theta \right] \right\} , \\{} & {} b_1^{\omega _1}=(\alpha +\gamma )(\theta +\gamma )\left[ \alpha ^2+2\alpha \gamma (1-\eta _w)+\gamma ^2(1-\eta _w^2)\right] , \\{} & {} b_2^{\omega _1}=(1-\eta _w)\left\{ \alpha ^2+\alpha \left[ 2\gamma (1+\eta _w)+\eta _w\theta \right] +\gamma (\gamma +\gamma \eta _w+2\eta _w\theta )\right\} , \\{} & {} b_3^{\omega _1}=\gamma \left\{ \alpha ^2+2\alpha (1-\eta _w)[\gamma +\eta _w(\theta +\gamma )]+\gamma (1-\eta _w)(\gamma +\gamma \eta _w+2\eta _w\theta )\right\} , \\{} & {} b_4^{\omega _1}=(\alpha +\gamma )(1-\eta _w)\left[ 2\alpha +(1+\eta _w)(\theta +\gamma )\right] , \end{aligned}$$

and

$$\begin{aligned} b_5^{\omega _1}=\alpha ^3+\alpha ^2\left[ \gamma (3-2\eta _w)+\theta \right] +(1-\eta _w)\gamma \left\{ \alpha \gamma (3+\eta _w) +(1+\eta _w)[2\theta \alpha +\gamma (\theta +\gamma )]\right\} . \end{aligned}$$

Since $\eta _w\in [0,1]$, all the above $b_i^{\omega _1}\ge 0$ for $0\le i\le 5$. That is to say $D_{\omega _1}\mathcal {R}_0\le 0$ for $\rho _i=\rho _s=1$, and $\eta _p=0$.

If further $\eta _w=0$, taking the derivative of $\mathcal {R}_0$ in $\alpha$ yields

$$\begin{aligned} D_\alpha \mathcal {R}_0=\frac{\beta \theta \omega _1\left[ \omega _1^2(\omega _2+\gamma ) a_{11}+\theta (a_{12}+\omega _2^2 a_{13})+\omega _1(\omega _2+\gamma ) a_{14}\right] }{\gamma (\omega _1+\gamma )\left[ \alpha \theta +\omega _1(\alpha +\theta )\right] ^2\left[ (\alpha +\gamma )(\theta +\gamma )+\omega _2(\alpha +\gamma +\theta )\right] ^2},\nonumber \\ \end{aligned}$$

(D.3)

where

$$\begin{aligned}{} & {} a_{11}=(\alpha +\gamma )^2(\theta +\gamma )+\omega _2[2\alpha ^2+2\alpha (\theta +\gamma )+\gamma (\theta +\gamma )], \\{} & {} a_{12}=\gamma (\alpha +\gamma )^2(\theta +\gamma )^2+\omega _2\gamma (\theta +\gamma )\left[ \alpha ^2+2(\alpha +\gamma )(\theta +\gamma +\alpha )\right] , \\{} & {} a_{13}=2\alpha \gamma (\theta +\gamma )+\gamma (\theta +\gamma )^2+\alpha ^2(2\gamma +\theta ), \end{aligned}$$

and

$$\begin{aligned} a_{14}=(\alpha +\gamma )^2(\theta +\gamma )^2+\omega _2\left[ 2\alpha (\theta +\gamma )^2+\gamma (\theta +\gamma )^2+\alpha ^2(2\gamma +3\theta )\right] . \end{aligned}$$

Clearly, $D_\alpha \mathcal {R}_0\ge 0$ since $a_{11},a_{12},a_{13},a_{14}\ge 0$ in (D.3). On the other hand, if $\eta _w=1$, then

$$\begin{aligned}{} & {} D_\alpha \mathcal {R}_0\nonumber \\{} & {} =\frac{-\omega _1\alpha \beta \theta \left\{ \alpha (\omega _2+\gamma )(\alpha +\gamma )^2\theta (\theta +\gamma )^2+\omega _1^2\left[ \omega _2a_{21}+a_{22}\right] +\omega _1\left[ a_{23}+\omega _2 a_{24}\right] \right\} }{\gamma (\omega _1+\gamma )(\alpha +\gamma )^2\left[ \alpha \theta +\omega _1(\alpha +\theta )\right] ^2\left[ (\alpha +\gamma )(\theta +\gamma )+\omega _2(\alpha +\gamma +\theta )\right] ^2},\nonumber \\ \end{aligned}$$

(D.4)

where

$$\begin{aligned}{} & {} a_{21}=(\alpha +\gamma +\theta )^2[2\gamma \theta +\alpha (\theta +\gamma )], \\{} & {} a_{22}=\gamma (\theta +\gamma )\left[ \alpha ^3+2\gamma \theta (\theta +\gamma )+2\alpha ^2(\gamma +2\theta )+\alpha (\gamma ^2+6\gamma \theta +2\theta ^2)\right] , \\{} & {} a_{23}=\gamma (\alpha +\gamma )(\theta +\gamma )\left[ 2\gamma \theta (\theta +\gamma )+\alpha ^2(\gamma +3\theta )+\alpha (\gamma ^2+5\gamma \theta +4\theta ^2)\right] , \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} a_{24}&=2\gamma ^2\theta (\theta +\gamma )^2+\alpha ^3(\gamma ^2+4\gamma \theta +2\theta ^2)+2\alpha ^2(\gamma ^3+5\gamma ^2\theta +5\gamma \theta ^2+\theta ^3)\\ {}&\quad +\alpha \gamma (\gamma ^3+8\gamma ^2\theta +12\gamma \theta ^2+5\theta ^3). \end{aligned} \end{aligned}$$

Therefore, $D_\alpha \mathcal {R}_0\le 0$ follows from $a_{21}, a_{22}, a_{23}, a_{24}\ge 0$ in (D.4).