Some new mathematical models of the fractional-order system of human immune against IAV infection

1.   Introduction

There are different patterns of rumor spreading depending on the presence or absence of online media [7], for example, the emergence of influential spreaders [2]. Before the development of online media, rumors were transmitted from person to person. With the development of online media such as social network service (SNS), personal broadcasting, blog, and group chatting, rumors can now spread in a variety of ways. In the past, offline media was the starting point and an important means of information delivery. Recently, it has become a social problem that offline media reproduces and delivers rumors from online media. This is a sign that information in online is rapidly being accepted by various social classes. In this paper, we study how the combination of classical interpersonal rumor spreading and online media influences rumor outbreak.

In order to consider the influence of online media, we denote by $I$ the density of people who do not know the rumor but are susceptible, $S$ is the density of people who spread the rumor, and $W$ is the amount of rumor in online generated by the group $S$, and $R$ is the density of people who know the rumor but are not interested in it or do not believe it. The process of rumor spreading is based on the following assumptions. (1) The group $I$ has an influx rate of $b$ and a natural decay rate of $\delta_i$. (2) Suppose the group $I$ meets $S$, then $I$ is converted to $S$ with an incidence rate of $\lambda_s$, and when the group $I$ encounters the rumor in online media, $I$ is also converted to $S$ with a rate of $\lambda_w$. (3) We assume that $S$ occurs only from $I$ and if $S$ encounters someone who knows the rumor, then they lose interest or do not believe in the rumor. In this case, let $\sigma_s$ and $\sigma_r$ denote the contact rates at which $S$ meets $S$ and $R$, respectively. (4) $S$ decreases with a natural decay rate of $\delta_s$ and becomes $R$ with the transmission rate $\mu$. (5) We assume that an online medium has its own natural decay rate of $\delta_w$ and is generated in proportion to the size of $S$ in offline. From the above assumptions, we can derive the following mean field equation.

Remark 1. (1) This rumor spreading process is a relatively short time process. Thus, we do not consider vertical transmission. See [8].

(2) If we take $b = \delta_i = \delta_s = \delta_r$ and $(I+S+R)(0) = 1$, then the total population density $I+S+R$ is conserved. Thus our model is a generalization of the model in [15].

Since the Daley-Kendall model [3], various studies on rumor spreading have been conducted. We briefly state the history of rumor spreading models associated with online media. See [12] for a general rumor spread, and [14] for threshold phenomena for general epidemic models. Since information transmission via online media developed in the late 1990s, intensive researches on rumors and online media began mainly in the early 2000s. In [1], the authors focused on the spread of computer-based rumors and analyzed the spread of rumors via computer-based communication in terms of information transmission. The authors in [7] noted the difference between online-based media and offline media. The study in [17] considered the spread of rumors through online networks by using the SIR model. The fast speed and unprofessional communication of online media is considered in [13]. See also [9]. In [11], a statistical rumor diffusion model is considered for online networks and it contained positive and negative bipolar reinforcement factors. [4,6,18] studied a rumor propagation model similar to the European fox rabies SIR model for the situation of changing online community number. In [10], the authors studied the rumor propagation phenomena for a model with two layers: online and offline. See also [19] for the SEIR type online rumor model.

This paper is organized as follows. In Section 2, we present the nonnegativity property of the solution to (1) and the stability of the rumor-free equilibrium. The basic reproduction number $\mathcal{R}_0$ is calculated by using a next-generation matrix. In Section 3, we provide the existence and uniqueness of endemic equilibrium and its global stability. In Section 4, we perform several numerical simulations to verify our analytical results.

2.   Elementary properties of the SIWR system and stability for rumor-free equilibrium

In this section, we consider the conservation of nonnegativity of the densities $I$, $S$, $W$, $R$ and the stability for a rumor-free equilibrium $E_0$.

2.1.   Nonnegativity of $I,S,W,R$

Lemma 2.1. Let $(I,S,W,R)$ be the unique global solution to (1). Assume that $\sigma_s$ and $\sigma_r$ are nonnegative constants and the rest of the coefficients are positive. If the initial data $(I(0),S(0),W(0),R(0))$ has only nonnegative components and satisfies

then the solution is nonnegative for all $t>0$ and $S(t), W(t)>0$ for $t>0$.

Proof. We take any positive $T>0$. By the continuity of the solution, there is $C(T)>0$ such that

By the first equation in (1) and the boundedness, if $I(0)\geq 0$, then for $0<t<T$,

We first prove that $S$ is nonnegative for $0<t<T$. Assume not, i.e., there is $t_{s}^-\in (0,T)$ such that

Let $t_0\in (0,t_{s}^-)$ be an entering time for $S$ into the negative region such that $S(t)\geq 0$ on $[0,t_0]$ and $S(t)< 0$ on $(t_0, t_0+\epsilon)$, where $\epsilon>0$ is a small constant. Note that by the second equation in (1),

This and the positivity of $I$ imply that if $W(t)\geq 0$ on $(0,s]$, $S(t)$ is nonnegative on $(0,s]$. Therefore, there is an entering time $t_0'\in (0, t_s^-)$ for $W$ into the negative region such that $W(t)\geq 0$ on $[0,t_0']$ and $W(t)< 0$ on $(t_0', t_0'+\epsilon')$, where $\epsilon'>0$ is a small constant. If $t_0'< t_0$, then $S(t)\geq 0$ and $W(t)<0$ on $(t_0',t_0)\cap (t_0',t_0'+\epsilon')$. Similarly, by the third equation in (1),

Thus, $W(t)$ is nonnegative on $(0,s]$ if $S\geq 0$ on $(0,s]$. This is a contradiction and we conclude that $t_0\geq t_0'$. Similarly, we can obtain that $t_0\leq t_0'$. Thus, $t_0 = t_0'$.

However, on $t\in [t_0,\infty)$,

is a solution to (1). By uniqueness of the solution, there is no $t_{s}^->0$ such that $S(t_{s}^-)<0$. Therefore, we prove that $S$ is nonnegative.

Similarly, we can also easily obtain that there is no $t_{w}^->0$ such that $W(t_{w}^-)<0$. Thus, for all $t>0$, $I,S,W\geq 0$. By the fourth equation in (1) and nonnegativity of $I$, $S$, $W$, we have that $R$ is also nonnegative on $(0,\infty)$. Therefore, we prove that the solution is nonnegative for all $t>0$.

Moreover, if $S(0)>0$, then for all $t>0$, $S(t)>0$ by (2). From (3), $W(t)>0$ on $(0,\infty)$. Similarly, if $W(0)>0$, then for all $t>0$, $W(t)>0$ by (3). By the virtue of the positivity of $I$ and (2), $S(t)>0$ on $(0,\infty)$. Thus, we conclude that if $S(0)>0$ or $W(0)>0$, then $S(t)>0$ and $W(t)>0$, $t\in (0,\infty)$.

2.2.   The basic reproduction number using a next-generation matrix

In this part, we calculate the basic reproduction number using a next-generation matrix. To consider the asymptotic behavior of the dynamics in (1), we determine the equilibrium point such that

If we assume that there is no rumor $(S = 0)$ in system (1) with (4), then the equilibrium point is unique and

The basic reproduction number $\mathcal{R}_0$ is generally a measure of the transmission of disease. This is usually expressed in terms of the rate of secondary transmission (or infection) and no transmission. $\mathcal{R}_0$ for more complex systems was calculated in [5,16] using the next-generation matrix methodology. Here, we follow the method of [16].

For the infected compartments, the next generation matrices at the rumor-free state $E_0 = \left(b/\delta_i,0,0,0\right)$ are given by

and hence

Here, $F$ and $V$ are related to the rate of new infections and transfer individuals, respectively. This yields

Therefore, we obtain the following formula for the basic reproduction number:

Here, $\rho(A)$ is the spectral radius of a matrix $A$.

2.3.   Stability for rumor-free equilibrium

For the linear stability, we consider the Jacobian matrix as follows.

Since the rumor-free equilibrium is

the Jacobian matrix at the rumor-free equilibrium is given by

Therefore, the corresponding characteristic equation is

Assume that $\mathcal{R}_0< 1$. Then by the definition of $\mathcal{R}_0$,

Thus,

Note that

Clearly, $-\delta_r$ and $-\delta_i$ are the eigenvalues of $J_{E_0}$ and are negative. The rest of the eigenvalues are zeros of the following polynomial.

Since $c_1>0$ and $c_2> 0$, zeros of $p_0(x)$ have only negative real part. Therefore, $E_0 = (b/\delta_i,0,0,0)$ is locally asymptotically stable if $\mathcal{R}_0<1$.

Clearly, if $\mathcal{R}_0 = 1$, then one of the eigenvalues has a zero real part. Thus, $E_0$ is locally stable but not asymptotically stable for the linearized system. Furthermore, if $\mathcal{R}_0> 1$, then $c_2< 0$. Therefore, $E_0$ is linearly unstable. We summarize the above argument to

Theorem 2.2. The rumor-free equilibrium $E_0$ of the system in (1) is linearly stable if $\mathcal{R}_0\leq 1$ and linearly unstable if if $\mathcal{R}_0> 1$. Moreover, $E_0$ is linearly asymptotically stable if $\mathcal{R}_0< 1$.

The rumor-free equilibrium $E_0$ is also a global attractive basin. We can use the standard methodology to obtain the global asymptotical behavior of the solution to (1).

Theorem 2.3. If $\mathcal{R}_0<1$, the rumor-free equilibrium $E_0$ is globally asymptotically stable on

Proof. Let

where $I_{rf} = b/\delta_i$. Since

and

we note that $V_0$ is nonnegative and radially unbounded. Then by elementary calculation,

Therefore, we have

Note that by Lemma 2.1 in Section 2, $I,S,R\geq 0$ and $S(t)>0$ for all $t>0$. This nonnegativity and (5) imply that if $(I(t),S(t))\ne (I_{rf},0)$ and $\mathcal{R}_0<1$, then

Therefore, $(I(t),S(t))$ converges to $(I_{rf},0)$ as $t$ goes to $\infty$ by Lyapunov stability theorem. By the third equation in (1), $W(t)$ converges to zero as $t$ goes to $\infty$. Similarly, by the fourth equation in (1), $R(t)$ converges to zero as $t$ goes to $\infty$.

Therefore, the rumor-free equilibrium $E_0$ is globally asymptotically stable.

3.   Stability analysis for endemic states

In this section, we present the existence and stability of endemic steady states for the rumor spreading model with an online reservoir. Endemic state refers to a nonzero steady state of $S$, i.e., the rumor is sustained. Since there is an influx $b$ for ignorant $I$, we can show that the unique endemic state exists as follows.

3.1.   Existence and uniqueness of the endemic equilibrium

To obtain the endemic equilibrium

we consider the following steady state equation:

Then the endemic equilibrium $E_* = (I_*,S_*,W_*,R_*)$ satisfies

We set

and

Then $U_* = S_*$ and

Note that the basic reproduction number satisfies

To find endemic equilibrium $E_*$, we set

The sum of all equations in (6) implies that

From the second equation in (6),

By (7)-(8),

Substituting $\tilde I_*$ into the first equation in (6) gives

Therefore, we have

If we obtain positive $S_*$, then by the first and third equations, we can derive $I_*$ and $R_*$ such that

and

Thus, if all components are nonnegative,

Theorem 3.1. If $\mathcal{R}_0>1$, then a unique positive endemic state $E_*$ exists, but if $\mathcal{R}_0\leq 1$, then there is no positive endemic state.

Proof. Assume that $\mathcal{R}_0>1$. Then there are three cases as follows.

● Case 1 $(\tilde \sigma_s-\tilde \sigma_r = 0)$: Since $\beta = 0$, we have

Condition (10) holds, which implies that a positive endemic state $E_*$ exists and is unique.

● Case 2 $(\tilde \sigma_s-\tilde \sigma_r>0)$: The equation (9) can be written as

Since $\mathcal{R}_0-1>0$ and $\tilde \sigma_s-\tilde \sigma_r>0$,

Therefore, there is a unique positive real root of the equation. To check the condition in (10), let

By elementary calculation,

Thus, $f(1/\tilde \sigma_r)>0$ and it follows that (10) holds, proving that there is a unique positive endemic equilibrium $E_*$.

● Case 3 $(\tilde \sigma_s-\tilde \sigma_r<0)$: Clearly, $\tilde S_*$ is a positive root of $f(x)$ in (11). The discriminant is

Since we assume that $\tilde \sigma_s-\tilde \sigma_r<0$, we need further analytical calculations to obtain $D>0$.

Let

Then the discriminant is represented as for $0\leq\tilde \sigma_s< \tilde \sigma_r$,

Note that $g$ is a quadratic function, therefore, $g$ has global minimum value at

Since we assume that $\mathcal{R}_0>1$,

Therefore, the minimum value of $g$ on $[0,\tilde \sigma_r)$ occurs at $x = 0$, thus, for $\tilde \sigma_s \in [0,\tilde \sigma_r )$,

We consider $g(0)$ as a function of $\tilde \sigma_r$, say $h(\tilde \sigma_r)$. Then $h(\tilde \sigma_r)$ is also a quadratic function of $\tilde \sigma_r$. Thus, $h(\tilde \sigma_r)$ has a global minimum as follows:

Therefore, $D>0$ and $f$ has two distinct real roots. Note that

Thus, $f$ has two distinct positive roots or two distinct negative roots.

By (12) and $\tilde \sigma_s-\tilde \sigma_r<0$,

Therefore, $f$ has two distinct positive roots and one is less than $1/\tilde \sigma_r$ and one is greater than $1/\tilde \sigma_r$. For small root, $\tilde R_*$ is positive and for large root, $\tilde R_*$ is negative.

For any case, we conclude that if $\mathcal{R}_0>1$, then a unique positive endemic state $E_*$ exists.

For the remaining part, we assume that $\mathcal{R}_0\leq1$. Similar to the previous proof, we have three cases.

● Case 1$'$ $(\tilde \sigma_s-\tilde \sigma_r = 0)$: From $\beta = 0$ and (9), it follows that

Thus there is no positive endemic state $E_*$.

● Case 2$'$ $(\tilde \sigma_s-\tilde \sigma_r>0)$: Note that $\tilde S_*$ is a positive root of $f(x)$ in (11). For $0\leq \tilde \sigma_r<\tilde \sigma_s$, the discriminant is

and $g$ has a global minimum value of $4 b \lambda _s^2 \left(\lambda _s+\sigma _s\right) \left(-b \lambda _s+\mu +1\right)$.

Since we assume that $\mathcal{R}_0<1$, the minimum value is positive, which yields that $D>0$. Moreover,

this implies that if $\mathcal{R}_0<1$, $f$ has two distinct positive roots or two distinct negative roots, and if $\mathcal{R}_0 = 1$, $0$ is a root of $f$.

Note that $f$ has a global minimum value at

Thus, $f$ has no positive root and there is no positive endemic equilibrium $E_*$.

● Case 3$'$ $(\tilde \sigma_s-\tilde \sigma_r<0)$: Note that

Since $\mathcal{R}_0-1\leq 0$ and $\tilde \sigma_s-\tilde \sigma_r<0$, we have

Therefore, there is at most one positive real root of the equation. However,

Thus, (10) does not hold. This implies that there is no positive endemic equilibrium $E_*$.

Therefore, we conclude that if $\mathcal{R}_0\leq 1$, then there is no positive endemic state.

3.2.   Stability for endemic equilibrium

In this part, we consider asymptotic stability for the endemic state $E_*$. Since the endemic state $E_*$ exists only for $\mathcal{R}_0>1$, we consider the case of $\mathcal{R}_0>1$.

Theorem 3.2. If $\mathcal{R}_0>1$, then the endemic equilibrium $E_*$ is globally asymptotically stable on

Proof. Let

In the same manner as Theorem 2.3, note that $V_*$ is nonnegative and radially unbounded.

We claim that if $(I(t),S(t),W(t),R(t))\ne (I_*,S_*,W_*,R_*)$ and $S(t)>0$, then

Since $(I_*,S_*,W_*,R_*)$ is the steady state of (1),

Therefore,

Similarly,

This implies that

Note that

We add the derivatives of $J_1$,$J_2$, and $J_3$ to obtain

Using $\delta_w W_* = \xi S_*$,

Using $\delta_w W_* = \xi S_*$ again,

Combining (13) and (14) with $\delta_w W_* = \xi S_*$,

Since

we have

Then we have

Note that

and

In conclusion, we have

From the first equation in (1), it follows that

This implies that

By the relationship between arithmetic and geometric means, if

then

If we assume that $S(0)>0$ or $W(0)>0$, then by the result in Section 2, $S(t)>0$ for $t>0$. Therefore, by Lyapunov stability theorem, we conclude that if $S(0)>0$ or $W(0)>0$, then $(I(t),S(t),W(t),R(t))$ converges to $E_*$ as $t$ goes to $\infty$ and this implies that the endemic equilibrium $E_*$ is globally asymptotically stable on

4.   Numerical simulation

In this section, we carry out some numerical simulations to verify the theoretical results. We use the fourth-order Runge-Kutta method with time step size $\Delta t = 0.01$. We assume that the initial condition is $(I(0),S(0),W(0),R(0)) = (1,1,0,0)$.

As shown before, the basic reproduction number is

We assume that the influx $b$ is $1$. Since the parameters $\sigma_s$, $\sigma_r$, and $\delta_r$ are not involved in the basic reproduction number $\mathcal{R}_0$, we fix these parameters as $\sigma_s = \sigma_r = \delta_r = 0.5$. If we take $\lambda_s = \lambda_w = \xi = 0.5$ and $\delta_i = \delta_s = \delta_w = \mu = 1$, then $\mathcal{R}_0 = 0.375$. In this case, as we proved in Theorem 2.3, the rumor-free equilibrium is $b/\delta_i = 1$ and is globally asymptotically stable. As seen in Figure 1(A), the population density of ignorants $I(t)$ converges to $1$ and the other densities $S(t)$, $W(t)$, and $R(t)$ converge to zero. On the other hand, if we set $\delta_i = \delta_s = \delta_w = \mu = 0.5$ and $\lambda_s = \lambda_w = \xi = 1$, then $\mathcal{R}_0 = 6$. As we proved in Theorem 3.2, the endemic equilibrium exists and is asymptotically stable. See Figure 1(B).

Now, we investigate the influence of an online reservoir. We change $\xi$ from $0$ to $2$ and all other parameters are fixed as $\lambda_s = \lambda_w = \sigma_s = \sigma_r = \delta_i = \delta_w = \delta_r = \mu = 0.5$ and $\delta_i = 1$. The final densities $I(T)$, $S(T)$, $W(T)$, and $R(T)$ when the final time $T = 10^3$ are given in Figure 2(A). We observe the phase transition when $\mathcal{R}_0 = 1$. That is, $\xi = 0.5$. We change $\lambda_w$ from $0$ to $2$ and all the other parameters are the same as the previous case. If we set $\xi = 0.5$, then we observe the effect of the trust rate $\lambda_w$ in Figure 2(B). In this case, the densities $S(t)$ and $R(t)$ are the same since $\xi = \delta_w$. As $\xi$ and $\lambda_w$ increase, the densities of the spreaders and stiflers also increase. Therefore, as online reservoirs become more active, rumors are more expansively spread.

Even though the contact rates $\sigma_s$ and $\sigma_r$ are not involved in $\mathcal{R}_0$, they affect the behavior of the solutions. See Figure 3. We choose $\sigma_s$ and $\sigma_r$ between $0.1$ and $2$. All other parameters are fixed to $0.5$. That is, $\lambda_s = \lambda_w = \delta_i = \delta_s = \mu = \xi = \delta_w = \delta_r = 0.5$ and hence $\mathcal{R}_0 = 2$. Since $\mathcal{R}_0>1$, the final time $T = 30$ is large enough. The bigger contact rates lead to a sharp reduction in the density of spreaders in a short time. Therefore, the final densities of the spreaders and stiflers are low if the contact rates are high. Furthermore, we confirm that the aggressive activity of the stiflers has an immense influence on the spread of a rumor. In Figure 4, we change $(\sigma_s,\sigma_r)$ on $[0,1]\times [0,1]$ and display the final densities $I(T )$, $S(T)$, $W(T)$, and $R(T)$ with $T = 30$.

We next compare the SIR and SIWR models. Without online an reservoir, the basic reproduction number of the SIR model is given by

If we fix the parameters such as $\sigma_s = \sigma_r = \delta_r = 0.5$ and $\lambda_s = \delta_i = \delta_s = \mu = 1$, then $\mathcal{R}_0^{SIR} = 0.5$. However, we introduce an online reservoir with $\delta_w = 0.5$ and $\lambda_w = \xi = 1$, then $\mathcal{R}_0 = 1.5$. Therefore, we conclude that an online reservoir promotes the spread of a rumor. The comparison of SIR and SIWR is given in Figure 5.

5.   Conclusion

In this paper, we consider a rumor spreading model with an online reservoir. By using a next-generation matrix, we calculated the basic reproduction number $\mathcal{R}_0$. We proved that a unique rumor-free equilibrium $E_0$ exists and if $\mathcal{R}_0 \leq 1$, then $E_0$ is linearly stable and if $\mathcal{R}_0 >1$, then $E_0$ is linearly unstable. For the asymptotic behavior, $E_0$ is globally asymptotically stable if $\mathcal{R}_0 < 1$. For the endemic equilibrium, if $\mathcal{R}_0 >1$, then there is a unique positive endemic state $E_*$ and if $\mathcal{R}_0 \leq1$, then there is no positive endemic state. Moreover, for a rumor spreading dynamics with an online reservoir, the endemic equilibrium $E_*$ is globally asymptotically stable if $\mathcal{R}_0 >1$. The presence of $\sigma_s$ and $\sigma_r$ does not affect the basic reproduction number and the asymptotic behaviors of steady states. We also investigated that the reproduction number $\mathcal{R}_0$ increases by the effect of the online reservoir. Thus, the development of online media promotes rumor propagation.

Acknowledgments

S.-H. Choi and H. Seo are partially supported by National Research Foundation (NRF) of Korea (no. 2017R1E1A1A03070692). H. Seo is partially supported by NRF of Korea (no. 2020R1I1A1A01069585).