The risk index for an SIR epidemic model and spatial spreading of the infectious disease

1.   Introduction

As we all know, Keller and Segel ^[1] first proposed the classical chemotaxis model (hereafter called K-S model), which has been widely applied in biology and medicine. The model can be given by the following:

where $v_{1}$ is the cell density, $v_{2}$ is the concentration of the chemical signal, and $f(v_{1})$ is the logistic source function. For the case of $\tau = 1$ and $f(v_{1}) = 0,$ it has been proven that the classical solutions to system (1.1) always remain globally bounded when $n = 1$ ^[2]. A critical mass phenomenon of system (1.1) has been shown in a two-dimensional space. Namely, if the initial data $v_{10}$ satisfies $\|v_{10}\|_{L^{1}(\Omega)} < \frac{4\pi}{\chi},$ then the solution $(v_{1}, v_{2})$ is globally bounded ^[3]. Alternatively, if the initial data $v_{10}$ satisfies $\|v_{10}\|_{L^{1}(\Omega)} > \frac{4\pi}{\chi},$ then the solution $(v_{1}, v_{2})$ is unbounded in finite or infinite time, provided $\Omega$ is simply connected ^[4,5]. In particular, for a framework of radially symmetric solutions in a planar disk, the solutions blow up in finite time if $\|v_{10}\|_{L^{1}(\Omega)} > \frac{8\pi}{\chi}$ ^[6]. When $f(v_{1}) = 0,$ Liu and Tao ^[7] changed $\tau v_{2t} = \Delta v_{2}-v_{2}+v_{1}$ to $v_{2t} = \Delta v_{2}-v_{2}+g(v_{1})$ with $0\leq g(v_{1})\leq Kv_{1}^{\alpha}$ for $K, \alpha > 0,$ and obtained the global well-posedness of model (1.1) provided that $0 < \alpha < \frac{2}{n}.$ Later on, the equation $\tau v_{2t} = \Delta v_{2}-v_{2}+v_{1}$ was changed to $0 = \Delta v_{2}-\varpi(t)+g(v_{1})$ with $\varpi(t) = \frac{1}{|\Omega|}\int_{\Omega}g(v_{1}(\cdot, t))$ for $g(v_{1}) = v_{1}^{\alpha}.$ Winkler ^[8] deduced that for any $v_{10},$ the model (1.1) is globally and classical solvable if $\alpha < \frac{2}{n};$ conversely, if $\alpha > \frac{2}{n},$ then the solutions are unbounded in a finite-time for any $\int_{\Omega}v_{10} = m > 0.$ For $\tau = 0,$ when $f(v_{1})\leq v_{1}(c-dv_{1})$ with $c, d > 0,$ Tello and Winkler ^[9] deduced the global well-posedness of model (1.1) provided that $d > \frac{n-2}{n}\chi.$ Afterwards, when $f(v_{1}) = cv_{1}-dv_{1}^{\epsilon}$ with $\epsilon > 1, \; c\geq 0, \; d > 0,$ Winkler ^[10] defined a concept of very weak solutions and observed that these solutions are globally bounded under some conditions. For more results on (1.1), the readers can refer to ^{[11,12,13,14]}.

Considering the volume filling effect ^[15], the self-diffusion functions and chemotactic sensitivity functions may have nonlinear forms of the cell density. The general model can be written as follows:

Here, $\psi(v_{1})$ and $\phi(v_{1})$ are nonlinear functions. When $\tau = 1$ and $f(v_{1}) = 0,$ for any $\int_{\Omega}v_{10} = M > 0,$ Winkler ^[16] derived that the solution $(v_{1}, v_{2})$ is unbounded in either finite or infinite time if $\frac{\phi(v_{1})}{\psi(v_{1})}\geq cv_{1}^{\alpha}$ with $\alpha > \frac{2}{n}, \; n\geq 2$ and some constant $c > 0$ for all $v_{1} > 1.$ Later on, Tao and Winkler ^[17] deduced the global well-posedness of model (1.5) provided that $\frac{\phi(v_{1})}{\psi(v_{1})}\leq cv_{1}^{\alpha}$ with $\alpha < \frac{2}{n}, \; n\geq 1$ and some constant $c > 0$ for all $v_{1} > 1.$ Furthermore, in a high-dimensional space where $n\geq5,$ Lin et al. ^[18] changed the equation $\tau v_{2t} = \Delta v_{2}-v_{2}+v_{1}$ to $0 = \Delta v_{2}-\varpi(t)+v_{1}$ with $\varpi(t) = \frac{1}{|\Omega|}\int_{\Omega}v_{1}(x, t)dx,$ and showed that the solution $(v_{1}, v_{2})$ is unbounded in a finite time.

Next, we introduce the chemotaxis model that involves an indirect signal mechanism. The model can be given by the following:

For $\tau = 1,$ when $\psi(v_{1}) = 1, \phi(v_{1}) = v_{1}$ and $f(v_{1}) = \lambda(v_{1}-v_{1}^{\alpha}),$ the conclusion in ^[19] implied that the system is globally classical solvable if $\alpha > \frac{n}{4}+\frac{1}{2}$ with $n\geq2.$ Furthermore, the authors in ^[20,21,22] extended the boundedness result to a quasilinear system. Ren ^[23] derived the global well-posedness of system (1.3) and provided the qualitative analysis of such solutions. For $\tau = 0,$ when $\psi(s)\geq c(s+1)^{\theta}$ and $|\phi(s)|\leq ds(s+1)^{\kappa-1}$ with $s\geq 0, \; c, d > 0$ and $\theta, \kappa\in R,$ Li and Li ^[24] obtained that the model (1.3) is globally classical solvable. Meanwhile, they also provided the qualitative analysis of such solutions. More results of the system with an indirect signal mechanism can be found in ^{[25,26,27,28]}.

Considering that the cell or bacteria populations have a tendency to move towards a degraded nutrient, the authors obtain another well-known chemotaxis-consumption system:

where $v_{1}$ denotes the cell density, and $v_{2}$ denotes the concentration of oxygen. If $0 < \chi\leq\frac{1}{6(n+1)\|v_{20}\|_{L^{\infty}(\Omega)}}$ with $n\geq2,$ then the results of ^[29] showed that the system (1.4) is globally classical solvable. Thereafter, Zhang and Li ^[30] deduced the global well-posedness of model (1.4) provided that $n\leq2$ or $0 < \chi\leq\frac{1}{6(n+1)\|v_{20}\| _{L^{\infty}(\Omega)}}, \; n\geq 3.$ In addition, for a sufficiently large $v_{10}$ and $v_{20},$ Tao and Winkler ^[31] showed that the defined weak solutions globally exist when $n = 3.$ Meanwhile, they also analyzed the qualitative properties of these weak solutions.

Based on the model (1.4), some researchers have considered the model that involves an indirect signal consumption:

where $w$ represents the indirect signaling substance produced by cells for degrading oxygen. Fuest ^[32] obtained the global well-posedness of model (1.5) provided that $n\leq 2$ or $\|v_{20}\|_{L^{\infty}(\Omega)}\leq\frac{1}{3n},$ and studied the convergence rate of the solution. Subsequently, the authors in ^[33] extended the boundedness conclusion of model (1.5) using conditions $n\geq 3$ and $0 < \|v_{20}\|_{L^{\infty}(\Omega)}\leq\frac{\pi}{\sqrt{n}}.$ For more results on model (1.5), the readers can refer to ^{[34,35,36,37,38,39]}.

Inspired by the work mentioned above, we find that there are few papers on the quasilinear chemotaxis model that involve the nonlinear indirect consumption mechanism. In view of the complexity of the biological environment, this signal mechanism may be more realistic. In this manuscript, we are interested in the following system:

where $\Omega \subset \mathbb{R}^{n}(n\geq 1)$ is a bounded and smooth domain, $\nu$ denotes the outward unit normal vector on $\partial \Omega,$ and $\chi, \; \lambda_{1}, \; \lambda_{2}, \; \theta > 0, \; 0 < \alpha\leq\frac{1}{\theta}, \; \beta\geq 2.$ Here, $v_{1}$ is the cell density, $v_{2}$ is the concentration of oxygen, and $w$ is the indirect chemical signal produced by $v_{1}$ to degrade $v_{2}.$ The diffusion functions $\psi, \; \phi\in C^{2}[0, \infty)$ are assumed to satisfy

for all $s\geq 0$ with $a_{0}, b_{0} > 0$ and $r_{1}, r_{2}\in\mathbb{R}.$ In addition, the initial data $v_{10}$ and $v_{20}$ fulfill the following:

Theorem 1.1. Assume that $\chi, \; \lambda_{1}, \; \lambda_{2}, \; \theta > 0, \; 0 < \alpha\leq\frac{1}{\theta},$ and $\; \beta\geq 2,$ and that $\Omega \subset \mathbb{R}^{n} (n\geq 1)$ is a smooth bounded domain. Let $\psi, \; \phi\in C^{2}[0, \infty)$ satisfy (1.7). Suppose that the initial data $v_{10}$ and $v_{20}$ fulfill (1.8). It has been proven that if $r_{1} > 2r_{2}+1,$ then the problem (1.6) has a nonnegative classical solution

which is globally bounded in the sense that

for all $t > 0,$ with $C > 0.$

Remark 1.2. Our main ideas are as follows. First, we obtain the $L^{\infty}$ bound for $v_{2}$ by the maximum principle of the parabolic equation. Next, we establish an estimate for the functional $y(t): = \frac{1}{p}\int_{\Omega}(v_{1}+1)^{p}+\frac{1}{2p}\int_{\Omega}|\nabla v_{2}|^{2p}$ for any $p > 1$ and $t > 0.$ Finally, we can derive the global solvability of model (1.6).

Remark 1.3. Theorem 1.1 shows that self-diffusion and logical source are advantageous for the boundedness of the solutions. In this manuscript, due to the indirect signal substance $w$ that consumes oxygen, the aggregation of cells or bacterial is almost impossible when self-diffusion is stronger than cross-diffusion, namely $r_{1} > 2r_{2}+1.$ We can control the logical source to ensure the global boundedness of the solution for model (1.6). Thus, we can study the effects of the logistic source, the diffusion functions, and the nonlinear consumption mechanism on the boundedness of the solutions.

2.   Preliminaries

In this section, we first state a lemma on the local existence of classical solutions. The proof can be proven by the fixed point theory. The readers can refer to ^[40,41] for more details.

Lemma 2.1. Let the assumptions in Theorem 1.1 hold. Then, there exists $T_{\max}\in(0, \infty]$ such that the problem (1.6) has a nonnegative classical solution $(v_{1}, v_{2}, w)$ that satisfies the following:

Furthermore, if $T_{\max} < \infty,$ then

Lemma 2.2. (cf. ^[42]) Let $\Omega \subset \mathbb{R}^{n} (n\geq 1)$ be a smooth bounded domain. For any $s\geq 1$ and $\epsilon > 0,$ one can obtain

for all $z\in C^{2}(\bar{\Omega})$ fulfilling $\frac{\partial z}{\partial \nu}\big{|}_{\partial \Omega} = 0,$ with $C_{\epsilon} = C(\epsilon, s, \Omega) > 0.$

Lemma 2.3. (cf. ^[43]) Let $\Omega \subset \mathbb{R}^{n} (n\geq 1)$ be a bounded and smooth domain. For $s\geq 1,$ we have

for all $z\in C^{2}(\bar{\Omega})$ fulfilling $\frac{\partial z}{\partial \nu}\big{|}_{\partial \Omega} = 0.$

Lemma 2.4. Let $\Omega \subset \mathbb{R}^{n} (n\geq 1)$ be a bounded and smooth domain. For any $z\in C^{2}(\Omega),$ one has the following:

where $D^{2}z$ represents the Hessian matrix of $z$ and $|D^{2}z|^{2} = \sum_{i, j = 1}^{n}z_{x_{i}x_{j}}^{2}.$

Proof. The proof can be found in [41, Lemma 3.1].

Lemma 2.5. (cf. ^[44,45]) Let $a_{1}, a_{2} > 0.$ The non-negative functions $f\in C([0, T))\cap C^{1}((0, T))$ and $y\in L_{loc}^{1}([0, T))$ fulfill

and

where $\tau = \min\{1, \frac{T}{2}\}$ and $T\in(0, \infty].$ Then, one deduces the following:

3.   Global boundedness of the solutions

In this section, we provide some useful Lemmas to prove Theorem 1.1.

Lemma 3.1. Let $\beta > 1,$ then, there exist $M, \; M_{1}, \; M_{2} > 0$ such that

and

Proof. By the parabolic comparison principle for $v_{2t} = \Delta v_{2}-w_{1}^{\theta}v_{2},$ we can derive (3.1). Invoking the integration for the first equation of (1.6), one has the following:

Invoking the Hölder inequality, we obtain the following:

We can apply the comparison principle to deduce the following:

Thereupon, we complete the proof.

Lemma 3.2. For any $\gamma > 1,$ we have the following:

where $C_{0} = \frac{2^{\gamma}}{1+\gamma} > 0.$

Proof. For $\gamma > 1,$ multiplying equation $0 = \Delta w-w+v_{1}^{\alpha}$ by $w^{\gamma-1},$ one obtain the following:

By Young's inequality, it is easy to deduce the following:

Thus, we arrive at (3.6) by combining (3.7) with (3.8).

Lemma 3.3. Let the assumptions in Lemma 2.1 hold. For any $p > \max\{1, \frac{1}{\theta}-1\},$ there exists $C > 0$ such that

for all $t\in (0, T_{\max})$.

Proof. Using the equation $v_{2t} = \Delta v_{2}-w_{1}^{\theta}v_{2}$, we obtain the following:

where we used the equality $\nabla v_{2}\cdot \nabla\Delta v_{2} = \frac{1}{2}\Delta|\nabla v_{2}|^{2}-|D^{2}v_{2}|^{2}.$ Testing (3.10) by $|\nabla v_{2}|^{2p-2}$ and integrating by parts, we derive the following:

Using Lemma 2.3 and (3.1), one has the following:

where $C_{1} = 2(4p^{2}+n)M^{2}.$ In virtue of Lemma 2.2, Young's inequality, and (3.12), an integration by parts produces the following:

with $C_{2}, C_{3} > 0.$ Due to $|\Delta v_{2}|\leq \sqrt{n}|D^{2}v_{2}|,$ we can conclude from (3.1) and the integration by parts that

with $C_{4} = (\sqrt{n}+2(p-2))M > 0$. Due to $p > \max\{1, \frac{1}{\theta}-1\},$ we have $\theta(p+1) > 1.$ With applications of Young's inequality, (3.12), and Lemma 3.2, we obtain the following from (3.14):

with $C_{5}, C_{6}, C_{7}, C_{8} > 0.$ Substituting (3.13) and (3.15) into (3.11), we derive the following:

for all $t\in (0, T_{\max}).$ Thereupon, we complete the proof.

Lemma 3.4. Let the assumptions in Lemma 2.1 hold. If $r_{1} > 2r_{2}+1,$ then for any $p > 1,$ we obtain the following:

for all $t\in (0, T_{\max}),$ with $C > 0.$

Proof. Testing the first equation of problem (1.6) by $(v_{1}+1)^{p-1},$ one can obtain the following:

for all $t\in(0, T_{\max}).$ In view of (1.7), the first term on the right-hand side of (3.18) can be estimated as follows:

For the second term on the right-hand side of (3.18), we can see that

We can obtain the following from Young's inequality:

with $C_{1} > 0.$ Utilizing Young's inequality and (3.12), one has the following:

where $C_{2} > 0.$ Due to $r_{1} > 2r_{2}+1,$ for any $p > 1 > \frac{r_{1}-2r_{2}}{2r_{2}-r_{1}+1},$ we can obtain $\frac{(p+1)(p+2r_{2}-r_{1})}{p} < p.$ Applying Young's inequality, we obtain the following:

where $C_{3} > 0.$ Hence, substituting (3.19)–(3.23) into (3.18), one obtains the following:

for all $t\in (0, T_{\max}),$ where $C_{4} > 0.$

Lemma 3.5. Let the assumptions in Lemma 2.1 hold. If $r_{1} > 2r_{2}+1,$ then for any $p > \max\{1, \frac{1}{\theta}-1\},$ we obtain the following:

where $C > 0.$

Proof. We can combine Lemma 3.3 with Lemma 3.4 to infer the following:

where $C_{1} > 0.$ Due to $0 < \alpha\leq\frac{1}{\theta}$ and $\beta\geq2,$ we can obtain $\theta\alpha(p+1)\leq p+1\leq p+\beta-1.$ Using Young's inequality, we can obtain the following:

where $C_{2} > 0.$ By the inequality $(w+s)^{\kappa}\leq 2^{\kappa}(w^{\kappa}+s^{\kappa})$ with $w, s > 0$ and $\kappa > 1,$ we deduce the following:

where $C_{3} > 0,$ where we have applied Young's inequality. Thus, we obtain the following:

where $C_{4} > 0.$ Therefore, we can obtain (3.25) by Lemma 2.5. Thereupon, we complete the proof.

The proof of Theorem 1.1. Recalling Lemma 3.5, for any $p > \max\{1, \frac{1}{\theta}-1\},$ and applying the $L^{p}-$estimates of elliptic equation, there exists $C_{1} > 0$ such that

The Sobolev imbedding theorem enables us to obtain the following:

with $C_{2} > 0.$ Besides, using the well-known heat semigroup theory to the second equation in system (1.6), we can find $C_{3} > 0$ such that

Therefore, using the Moser-iteration^[17], we can find $C_{4} > 0$ such that

Based on (3.31)–(3.33), we can find $C_{5} > 0$ that fulfills the following:

for all $t\in (0, T_{\max}).$ According to Lemma 2.1, we obtain $T_{\max} = \infty.$ Thereupon, we complete the proof of Theorem 1.1.

4.   Conclusions and outlook

In this manuscript, based on the model established in ^[35], we further considered that self-diffusion and cross-diffusion are nonlinear functions, as well as the mechanism of nonlinear generation and consumption of the indirect signal substance $w.$ We mainly studied the effects of diffusion functions, the logical source, and the nonlinear consumption mechanism on the boundedness of solutions, which enriches the existing results of chemotaxis consumption systems. Compared with previous results ^[29,32], the novelty of this manuscript is that our boundedness conditions are more generalized and do not depend on spatial dimension or the sizes of $\|v_{20}\|_{L^{\infty}(\Omega)}$ established in ^[32], which may be more in line with the real biological environment. In addition, we will further explore interesting problems related to system (1.6) in our future work, such as the qualitative analysis of system (1.6), the global classical solvability for full parabolic of system (1.6), and so on.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (No. 12271466, 11871415).

Conflict of interest

The authors declare there is no conflict of interest.