Abstract
Caspase-1-mediated pyroptosis is the predominance for driving CD4\(^{+}\) T cells death. Dying infected CD4\(^{+}\) T cells can release inflammatory signals which attract more uninfected CD4\(^{+}\) T cells to die. This paper is devoted to developing a diffusive mathematical model which can make useful contributions to understanding caspase-1-mediated pyroptosis by inflammatory cytokines IL-1\(\beta \) released from infected cells in the within-host environment. The well-posedness of solutions, basic reproduction number, threshold dynamics are investigated for spatially heterogeneous infection. Travelling wave solutions for spatially homogeneous infection are studied. Numerical computations reveal that the spatially heterogeneous infection can make \(\mathscr {R}_0>1\), that is, it can induce the persistence of virus compared to the spatially homogeneous infection. We also find that the random movements of virus have no effect on basic reproduction number for the spatially homogeneous model, while it may result in less infection risk for the spatially heterogeneous model, under some suitable parameters. Further, the death of infected CD4\(^{+}\) cells which are caused by pyroptosis can make \(\mathscr {R}_0<1\), that is, it can induce the extinction of virus, regardless of whether or not the parameters are spatially dependent.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11371230), Shandong Provincial Natural Science Foundation, China (No. ZR2015AQ001), a Project for Higher Educational Science and Technology Program of Shandong Province of China (No. J13LI05), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.
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Appendices
Appendix A
1.1 Proof of Theorem 2.1
For any \(\phi =(\phi _1,\phi _2,\phi _3,\phi _4)\in [\mathbf 0 ,\mathbf M ]_\mathbb {C}\) and any \(\acute{k}\ge 0\), we easily obtain that
For
we have that
Therefore, it then follows that \(\phi +\acute{k}\mathscr {F}(\phi )\in [\mathbf 0 ,\mathbf M ]_{\mathbb {C}}\). This implies that
for all \(\phi \in [\mathbf 0 ,\mathbf M ]_{\mathbb {C}}.\) From Corollary 4 in Martin and Smith (1990) (see, also Wu 1996, Corollary 8.1.3), we obtain the conclusion. The proof is completed. \(\square \)
Appendix B
1.1 Proof of Lemma 4.1
For any fixed \(\phi =(\phi _1,\phi _2,\phi _3,\phi _4) \in \mathbb {C_{+}}\), from the first equation of model (3), we obtain that
Then, the comparison principle implies there exists \(t_1(\phi )>0\) such that \(U(x,t)\le \frac{2\overline{\xi }}{d_U}=:B_1\) for \(t>t_1\).
From the second equation of model (3), it then follows that
Again, the comparison principle implies there exists \(t_2(\phi )>0\) such that \(V(x,t)\le \frac{2\overline{\beta }B_1}{a(\underline{d}_V+\underline{\alpha }_1)}=:B_2\) for \(t>t_2\).
From the third equation of model (3), we get that
Again, there exists \(t_3(\phi )>0\) such that \(M(x,t)\le \frac{2\overline{\alpha }_2B_2}{d_M}=:B_3\) for \(t>t_3\).
From the fourth equation of model (3), we obtain that
Again, there exists \(t_4(\phi )>0\) such that \(\omega (x,t)\le \frac{2\overline{k}B_2}{d_\omega } =:B_4\) for \(t>t_4\).
Therefore, the solutions of model (3) are ultimately bounded with respect to the maximum norm. The solution semiflow \(\varPhi (t)=u(t,.):\ \mathbb {C_{+}}\rightarrow \mathbb {C_{+}}\) is point dissipative. By Theorem 2.2.6 in Wu (1996), we get that \(\varPhi (t)\) is compact for any \(t > \tau \). Therefore, from Theorem 3.4.8 in Hale (1988), we know that \(\varPhi (t)=u(t,.)\) has a compact global attractor in \(\mathbb {C_{+}}\). The proof is completed. \(\square \)
1.2 Proof of Lemma 4.2
From model (3), we easily obtain that
If \(V(t_0,\cdot , \phi )\not \equiv 0\), \(M(t_0,\cdot , \phi )\not \equiv 0\) and \(\omega (t_0,\cdot , \phi )\not \equiv 0\), the comparison principle implies that \(V(t,\cdot , \phi )>0\), \(M(t,\cdot , \phi )>0\), and \(\omega (t,\cdot , \phi )>0\) for \(t> t_0\), \(x\in \overline{\varOmega }\), that is, the conclusion (i) holds.
From model (3), we easily obtain that
Let \(v(x,t,\phi )\) be the solution of
It then follows that \(U(t,x,\phi )\ge v(t,x,\phi )>0\) for all \(t>0\), and \(x\in \overline{\varOmega }\). Moreover, by Lemma 3.1 and the comparison principle, we obtain that
uniformly for \(x\in \overline{\varOmega }.\) The proof is completed. \(\square \)
1.3 Proof of Lemma 4.3
For any given \(\phi \in \mathbb {C_{+}}\) with \(\phi _2(0)\not \equiv 0\) and \(\phi _4(0)\not \equiv 0\), let \(u(t,x,\phi )=(U(t,x),V(x,t), M(t,x),\omega (t,x))\). In view of the parabolic maximum principle and Lemma 4.2, from model (3), we easily get that
By Lemma 3.3, there exists \(\tau _1>0\) such that \(\overline{\lambda }_0(\widehat{U}(x)-\tau _1)>0\). Suppose, by contradiction, we assume that there exists some \(\phi \in \mathbb {C_{+}}\) with \(\phi _2(0)\not \equiv 0\) and \(\phi _4(0)\not \equiv 0\) such that
Then, there exists a sufficiently large positive number \(T_1\) such that
Therefore, we obtain the following model
Let \(\psi \) be the positive eigenfunction associated with \(\overline{\lambda }_0(\widehat{U}(x)-\tau _1)\). Then, we obtain that
admits a solution \(u(t,x)=e^{\overline{\lambda }_0(\widehat{U}(x)-\tau _1) t}(\psi _1(x),\ \psi _2(x))\). Since \(u(t,x,\phi _0)\gg 0\) for all \(t>0\) and \(x\in \overline{\varOmega }\), there exists \(\eta >0\) such that
By the comparison principle, we have that
Since \(\lambda _0(\widehat{U}(x)-\tau _1)>0\), we obtain that \(\mathop {\lim }\nolimits _{t \rightarrow +\infty }V(t,x)=\infty ,\ \mathop {\lim }\nolimits _{t \rightarrow +\infty }\omega (t,x)=\infty ,\) which is a contradiction. We complete the proof. \(\square \)
Appendix C
1.1 Proof of Lemma 5.3
If \(t>-\frac{1}{\varepsilon }\ln q_1\), then \(\underline{\phi }(t)=0\). It is easy to obtain the result. If \(t\le -\frac{1}{\varepsilon }\ln q_1\), it then follows that
From (16) and Lemma 5.1, it then follows that \(h_1(\lambda _c+\varepsilon ,c)\eta _1 +\frac{\beta \xi \eta _4}{d_U}<0\). Then for sufficiently large \(q_1\), we obtain that the last inequality holds.
If \(t>-\frac{1}{\varepsilon }\ln q_1\), then \(\underline{\varphi }(t)=0\). We easily obtain the result. If \(t\le -\frac{1}{\varepsilon }\ln q_1\), it then follows that
Similarly, from (16) and Lemma 5.1, it then follows that \(h_2(\lambda _c+\varepsilon ,c)\eta _2 +\frac{\beta \xi \eta _4e^{\tau (d_2(\lambda _c+\varepsilon )^{2}-c (\lambda _c+\varepsilon ))}}{d_U}<0\). Then for sufficiently large \(q_1\), we obtain that the last inequality holds.
It can be similarly shown that
We complete the proof. \(\square \)
1.2 Proof of Lemma 5.4
By Lemma 5.2 and simple computations, we easily obtain that
For \(t>\frac{1}{\lambda _c}\ln \frac{\xi }{d_U \eta _1}\triangleq t_1\), by using integration of parts twice, it then follows that
By employing the similar method above, we can prove \(F_1(\phi ,\varphi ,\psi ,\gamma )(t)\le \overline{\phi }(t)\) for \(t\le t_1\). Thus, for any \(t\ge 0\), we have that \(F_1(\phi ,\varphi ,\psi ,\gamma )(t)\le \overline{\phi }(t)\).
If \(t<t^*\), it then follows that
If \(t\ge t^*\), it then follows that
Therefore, by Lemma 3.2 in Wang et al. (2012a), it then follows that \(F_1(\phi ,\varphi ,\psi ,\gamma )\ge D_1^{-1}(D_1\underline{\phi }(t))\ge \underline{\phi }(t).\) From the discussions above, for any \(t\ge 0\), we easily obtain that \( \underline{\phi }(t)\le F_1(\phi ,\varphi ,\psi ,\gamma )\le \overline{\phi }(t). \)
Similarly, it then follows that
Consequently, by Lemma 3.2 in Wang et al. (2012a), we have \(F_2(\phi ,\varphi ,\psi ,\gamma )\le D_2^{-1}\Big (D_2\overline{\varphi }(t)\Big )=\overline{\varphi }(t)\).
If \(t<t^*\), it then follows that
If \(t\ge t^*\), it then follows that
A combination of the above two inequalities yields \(F_2(\phi ,\varphi ,\psi ,\gamma )\ge D_2^{-1}(D_2\underline{\varphi }(t))\ge \underline{\varphi }(t).\) Similarly, we can show that \(\underline{\psi }\le F_3(\phi ,\varphi ,\psi ,\gamma )\le \overline{\psi },\) \(\underline{\gamma }\le F_4(\phi ,\varphi ,\psi ,\gamma )\le \overline{\gamma }.\) We complete the proof. \(\square \)
1.3 Proof of Lemma 5.5
We firstly check that \(H=(H_1,H_2,H_3,H_4):\varPi \rightarrow \varPi \) is continuous with respect to the norm \(|{\cdot }|_{\mu }\) in \(B_{\mu }(\mathbb {R},\mathbb {R}^{4}).\) In fact, for any \(\varPhi _{11}, \varPsi _{11} \in C(\mathbb {R},\mathbb {R}^{4})\), it then follows that
For any given \(\varepsilon >0\), choose \(0<\delta _1<\displaystyle \frac{\varepsilon }{d_U+\beta _1+\frac{\beta }{a}+\frac{q}{b}+\frac{2\beta \xi }{d_U}+\frac{2q\xi }{d_U}}\), as \(|\varPhi _{11}-\varPsi _{11}|_{\mu }<\delta _1\), then we have \(|H_1(\varPhi _{11})(t)-H_1(\varPsi _{11})(t)|e^{-\mu |t|}<\varepsilon .\) Similarly, we have that
For any given \(\varepsilon >0\), choose \(0<\delta _2=\displaystyle \frac{\varepsilon }{e^{\tau (d_2\mu ^{2}+c\mu )}\Big (\frac{\beta }{a}+\frac{2\beta \xi }{d_U}\Big )+\beta _2+d_V+\alpha _1}\), as \(|\varPhi _{11}-\varPsi _{11}|_{\mu }<\delta _2\), then we have \(|H_2(\varPhi _{11})(t)-H_2(\varPsi _{11})(t)|e^{-\mu |t|}<\varepsilon .\) Hence, \(H_2\) is continuous with respect to the norm \(|{\cdot }|_{\mu }\) in \(B_{\mu }(\mathbb {R},\mathbb {R}^{4}).\) Similarly, we can show that \(H_3\) and \(H_4\) is continuous with respect to the norm \(|{\cdot }|_{\mu }\) in \(B_{\mu }(\mathbb {R},\mathbb {R}^{4}).\)
Applying the method similar to Ma (2001), Lemma 2.4 (see, also, Li et al. (2006) Lemma 3.4), we show that F is continuous with respect to the norm \(B_{\mu }(\mathbb {R},\mathbb {R}^{4}).\) By simple computations, we have that
where \(P=d_U+\beta _1+\frac{\beta }{a}+\frac{q}{b}+\frac{2\beta \xi }{d_U}+\frac{2q\xi }{d_U}.\) If \(t<0\), it then follows that
If \(t\ge 0\), we have that
Hence, it then follows that
where
Similarly, we can also prove that \(F_2,\ F_3\), and \(F_4\) are continuous with respect to the norm |.| in \(B_{\mu }(\mathbb {R},\mathbb {R}^{4})\). The proof is completed. \(\square \)
1.4 Proof of Lemma 5.6
By employing Arzela–Ascoli theorem, we show that the operator \(F:\varPi \rightarrow \varPi \) is compact with respect to the norm |.| in \(B_{\mu }(\mathbb {R},\mathbb {R}^{4})\). Let \(I_\mathbf k =[-\,\mathbf k ,\mathbf k ]\) with \(\mathbf k \in N\) be a compact interval on \(\mathbb {R}\). We regard \(\varPi \) as a bounded subset of \(C(I_\mathbf k ,\mathbb {R}^{4})\) equipped with the maximum norm. From Lemma 5.4, it then follows that F is uniformly bounded equipped with the maximum norm. In the following, we employ the following equalities to show that F is equi-continuous. In terms of the definition of \(F_i\) in (18) and integral representation for the derivative of \(D_i^{-1}\ (i=1,2,3,4)\), for any \((\phi ,\varphi ,\psi ,\gamma )\in \varPi \), it then follows that
Similarly, we also have that
where
Let \(\mathbf{u _n}\) be a sequence of \(\varPi \), which is viewed as a bounded subset of \(C(I_\mathbf k )\) with \(I_\mathbf k :=[-\,\mathbf k ,\mathbf k ]\). From the discussions above, we conclude that F is uniformly bounded and equi-continuous. According to the Arzela–Ascoli theorem, there exists a subsequence \(\mathbf{u _{n_\mathbf k }}\) such that \(\mathbf v _{n_\mathbf k }=F\mathbf u _{n_\mathbf k }\) converges in \(C(I_\mathbf k )\) for any \(\mathbf k \in N\). Let \(\mathbf v \) be the limit of \(\mathbf v _{n_\mathbf k }\). It is easily observe that \(\mathbf v \in C(\mathbb {R},\mathbb {R}^{4})\). Furthermore, from Lemma 5.4 and \(\varPi \) is closed, we have that \(\mathbf v \in \varPi \). Since \(\mu>\lambda _c>0\), it can be obtained that \(e^{-\mu \mid t\mid }\overline{\phi }(t)\), \(e^{-\mu \mid t\mid }\overline{\varphi }(t)\), \(e^{-\mu \mid t\mid }\overline{\psi }(t)\) and \(e^{-\mu \mid t\mid }\overline{\gamma }(t)\) are uniformly bounded on \(\mathbb {R}\). Consequently, \(\varPi \) is uniformly bounded with respect to the norm \(|{\cdot }|_{\mu }\). Therefore, it follows that the norm \(|\mathbf v _{n_\mathbf k }-\mathbf v |_{\mu }\) is uniformly bounded for all \(\mathbf k \in N\). For any \(\varepsilon >0\), there exists an integer \(Z>0\), independent of \(\mathbf v _{n_\mathbf k }\) such that \(e^{-\mu \mid t\mid }|\mathbf v _{n_\mathbf k }(x)-\mathbf v (x)|<\varepsilon \) for \(|t|>Z\) and \(\mathbf k \in N\). Due to the fact that \(\mathbf v _{n_\mathbf k }\) converges to \(\mathbf v \) on the compact interval \([-\,Z,Z]\) with respect to the maximum norm, it then follows that there exists \(K\in N\), such that \(e^{-\mu \mid t\mid }|\mathbf v _{n_\mathbf k }(x)-\mathbf v (x)|<\varepsilon \) for \(|t|\le Z\) and \(\mathbf k > K\). Consequently, it can be concluded that \(\mathbf v _{n_\mathbf k }\) converges to \(\mathbf v \) with respect to the norm \(|{\cdot }|_{\mu }\). The proof is completed. \(\square \)
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Wang, W., Zhang, T. Caspase-1-Mediated Pyroptosis of the Predominance for Driving CD4\(^{+}\) T Cells Death: A Nonlocal Spatial Mathematical Model. Bull Math Biol 80, 540–582 (2018). https://doi.org/10.1007/s11538-017-0389-8
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DOI: https://doi.org/10.1007/s11538-017-0389-8
Keywords
- Caspase-1-mediated pyroptosis
- Inflammatory cytokines IL-1\(\beta \)
- Basic reproduction number
- Threshold dynamics
- Travelling wave solutions