Immune response in virus model structured by cell infection-age

  • Received: 01 October 2015 Accepted: 29 June 2018 Published: 01 July 2016
  • MSC : Primary: 92C60; Secondary: 35B40.

  • This paper concerns modeling the coupled within-host population dynamics of virus and CTL (Cytotoxic T Lymphocyte) immune response. There is substantial evidence that the CTL immune response plays a crucial role in controlling HIV in infected patients. Recent experimental studies have demonstrated that certain CTL variants can recognize HIV infected cells early in the infected cell lifecycle before viral production, while other CTLs only detect viral proteins (epitopes) presented on the surface of infected cells after viral production. The kinetics of epitope presentation and immune recognition can impact the efficacy of the immune response. We extend previous virus models to include cell infection-age structure in the infected cell compartment and immune response killing/activation rates of a PDE-ODE system. We characterize solutions to our system utilizing semigroup theory, determine equilibria and reproduction numbers, and prove stability and persistence results. Numerical simulations show that ``early immune recognition'' precipitates both enhanced viral control and sustained oscillations via a Hopf bifurcation. In addition to inducing oscillatory dynamics, considering immune process rates to be functions of cell infection-age can also lead to coexistence of multiple distinct immune effector populations.

    Citation: Cameron Browne. Immune response in virus model structured by cell infection-age[J]. Mathematical Biosciences and Engineering, 2016, 13(5): 887-909. doi: 10.3934/mbe.2016022

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  • This paper concerns modeling the coupled within-host population dynamics of virus and CTL (Cytotoxic T Lymphocyte) immune response. There is substantial evidence that the CTL immune response plays a crucial role in controlling HIV in infected patients. Recent experimental studies have demonstrated that certain CTL variants can recognize HIV infected cells early in the infected cell lifecycle before viral production, while other CTLs only detect viral proteins (epitopes) presented on the surface of infected cells after viral production. The kinetics of epitope presentation and immune recognition can impact the efficacy of the immune response. We extend previous virus models to include cell infection-age structure in the infected cell compartment and immune response killing/activation rates of a PDE-ODE system. We characterize solutions to our system utilizing semigroup theory, determine equilibria and reproduction numbers, and prove stability and persistence results. Numerical simulations show that ``early immune recognition'' precipitates both enhanced viral control and sustained oscillations via a Hopf bifurcation. In addition to inducing oscillatory dynamics, considering immune process rates to be functions of cell infection-age can also lead to coexistence of multiple distinct immune effector populations.


    In this paper, we shall only consider graphs without multiple edges or loops. Let G=(V(G),E(G)) be a graph, vV(G), the neighborhood of v in G is denoted by N(v). That is to say N(v)={u|uvE(G),uV(G)}. The degree of a vertex v is denoted by d(v), i.e. d(v)=|N(v)|. A graph is trivial if it has a single vertex. The maximum degree and the minimum degree of a graph G are denoted by Δ(G) and δ(G), respectively. Denote by Kn the complete graph on n vertices.

    A subset D of the vertex set of a graph G is a dominating set if every vertex not in D has at least one neighbor in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A dominating set D of G with |D|=γ(G) is called a γ-set of G.

    Roman domination of graphs is an interesting variety of domination, which was proposed by Cockayne et al. [6]. A Roman dominating function (RDF) of a graph G is a function f:V(G){0,1,2} such that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight w(f) of a Roman dominating function f is the value w(f)=uV(G)f(u). The minimum weight of an RDF on a graph G is called the Roman domination number γR(G) of G. An RDF f of G with w(f)=γR(G) is called a γR-function of G. The problems on domination and Roman domination of graphs have been investigated widely, for example, see list of references [8,9,10,13] and [3,7,12], respectively.

    In 2016, Chellali et al. [5] introduced a variant of Roman dominating functions, called Roman {2}-dominating functions. A Roman {2}-dominating function (R{2}DF) of G is a function f:V{0,1,2} such that uN(v)f(u)2 for every vertex vV with f(v)=0. The weight of a Roman {2}-dominating function f is the sum vVf(v). The Roman {2}-domination number γ{R2}(G) is the minimum weight of an R{2}DF of G. Note that if f is an R{2}DF of G and v is a vertex with f(v)=0, then either there is a vertex uN(v) with f(u)=2, or at least two vertices x,yN(v) with f(x)=f(y)=1. Hence, an R{2}DF of G is also an RDF of G, which is also mentioned by Chellali et al [5]. Moreover, they showed that the decision problem for Roman {2}-domination is NP-complete, even for bipartite graphs.

    In fact, a Roman {2}-dominating function is essentially the same as a weak {2}-dominating function, which was introduced by Brešar et al. [1] and studied in literatures [2,11,14,15].

    For a mapping f:V(G){0,1,2}, let (V0,V1,V2) be the ordered partition of V(G) induced by f such that Vi={x:f(x)=i} for i=0,1,2. Note that there exists a 1-1 correspondence between the function f and the partition (V0,V1,V2) of V(G), so we will write f=(V0,V1,V2).

    Chellali et al. [4] obtained the following lower bound of Roman domination number.

    Lemma 1. (Chellali et al. [4]) Let G be a nontrivial connected graph with maximum degree Δ. Then γR(G)Δ+1Δγ(G).

    In this paper, we generalize this result on nontrivial connected graph G with maximum degree Δ and minimum degree δ. We prove that γR(G)Δ+2δΔ+δγ(G). As a corollary, we obtain that 32γ(G)γR(G)2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)2γ{R2}(G)1 for every graph G and there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k1 for any integer k2.

    Lemma 2. (Cockayne et al. [6]) Let f=(V0,V1,V2) be a γR-function of an isolate-free graph G with |V1| as small as possible. Then

    (i) No edge of G joins V1 and V2;

    (ii) V1 is independent, namely no edge of G joins two vertices in V1;

    (iii) Each vertex of V0 is adjacent to at most one vertex of V1.

    Theorem 3. Let G be a nontrivial connected graph with maximum degree Δ(G)=Δ and minimum degree δ(G)=δ. Then

    γR(G)Δ+2δΔ+δγ(G). (2.1)

    Moreover, if the equality holds, then

    γ(G)=n(Δ+δ)Δδ+Δ+δandγR(G)=n(Δ+2δ)Δδ+Δ+δ.

    Proof. Let f=(V0,V1,V2) be a γR-function of G with V1 as small as possible. By Lemma 2, we know that N(v)V0 for any vV1 and N(v1)N(v2)= for any v1,v2V1. So we have

    |V1||V0|δ (2.2)

    Since G is nontrivial, it follows that V2. Note that every vertex in V2 is adjacent to at most Δ vertices in V0; we have

    |V0|Δ|V2| (2.3)

    By Formulae (2.2) and (2.3), we have

    |V1|Δδ|V2| (2.4)

    By the definition of an RDF, every vertex in V0 has at least one neighbor in V2. So V1V2 is a dominating set of G. Together with Formula (2.4), we can obtain that

    γ(G)|V1|+|V2|Δδ|V2|+|V2|=Δ+δδ|V2|.

    Note that f is a γR-function of G; we have

    γR(G)=|V1|+2|V2|=(|V1|+|V2|)+|V2|γ(G)+δΔ+δγ(G)=Δ+2δΔ+δγ(G).

    Moreover, if the equality in Formula (2.1) holds, then by previous argument we obtain that |V1|=|V0|δ, |V0|=Δ|V2|, and V1V2 is a γ-set of G. Then we have

    n=|V0|+|V1|+|V2|=|V0|+|V0|δ+|V0|Δ=Δδ+Δ+δΔδ|V0|.

    Hence, we have

    |V0|=nΔδΔδ+Δ+δ,|V1|=nΔΔδ+Δ+δ, and |V2|=nδΔδ+Δ+δ.

    So

    γR(G)=|V1|+2|V2|=n(Δ+2δ)Δδ+Δ+δ and γ(G)=|V1|+|V2|=n(Δ+δ)Δδ+Δ+δ

    since V1V2 is a γ-set of G. This completes the proof.

    Now we show that the lower bound in Theorem 3 can be attained by constructing an infinite family of graphs. For any integers k2, δ2 and Δ=kδ, we construct a graph Hk from K1,Δ by adding k news vertices such that each new vertex is adjacent to δ vertices of K1,Δ with degree 1 and no two new vertices has common neighbors. Then add some edges between the neighbors of each new vertex u such that δ(Hk)=δ and the induced subetaaph of N(u) in Hk is not complete. The resulting graph Hk is a connected graph with maximum degree Δ(G)=Δ and maximum degree δ(G)=δ. It can be checked that γ(Hk)=k+1 and γR(Hk)=k+2=Δ+2δΔ+δγ(G).

    For example, if k=2, δ=3 and Δ=kδ=6, then the graph H2 constructed by the above method is shown in Figure 1, where u1 and u2 are new vertices.

    Figure 1.  An example to illustrate the construction of Hk.

    Furthermore, by Theorem 3, we can obtain a lower bound of the Roman domination number on regular graphs.

    Corollary 4. Let G be an r-regular graph, where r1. Then

    γR(G)32γ(G) (2.5)

    Moreover, if the equality holds, then

    γ(G)=2nr+2andγR(G)=3nr+2.

    Proof. Since G is r-regular, we have Δ(G)=δ(G)=r. By Theorem 3 we can obtain that this corollary is true.

    For any integer n2, denote by G2n the (2n2)-regular graph with 2n vertices, namely G2n is the graph obtained from K2n by deleting a perfect matching. It can be checked that γ(G2n)=2 and γR(G2n)=3=32γ(G) for any n2. Hence, the bound in Corollary 4 is attained.

    Note that γR(G)2γ(G) for any graph G; we can conclude the following result.

    Corollary 5. Let G be an r-regular graph, where r1. Then

    32γ(G)γR(G)2γ(G).

    Chellali et al. [5] obtain the following bounds for the Roman {2}-domination number of a graph G.

    Lemma 6. (Chellali et al. [5]) For every graph G, γ(G)γ{R2}(G)γR(G)2γ(G).

    Lemma 7. (Chellali et al. [5]) If G is a connected graph of order n and maximum degree Δ(G)=Δ, then

    γ{R2}(G)2nΔ+2.

    Theorem 8. For every graph G, γR(G)2γ{R2}(G)1. Moreover, for any integer k2, there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k1.

    Proof. Let f=(V0,V1,V2) be an γ{R2}-function of G. Then γ{R2}(G)=|V1|+2|V2| and γR(G)2|V1|+2|V2| since V1V2 is a dominating set of G. If |V2|1, then γR(G)2|V1|+2|V2|=2γ{R2}(G)2|V2|2γ{R2}(G)2. If |V2|=0, then every vertex in V0 is adjacent to at least two vertices in V1. So for any vertex uV1, f=(V0,{u},V1{u}) is an RDF of G. Then we have γR(G)1+2|V1{u}|=2|V1|1=2γ{R2}(G)1.

    For any integer k2, let Ik be the graph obtained from Kk by replacing every edge of Kk with two paths of length 2. Then Δ(Ik)=2(k1) and δ(Ik)=2. We first prove that γ{R2}(Ik)=k. Since V(Ik)=|V(Kk)|+2|E(Kk)|=k+2k(k1)2=k2, by Lemma 7 we can obtain γ{R2}(Ik)2|V(Ik)|Δ(Ik)+2=2k22(k1)+2=k. On the other hand, let f(x)=1 for each xV(Ik) with d(x)=2(k1) and f(y)=0 for each yV(Ik) with d(y)=2. It can be seen that f is an R{2}DF of Ik and w(f)=k. Hence, γ{R2}(Ik)=k.

    We now prove that γR(Ik)=2k1. Let g={V1,V2,V3} be a γR-function of Ik such that |V1| is minimum. For each 4-cycle C=v1v2v3v4v1 of Ik with d(v1)=d(v3)=2(k1) and d(v2)=d(v4)=2, we have wg(C)=g(v1)+g(v2)+g(v3)+g(v4)2. If wg(C)=2, then by Lemma 2(iii) we have g(vi){0,2} for any i{1,2,3,4}. Hence, one of v1 and v3 has value 2 and g(v2)=g(v4)=0. If wg(C)=3, then by Lemma 2(i) we have {g(v1),g(v3)}={1,2} or {g(v2),g(v4)}={1,2}. When {g(v2),g(v4)}={1,2}, let {g(v1),g(v2)}={1,2}, g(v2)=g(v4)=0 and g(x)=g(x) for any xV(Ik){v1,v2,v3,v4}. Then g is also a γR-function of Ik. If wg(C)=4, then exchange the values on C such that v1,v3 have value 2 and v2,v4 have value 0. So we obtain that Ik has a γR-function h such that h(y)=0 for any yV(Ik) with degree 2. Note that any two vertices of Ik with degree 2(k1) belongs to a 4-cycle considered above; we can obtain that there is exactly one vertex z of Ik with degree 2(k1) such that h(z)=1. Hence, γR(Ik)=w(h)=2k1.

    Note that the graph Ik constructed in Theorem 8 satisfies that γ(Ik)=k=γ{R2}(Ik). By Theorem 8, it suffices to prove that γ(Ik)=k. Let A={v:vV(Ik),d(v)=2(k1)} and B=V(Ik)A. We will prove that Ik has a γ-set containing no vertex of B. Let D be a γ-set of Ik. If D contains a vertex uB. Since the degree of u is 2, let u1 and u2 be two neighbors of u in Ik. Then d(u1)=d(u2)=2(k1) and, by the construction of Ik, u1 and u2 have two common neighbors u,u with degree 2. Hence, at least one of u,u1, and u2 belongs to D. Let D=(D{u,u}){u1,u2}. Then D is also a γ-set of Ik. Hence, we can obtain a γ-set of Ik containing no vertex of B by performing the above operation for each vertex vDB. So A is a γ-set of Ik and γ(Ik)=|A|=k.

    By Lemma 6 and Theorem 8, we can obtain the following corollary.

    Corollary 9. For every graph G, γ{R2}(G)γR(G)2γ{R2}(G)1.

    Theorem 10. For every graph G, γR(G)γ(G)+γ{R2}(G)1.

    Proof. By Lemma 6 we can obtain that γR(G)2γ(G)γ(G)+γ{R2}(G). If the equality holds, then γR(G)=2γ(G) and γ(G)=γ{R2}(G). So γR(G)=2γ{R2}(G), which contradicts Theorem 8. Hence, we have γR(G)γ(G)+γ{R2}(G)1.

    In this paper, we prove that γR(G)Δ+2δΔ+δγ(G) for any nontrivial connected graph G with maximum degree Δ and minimum degree δ, which improves a result obtained by Chellali et al. [4]. As a corollary, we obtain that 32γ(G)γR(G)2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)2γ{R2}(G)1 for every graph G and the bound is achieved. Although the bounds in Theorem 3 and Theorem 8 are achieved, characterizing the graphs that satisfy the equalities remain a challenge for further work.

    The author thanks anonymous referees sincerely for their helpful suggestions to improve this work. This work was supported by the National Natural Science Foundation of China (No.61802158) and Natural Science Foundation of Gansu Province (20JR10RA605).

    The author declares that they have no conflict of interest.

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