
We examined intraspecific infectious rivalry in a dynamic contagious disease model. A non-linear dynamic model that considers multiple individual categories was used to study the transmission of infectious diseases. The combined effect of parameter sensitivities on the model was simulated using system sensitivities. To investigate the dynamic behavior and complexity of the model, the Caputo-Fabrizio (C-F) fractional derivative was utilized. The behavior of the proposed model around the parameters was examined using sensitivity analysis, and fractional solutions included more information than the classical model. Fixed point theory was used to analyze the existence and uniqueness of the solution. The Ulam-Hyers (U-H) criterion was used to examine the stability of the system. A numerical approach based on the C-F fractional operator was utilized to improve comprehension and treatment of the infectious disease model. A more precise and valuable technique for solving the infectious disease model was used in MATLAB numerical simulations to demonstrate. Time series and phase diagrams with different orders and parameters were generated. We aimed to expedite patient recovery while reducing the frequency of disease transmission in the community.
Citation: Parveen Kumar, Sunil Kumar, Badr Saad T Alkahtani, Sara S Alzaid. A mathematical model for simulating the spread of infectious disease using the Caputo-Fabrizio fractional-order operator[J]. AIMS Mathematics, 2024, 9(11): 30864-30897. doi: 10.3934/math.20241490
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We examined intraspecific infectious rivalry in a dynamic contagious disease model. A non-linear dynamic model that considers multiple individual categories was used to study the transmission of infectious diseases. The combined effect of parameter sensitivities on the model was simulated using system sensitivities. To investigate the dynamic behavior and complexity of the model, the Caputo-Fabrizio (C-F) fractional derivative was utilized. The behavior of the proposed model around the parameters was examined using sensitivity analysis, and fractional solutions included more information than the classical model. Fixed point theory was used to analyze the existence and uniqueness of the solution. The Ulam-Hyers (U-H) criterion was used to examine the stability of the system. A numerical approach based on the C-F fractional operator was utilized to improve comprehension and treatment of the infectious disease model. A more precise and valuable technique for solving the infectious disease model was used in MATLAB numerical simulations to demonstrate. Time series and phase diagrams with different orders and parameters were generated. We aimed to expedite patient recovery while reducing the frequency of disease transmission in the community.
We only consider finite and undirected graphs throughout this paper. Let G=(V(G),E(G)) be a graph with n=|V(G)| vertices and m=|E(G)| edges. For any vertex u∈V(G), we use dG(u) (or du when no confusion can arise) to denote the degree of u in G, which is the number of edges incident to u. Such a maximal number (resp. minimal number) is called the maximal degree Δ(G) (resp. minimal degree δ(G)). For any vertex u in G, we use NG(u) to denote the set of all vertices adjacent with u, and the elements of NG(u) are called neighbors of u. A sequence of positive integers π=(d1,d2,…,dn) is called the degree sequence of G if di=dvi for any vertex vi∈V(G), where i=1,2,…,n.
The join of two graphs G1 and G2, denoted by G1+G2, is the graph with the vertex set V(G1)∪V(G2) and edge set E(G1)∪E(G2)∪{xy|x∈V(G1),y∈V(G2)}. The cyclomatic number of G is the minimum number of edges in it whose removal makes it acyclic, denoted by γ=γ(G). Let Gγn be the set of n-vertex graphs with cyclomatic number γ. We use Kn and Pn to denote the complete graph and path of n vertices, respectively. As usual, we use the symbol ℓ(Pn) to denote the length of the path Pn, which equals to the number of edges in Pn. The cyclomatic number, denoted by γ, of a graph G is the minimum number of edges of G whose removal makes G acyclic. Let Gγn be the class of all connected graphs with order n and cyclomatic number γ. We use [4] for terminology and notation not defined here.
The topological index is a real number that can be used to characterize the properties of the molecule graph. Nowadays, hundreds of topological indices have been considered and used in quantitative structure-activity and quantitative structure-property relationships. One of the well-known topological indices is the general Randić index, which was defined by Bollobás and Erdös [5] and Amic [1] independently:
Rα(G)=∑uv∈E(G)[dudv]α, |
where α is a nonzero real number. This topological index has been extensively investigated. We encourage interested readers to consult [3,6,7,10,11,13] for more mathematical properties and their applications.
Even though the mathematical and chemical theory of the general Randić index has been well considered, some extremal graph-theoretical problems concerning this graph invariant are still open. In this paper, we focus on exploring the extremal graphs in Gγn with respect to the general Randić index.
It is interesting to explore the extremal graphs for some topological indices in the class of graphs with a given cyclomatic number. In this section, we focus on determining the extremal graphs in Gγn with the minimum general Randić index. Before proceeding, we shall prove or list several facts as preliminaries.
Lemma 2.1. The function P(x,α)=2αxα+1−(x−1)α[2α(x−2)+3α]+xα(2α−3α)−6α>0 for x≥4 and 1≤α≤3925.
Proof. It is routine to check that
P(x,α)=2αxα+1−(x−1)α[2α(x−2)+3α]+xα(2α−3α)−6α=2αxα+1−(x−1)α[2α(x−1)−2α+3α]+xα(2α−3α)−6α=2α[xα+1−(x−1)α+1]+(2α−3α)(x−1)α+(2α−3α)xα−6α=2α[xα+1−(x−1)α+1]+(2α−3α)[xα+(x−1)α]−6α=2α[xα+(x−1)α]+2αx(x−1)[xα−1−(x−1)α−1]+(2α−3α)[xα+(x−1)α]−6α=(2⋅2α−3α)[xα−(x−1)α]+2αx(x−1)[xα−1−(x−1)α−1]−6α. |
Note that ρ(t)=tα−(t−1)α is an increasing function for t∈[4,+∞), and 2⋅2α>3α if, and only if, α<ln2ln3−ln2≈1.709, then we have
P(x,α)=(2⋅2α−3α)[xα−(x−1)α]+2αx(x−1)[xα−1−(x−1)α−1]−6α≥(2⋅2α−3α)(4α+3α)+2α⋅12⋅(4α−1−3α−1)−6α=5⋅8α−3⋅6α−9α−12α. |
For simplicity, let H(α)=5⋅8α−3⋅6α−9α−12α. To continue our proof, we first verify the following fact.
Claim 1. The function ϱ(t)=k1at−k2bt−k3ct has a unique zero point in the interval [0,+∞) for any positive real numbers k1,k2,k3,a,b,c such that k1−k2−k3>0 and 1<a<b<c.
Proof of Claim 1. It is routine to check that ϱ′(t)=k1lna⋅at−k2lnb⋅bt−k3lnc⋅ct. Note that ϱ(0)=k1−k2−k3>0 and ϱ(M)=at[k1−k2(ba)t−k3(ca)t]t=M→−∞, and it follows that ϱ(t) has zero points in the interval [0,+∞). Without loss of generality, we assume that t1,t2=t1+h∈[0,+∞) are the two distinct zero points of ϱ(t) for h>0, which is equivalent to k1at1−k2bt1−k3ct1=0 and k1at2−k2bt2−k3ct2=0. Besides, we know that ϱ′(t)=k1lna⋅at−k2lnb⋅bt−k3lnc⋅ct, which implies that
ϱ′(t1)=k1lna⋅at1−k2lnb⋅bt1−k3lnc⋅ct1<lna(k1at1−k2bt1−k3ct1)=0. |
In addition, we have
ϱ(t2)=ϱ(t1+h)=k1at1ah−k2bt1bh−k3ct1ch=(k2bt1+k3ct1)ah−k2bt1bh−k3ct1ch<(k2bt1+k3ct1)ah−(k2bt1+k3ct1)bh=(k2bt1+k3ct1)(ah−bh)<0, |
which contradicts to the fact that ϱ(t2)=0. Hence, there must exist a unique number t0∈[0,+∞) such that ϱ(t0)=0. As desired, we have completed the proof of Claim 1.
Claim 2. The function H(α)=5⋅8α−3⋅6α−9α−12α has a unique zero point in the interval (1,2).
Proof of Claim 2. It is routine to check that H′(α)=5ln8⋅8α−3ln6⋅6α−ln9⋅9α−ln12⋅12α. Note that H(1)=1>0 and H(2)=−13<0, and it follows that H(α) has zero points in the interval (1,2). Without loss of generality, we assume that α0,α1,…,αl are the zero points of H(α) such that 1<α0<α1<…<αl. Hence, H(α0)=5⋅8α0−3⋅6α0−9α0−12α0=0. Furthermore,
H′(α0)=5ln8⋅8α0−3ln6⋅6α0−ln9⋅9α0−ln12⋅12α0=ln8(3⋅6α0+9α0+12α0)−3ln6⋅6α0−ln9⋅9α0−ln12⋅12α0=3(ln8−ln6)⏟k16α0−(ln9−ln8)⏟k29α0−(ln12−ln8)⏟k312α0. |
It follows from Claim 1 that ϱ(t)|a=6,b=9,c=12 has a unique zero point in the interval t0∈[0,+∞). Consequently, we know that the unique zero point of ϱ(t)|a=6,b=9,c=12 must lie in the interval (0,1) since ϱ(0)|a=6,b=9,c=12>0 and ϱ(1)|a=6,b=9,c=12=−0.7473<0. Hence, ϱ(t)|a=6,b=9,c=12<0 always holds for any real number t≥1. This implies that H′(αi)=ϱ(αi)|a=6,b=9,c=12<0 for αi>1 and i=0,1,…,l, which contradicts to the continuity of the function H(α). As desired, we have completed the proof of Claim 2.
Now, we continue to our proof. Note that H(3925)=5⋅83925−3⋅63925−93925−123925≈0.01857>0 and H(1.57)=5⋅81.57−3⋅61.57−91.57−121.57≈−0.07428<0. Hence, P(x,α)≥H(α)>0 for α∈[1,3925]. As desired, we have completed the proof of Lemma 2.1.
Lemma 2.2. The function Q(x,α)=3α(xα+1−(x−1)α+1)−9α+2xα(2α−3α)>0 for x≥4 and α≥1.
Proof. For simplicity, we distinguish the following two cases.
Case 1. α∈[1,3).
Note that h(t)=tα is an increasing function in the interval [1−1x,1] for α≥1, and it follows from Lagrange's mean value formula that h(1)−h(1−1x)=[1−(1−1x)]h′(ξ)=1xh′(ξ)=1xαξα−1>0, where ξ∈(1−1x,1). Hence, x[1−(1−1x)α]=x[h(1)−h(1−1x)]=αξα−1. Thus, we have
Qx(x,α)=3α(α+1)[xα−(x−1)α]+2αxα−1(2α−3α)=xα−1{3α(α+1)x[1−(1−1x)α]+2α(2α−3α)}=xα−1[3αα(α+1)ξα−1+2α(2α−3α)]=αxα−1[3α(α+1)ξα−1+2(2α−3α)]. |
By our initial hypothesis, it is routine to check that ξα−1>(1−1x)α−1, then we have
Qx(x,α)>αxα−1[3α(α+1)(1−1x)α−1+2(2α−3α)]>αxα−1[3α(α+1)(34)α−1+2(2α−3α)](because1−1x>34)=2α3αxα−1[12(α+1)(34)α−1+(23)α−1]>2α3αxα−1[932(α+1)+(23)α−1](because(34)α−1>(34)2). |
Let p(α)=932(α+1)+(23)α−1, then we have p′(α)=932+(23)αln23 and p″(α)=(23)α(ln23)2>0. Hence, p′(α)≥p′(1)=932+23ln(23)≥1100>0, which implies that p(α) is increasing in the interval [1,+∞). Hence, p(α)≥p(1)=1148>0. It immediately yields that Qx(x,α)>2α3αxα−1p(α)>0. Therefore, we have
Q(x,α)≥f(4,α)=3α(4α+1−3α+1)−9α+2⋅4α(2α−3α)=2⋅9α[(129)α+(89)α−2]>0, |
as desired, and we have completed the proof.
Case 2. α∈[3,+∞).
Note that
Qx(x,α)=3α(α+1)[xα−(x−1)α]+2αxα−1(2α−3α)=xα−1{3α(α+1)x[1−(1−1x)α]+2α(2α−3α)}. |
Let g(α)=1−2x−(1−1x)α be a function defined in the interval [3,+∞), then we have g′(α)=(1−1x)αln(1+1x−1)>0. Hence, g(α)≥g(3)=1−2x−(1−1x)3=x2−3x+1x3>0, implying that 1−(1−1x)α>2x. Thus, we have
Qx(x,α)=xα−1{3α(α+1)x[1−(1−1x)α]+2α(2α−3α)}>2αxα−1[3αα+1α−(3α−2α)](becauseα+1α>1)>0. |
Let l(α)=(129)α+(89)α−2. It is routine to check that l′(α)=(89)α[(128)αln(129)+ln(89)]>(89)α[ln(129)+ln(89)]=(89)αln15>0. Hence, l(α)≥l(3)=782729>0. It then follows that
Q(x,α)≥f(4,α)=3α(4α+1−3α+1)−9α+2⋅4α(2α−3α)=2⋅9α[(129)α+(89)α−2]>0, |
as desired, and we have completed the proof.
Proposition 2.3. Let G∈Gγn be a graph with γ≥3 and n≥2(γ−1), then there is no pendent vertex in G if it has a minimum general Randić index for α≥1.
Proof. Suppose to the contrary that there exists a pendent vertex in G. Let u be a vertex of degree at least three and NG(u)={u1,u2,…,uk}. In what follows, we use P=uu1^u2…^ur to denote a pendent path in G. Assume that u2≠u1 is another neighbor of u with du2≥2. We consider the graph ^G1=G−uu2+u2^ur (depicted in Figure 2), which is an element of Gγn. Let l−2 be the number of vertices in {u3,u4,…,uk}, whose degree is greater than or equal to two. Clearly, l≥2 and k∑i=3dαui≥2α(l−2). For simplicity, we distinguish the following two cases:
Case 1. ℓ(P)=1.
Direct calculations show that
Rα(G)−Rα(^G1)=dαudαu2+dαu−2α(du−1)α−2αdαu2+[dαu−(du−1)α]k∑i=3dαui≥dαudαu2+dαu−2α(du−1)α−2αdαu2+2α(l−2)[dαu−(du−1)α]=[(dαu2−2α+1)(dαu−2α)]⏟A1+2α{(l−1)[dαu−(du−1)α]+(1−2α)}⏟A2. |
It is not difficult to find the first term of the previous equality A1=[(dαu2−2α+1)(dαu−2α)]>0 for α≥1, du≥3 and du2≥2. To continue the proof, it remains to verify that A2>0. For simplicity, we let H(x)=(l−1)[xα−(x−1)α]−(2α−1) for α≥1 and x≥3. It is routine to check that H(x)>[(xα−(x−1)α)−(3α−2α)]+[(3α−2α)−(2α−1)] since l≥3. Note that f1(x)=xα−(x−1)α is increasing in the interval [3,△], then we have xα−(x−1)α≥3α−2α. In addition, we know that 3α−2α≥2α−1 always holds for α≥1. Hence, H(x)>0 and, consequently, we have A2>0. It then immediately deduces that Rα(G)−Rα(^G1)>0, a contradiction. This implies that there is no pendent vertex in G.
Case 2. ℓ(P)≥2.
Direct calculations show that
Rα(G)−Rα(^G1)=dαudαu2−(du−1)α2α+2αdαu−2α(du−1)α+2α(1−2α)−2αdαu2+[dαu−(du−1)α]k∑i=3dαui≥dαudαu2−(du−1)α2α+2αdαu−2α(du−1)α+2α(l−2)[dαu−(du−1)α]+2α(1−2α)−2αdαu2=dαu2(dαu−2α)⏟A3+2α(l−1)[dαu−(du−1)α]⏟A4+2α[dαu−(du−1)α+1−2α]⏟A5. |
Note that A3>0 and A4>0, and it is also not difficult to find that A5=1l−1A2 is positive under the initial assumptions. Hence, Rα(G)−Rα(^G1)>0. Again a contradiction. This implies that there is no pendent vertex in G.
As desired, we complete the proof of Proposition 2.3.
Proposition 2.4. Let G∈Gγn be a graph with γ≥3 and n≥2(γ−1), then the maximum vertex degree is three in G if it has minimum general Randić index for 1≤α≤3925.
Proof. It follows from Proposition 2.3 that G contains at least one cycle as its induced subgraph, and the n-vertex cycle is the only connected graph for the which minimum and maximum vertex degree is two. Hence, in conjunction with the assumption γ≥3, we have Δ=Δ(G)≥3. To complete the proof, it suffices to show that Δ=3. If Δ>3, then it is routine to check that
n=∑2≤i≤Δni≥2(γ−1)=2(m−n)=2(∑2≤i≤Δini2−∑2≤i≤Δni), |
which is equivalent to
n2≥∑4≤i≤Δ(i−3)ni>∑4≤i≤Δ(4−3)ni>0. |
Hence, there at least exists a vertex of degree two. For simplicity, we suppose that u is the vertex in G with maximum degree and NG(u)={u1,u2,…,uΔ}. We distinguish the following two cases.
Case 1. ∃i∈{1,2,…,Δ} such that dui=2.
For convenience, we suppose that u1 is the neighbor of u with degree two and du2≥du3≥…≥2.
Subcase 1.1. du2=2 and u1 is not adjacent to u2.
Let ^G2=G−uu2+u1u2∈Gγn. t is the neighbor of u1, different from u, depicted in Figure 3. Hence, we have
Rα(G)−Rα(^G2)=dαu2α+1−(du−1)α3α−6α+dαt(2α−3α)+[dαu−(du−1)α]Δ∑i=3dαui=dαu2α+1−(du−1)α3α−6α+dαt(2α−3α)+2α(du−2)[dαu−(du−1)α]=2αdα+1u−(du−1)α[2α(du−2)+3α]+dαt(2α−3α)−6α≥2αdα+1u−(du−1)α[2α(du−2)+3α]+dαu(2α−3α)−6α. |
For simplicity, we let f2(x)=2αxα+1−(x−1)α[2α(x−2)+3α]+xα(2α−3α)−6α. It follows from Lemma 2.1 that f2(x)=P(x,α)>0 for x≥4 and 1≤α≤3925. Hence, Rα(G)−Rα(^G2)>0, which contradicts to the choice of G. Hence, the maximum vertex degree of G is three.
Subcase 1.2. du2=2 and u1 is adjacent to u2.
Let ^G3=G−uu3+u1u3∈Gγn, depicted in Figure 4. Hence, we have
Rα(G)−Rα(^G3)=3×dαu2α+4α−(du−1)α3α−2×6α−2α(du−1)α+[dαu−(du−1)α]Δ∑i=4dαui=3×dαu2α+4α−(du−1)α3α−2×6α−2α(du−1)α+2α(du−3)[dαu−(du−1)α]=2αdα+1u+4α−(du−1)α[2α(du−2)+3α]−2×6α. |
Let f3(x)=2αxα+1+4α−(x−1)α[2α(x−2)+3α]−2×6α, and then we have f3(x)=f2(x)+(3α−2α)(xα−2α)>0. Hence, Rα(G)−Rα(^G3)>0 for x≥4 and 1≤α≤3925, which contradicts to the choice of G. Hence, the maximum vertex degree of G is three.
Subcase 1.3. du2>2 and u1 is adjacent to u2 and du3>2.
Let ^G4=G−uu3+u1u3∈Gγn, depicted in Figure 5. Hence, we have
Rα(G)−Rα(^G4)=dαu2α+2αdαu2+dαudαu2+dαudαu3−(du−1)α3α−3αdαu2−dαu2(du−1)α−3αdαu3+[dαu−(du−1)α]Δ∑i=4dαui≥dαu2α+dαu2(2α−3α)+dαu3(dαu−3α)−(du−1)α3α+dαu2(dαu3−3α)−(du−1)α3α+2α(du−3)[dαu−(du−1)α]>2αdα+1u−(du−1)α[2α(du−2)+3α]+dαu(2α−3α)−6α>0, |
and the last inequality holds because f2(x)>0 for x≥4 and 1≤α≤3925. Hence, Rα(G)−Rα(^G4)>0. Again, a contradiction. Hence, the maximum vertex degree of G is three.
Subcase 1.4. du2>2, u1 is adjacent to u2 and du3=2.
Let ^G5=G−uu3+u1u3∈Gγn, depicted in Figure 6. Hence, we have
Rα(G)−Rα(^G5)=dαu2α+1+2αdαu2+dαudαu2−6α−(du−1)α3α−dαu23α−(du−1)αdαu2+[dαu−(du−1)α]Δ∑i=4dαui≥dαu2α+1−6α−(du−1)α3α+dαu2[(2α−3α+dαu−(du−1)α)]+2α(du−3)[dαu−(du−1)α]>2αdα+1u−(du−1)α[2α(du−2)+3α]+4α−2×6α>0, |
and the last inequality holds because f3(x)>0 for x≥4 and 1≤α≤3925. Hence, Rα(G)−Rα(^G5)>0. Again, a contradiction. Thus, we have completed that the maximum vertex degree of G is three.
Subcase 1.5. du2>2, u1 is not adjacent to u2.
Let ^G6=G−uu2+u1u2∈Gγn, depicted in Figure 7. Hence, we have
Rα(G)−Rα(^G6)=dαu2α+2αdαt+dαudαu2−(du−1)α3α−3αdαt−3αdαu2+[dαu−(du−1)α]Δ∑i=3dαui≥dαu2α+dαu(2α−3α)+2α(dαu−3α)−(du−1)α3α+2α(du−2)[dαu−(du−1)α]=2αdα+1u−(du−1)α[2α(du−2)+3α]+dαu(2α−3α)−6α>0, |
and the last inequality holds because f2(x)>0 for x≥4 and 1≤α≤3925. Hence, Rα(G)−Rα(^G6)>0, a contradiction to the choice of G. Hence, the maximum vertex degree of G is three.
Case 2. ∀i∈{1,2,…,Δ} such that dui>2.
Note that there is a vertex u0∈V(G)∖NG(u) of degree two, which is not adjacent to at least one neighbor, say u1, of u. Let ^G7=G−uu1+u0u1∈Gγn (depicted in Figure 8). Hence,
Rα(G)−Rα(^G7)=dαu1(dαu−3α)+(2α−3α)∑z∈NG(u0)dαz+[dαu−(du−1)α]Δ∑i=2dαui≥3α(dαu−3α)+2×dαu(2α−3α)+3α(du−1)[dαu−(du−1)α]=3α[dα+1u−(du−1)α+1]−9α+2×dαu(2α−3α). |
For simplicity, we let f4(x)=3α[xα+1−(x−1)α+1]−9α+2xα(2α−3α)=Q(x,α), which is positive for x≥4 and α≥1 by Lemma 2.2. Hence, we have Rα(G)−Rα(^G7)>0. Thus, there would be a contradiction to the choice of G, and the maximum vertex degree of G is three.
This completes the proof of Proposition 2.4.
Let φij be the number of edges in G joining the vertices of degree i and j, and we use ni and nj to denote the number of vertices of degree i and j, respectively.
Proposition 2.5.. ([2]) Let G∈Gγn, γ≥3, be a graph such that it contains only vertices of degrees two and three, then the following holds:
(i) at least two vertices of degree two are adjacent if n>5(γ−1).
(ii) φ22=0 implies φ33=0 (or φ33=0 implies φ22=0) if n=5(γ−1).
(iii) at least two vertices of degree three are adjacent if 2(γ−1)≤n≤5(γ−1).
Proposition 2.6. Let G∈Gγn be a graph with γ≥3 and n>5(γ−1), then at least one of the vertices x and y for any edge e=xy has the degree two in G if it has a minimum general Randić index for 1≤α≤3925.
Proof. By Proposition 2.3 and Proposition 2.4, we know 2≤du≤3 holds for any vertex u in G. Simultaneously, it follows from Proposition 2.5 that there at least exist two vertices, say u1 and u2, such that φ22>0. Suppose to the contrary that there exists two adjacent vertices v1 and v2 of degree three (i.e., φ33>0). Let u0≠u1 be the vertex adjacent with u2, which may coincide with v1 or v2. For convenience, we distinguish the following two cases.
Case 1. NG(u1)∩NG(u2)=∅.
Let ^G8=G−{u1u2,u2u0,v1v2}+{u1u0,v1u2,v2u2} (depicted in Figure 9), which is an element in Gγn. By direct calculations, we have Rα(G)−Rα(^G8)=4α+9α−2×6α>0. This contradicts to the assumption of G. Hence, φ33=0. As desired, we have completed the proof.
Case 2. NG(u1)∩NG(u2)≠∅.
Without loss of generality, we let u0∈NG(u1)∩NG(u2)≠∅. In what follows, we consider the following three subcases.
If u0≠{v1,v2} and u0 is not adjacent to v1 and v2, we let ^G9=G−{u2u0,v1v2}+{u2v2,v1u0} (depicted in Figure 10), which is an element in Gγn. By direct calculations, we have Rα(G)−Rα(^G9)=0. Note that N^G9(u1)∩N^G9(u2)=∅, by the analogous method as in Case 1, and there exists a new graph ~G1 such that Rα(G)−Rα(~G1)=Rα(^G9)−Rα(~G1)>0. This contradicts to the assumption of G. As desired, we have completed the proof.
If u0≠{v1,v2} and u0 is adjacent to v1, we let ^G10=G−{u1u2,v1v2}+{u1v1,u2v2}, which is an element in Gγn. By direct calculations, we have Rα(G)−Rα(^G10)=4α+9α−2⋅6α>0. This contradicts to the assumption of G. As desired, we have completed the proof.
If u0=v2, then we consider a neighbor ˜v of v1 different from v2. Let ^G11=G−{u2v2,˜vv1}+{v1u2,˜vv2}∈Gγn, and again we obtain that Rα(G)−Rα(^G11)=0. Note that N^G11(u1)∩N^G11(u2)=∅, by the analogous method as in Case 1, and there exists a new graph ~G2 such that Rα(G)−Rα(~G2)=Rα(^G11)−Rα(~G2)>0. This contradicts to the assumption of G. We have completed the proof.
Proposition 2.7. Let G∈Gγn be a graph with γ≥3 and n=5(γ−1), then one of the vertices x and y for any edge e=xy has the degree two and the other has the degree three in G if it has the minimum general Randić index for 1≤α≤3925.
Proof. It follows from Propositions 2.3 and 2.4 that 2≤du≤3 holds for any vertex u in G. If φ22=0 and φ33≠0, then one can find that φ23=0 by Proposition 2.5, a contradiction. If φ22≠0 and φ33≠0, then from the proof of Proposition 2.6 we conclude that there exists a graph ^G12∈Gγn such that Rα(G)−Rα(^G12)>0. This contradicts to the initial assumption of G. Hence, φ22=φ33=0. As desired, we complete the proof of Proposition 2.7.
In a similar way, we obtain the following fact.
Proposition 2.8. Let G∈Gγn be a graph with γ≥3 and 2(γ−1)<n<5(γ−1), then G does not contain any edge connecting the vertices of degree two if it has a minimum general Randić index for 1≤α≤3925.
Denote by ¯Gij[φij≠0] that the graphs contain only vertices of degree i and j, such that for every edge in a one end-vertex has the degree i and the other end-vertex has the degree j, and we use ¯Gij[φii=0] (resp. ¯Gij[φjj=0]) to denote the graphs containing only vertices of degree i and j, such that no vertices of degree i (resp. j) are adjacent.
Theorem 2.9. Let G∈Gγn be a graph with γ≥3 and n vertices, then the following holds for 1≤α≤3925:
(i) Rα(G)≥9α(n+γ−1) for n=2(γ−1), with equality if, and only if, G is isomophic to cubic graphs.
(ii) Rα(G)≥6α(2n−4γ+4)−9α(n−5γ+5) for 2(γ−1)<n<5(γ−1), with equality if, and only if, G is isomophic to ¯G23[φ22=0].
(iii) Rα(G)≥6α(n+γ−1) for n=5(γ−1), with equality if, and only if, G is isomophic to ¯G23[φ23≠0].
(iv) Rα(G)≥4α(n−5γ+5)+6α(6γ−6) for n>5(γ−1), with equality if, and only if, G is isomophic to ¯G23[φ33=0].
Proof. Let ^G13∈Gγn be a graph that achieves the minimum general Randić index. We only give the proof of (ii); the rest could be proved in a similar way. It follows from Propositions 2.3 and 2.4 that 2≤du≤3 holds for any vertex u in ^G13. Hence, we have n2+n3=n and 2n2+3n3=2(n+γ−1) by the Handshaking Theorem. Besides, by Proposition 2.8, it is easily seen that φ22=0. Hence, φ23=2n2 and φ23+2φ33=3n3. Direct calculations show that φ23=2n−4γ+4 and φ33=5γ−n−5. Thus, Rα(G)≥Rα(^G13)=6α(2n−4γ+4)−9α(n−5γ+5). The corresponding extremal graph is ¯G23[φ22=0].
The second Zagreb index is another well-known vertex degree-based graph invariant in chemical graph theory, which was introduced in 1972 by Gutman and Trinajstić [8]. We encourage the interested reader to consult [9,12] for more information for this graph invariant. Undoubtedly, the second Zagreb index is the special case of the general Randić index when α=1. It is easily seen that Theorem 2.9 extends one of the main results proved by Ali et al. [2].
We begin with the following auxiliary result, which plays an important part in our proofs.
Proposition 3.1. Let G∈Gγn be a graph with a maximum general Randić index for α≥1, then Δ(G)=n−1.
Proof. Suppose to the contrary that there exists a vertex u in G with Δ(G)=du<n−1. Note that there exists v∈V(G) such that u≠v, du≥dv and NG(v)∖NG(u)={v1,v2,…,vp}≠∅. We can construct a new graph ^G14 the following way
^G14=G−{vv1,vv2,…,vvp}+{uv1,uv2,…,uvp}∈Gγn. |
It is routine to check that
Rα(^G14)−Rα(G)=∑x∈NG(u)∖NG(v)[(du+p)α−dαu]dαx+p∑i=1[(du+p)α−dαv]dαvi+∑y∈NG(u)∩NG(v)[(du+p)α+(dv−p)α−dαu−dαv]dαy. |
Note that H(t)=tα is an increasing function for α≥1, and the first and second terms of the previous equality are nonnegative. By the Lagrange mean value theorem, we have (du+p)α−dαu=αpξα−1 (resp. dαv−(dv−p)α=αpηα−1) for ξ∈(du,du+p) (resp. η∈(dv−p,dv)). Hence, A6=(du+p)α+(dv−p)α−dαu−dαv>0, and, consequently, we have Rα(^G14)>Rα(G), a contradiction. This completes the proof.
Proposition 3.2. ([14]) Let x1,x2,…,xn,p,t≥1 be integers, α be any real number such that α∉{0,1} and x1+x2+…+xn=p.
(1) The function f(x1,x2,…,xn)=∑ni=1xαi is the minimum for α<0 or α>1 (maximum for 0<α<1, respectively) if, and only if, x1,x2,…,xn are almost equal, or |xi−xj|≤1 for every i,j=1,2,…,n.
(2) If x1≥x2≥t, the maximum of the function f(x1,x2,…,xn) is reached for α<0 or α>1 (minimum for 0<α<1,respectively) only for x1=p−t−n+2,x2=t,x3=x4=…=xn=1. The second maximum (the second minimum, respectively) is attained only for x1=p−t−n+1,x2=t+1,x3=x4=…=xn=1.
Theorem 3.3. Let G∈Gγn be a graph with \gamma = {k-1\choose 2} and k\geq 4 , then for \alpha\geq 1 we have
\begin{equation*} \begin{split} R_{\alpha}(G)\leq {k-1\choose 2}(k^{2}-2k+3)^{2\alpha}+(n-1)^{\alpha}(k^{2}-2k+3)^{\alpha}+(n-2)(n-1)^{\alpha}, \end{split} \end{equation*} |
with equality if, and only if, G\cong(K_{1}^{\gamma}\cup (n-2)K_{1})+K_{1}\cong K_{n}^{\gamma} , depicted in Figure 11.
Proof. It follows from Proposition 3.1 that there at least exists one vertex with a maximum degree n-1 . Hence, we have G = \widehat{G_{15}}+K_{1} , which contains |V(\widehat{G_{15}})| = n-1 vertices and m(\widehat{G_{15}}) = {k-1\choose 2} edges. For simplicity, let \pi = (d_{1}, d_{2}, \ldots, d_{n}) and \widehat{\pi} = (\widehat{d_{1}}, \widehat{d_{2}}, \ldots, \widehat{d_{n-1}}) be the nonincreasing degree sequence of G and \widehat{G_{15}} , respectively. Hence, d_{i} = \widehat{d_{i-1}}+1 for i = 2, 3, \ldots, n and d_{1} = n-1. Thus, we have
\begin{equation*} \begin{split} R_{\alpha}(G) = &\sum\limits_{v_{1}v_{j}\in E(G)}\left[d_{1}d_{j}\right]^{\alpha}+\sum\limits_{v_{i}v_{j}\in E(G), 2\leq i < j\leq n}\left[d_{i}d_{j}\right]^{\alpha}\\ = &(n-1)^{\alpha}\sum\limits_{i = 2}^{n}d_{i}^{\alpha}+\sum\limits_{v_{i}v_{j}\in E(\widehat{G_{15}})}\left[\left(\widehat{d_{i}}+1\right)\left(\widehat{d_{j}}+1\right)\right]^{\alpha}. \end{split} \end{equation*} |
By Proposition 3.2 for \alpha\geq 1 , we have
\begin{equation*} \begin{split} \mathscr{A}_{7} = &\sum\limits_{i = 1}^{n}d_{i}^{\alpha}-d_{1}^{\alpha}\\ \leq&1\cdot(n-1)^{\alpha}+1\cdot t^{\alpha}+(n-2)\cdot 1^{\alpha}-1\cdot(n-1)^{\alpha}\\ = &(n-1)^{\alpha}+(k^{2}-3k+3)^{\alpha}+(n-2)-(n-1)^{\alpha}\\ = &(k^{2}-3k+3)^{\alpha}+(n-2), \end{split} \end{equation*} |
where d_{1} = n-1 = 2m-t-n+2, d_{2} = t = 2\gamma+1 and d_{3} = d_{4} = \ldots = d_{n} = 1 . In addition, we find the maximum value of
\begin{equation*} \begin{split} \mathscr{A}_{8} = &\sum\limits_{v_{i}v_{j}\in E(\widehat{G_{15}})}\left[\left(\widehat{d_{i}}+1\right)\left(\widehat{d_{j}}+1\right)\right]^{\alpha}\\ \leq& {k-1\choose 2}\left[\left(k^{2}-3k+3\right)\left(k^{2}-3k+3\right)\right]^{\alpha}, \end{split} \end{equation*} |
with equality if, and only if, \widehat{G_{15}}\cong K_{1}^{\gamma}\cup (n-2)K_{1} .
It follows from the previous that
\begin{equation*} \begin{split} R_{\alpha}(G) = &(n-1)^{\alpha}\mathscr{A}_{7}+\mathscr{A}_{8}\\ \leq &{k-1\choose 2}(k^{2}-2k+3)^{2\alpha}+(n-1)^{\alpha}(k^{2}-2k+3)^{\alpha}+(n-2)(n-1)^{\alpha}, \end{split} \end{equation*} |
Hence, G = (K_{1}^{\gamma}\cup (n-2)K_{1})+K_{1}\cong K_{n}^{\gamma} . This completes the proof of Theorem 3.3.
The authors declare they have not used Artificial Intelligence(AI) tools in the creation of this article.
We would like to show our great gratitude to the anonymous referees for carefully reading the manuscript and improving its presentation and accuracy. The first author is supported by the Natural Science Foundation of Beijing (No.1222012) and National Key Research and Development Project (No.2019YFB2006602).
The authors declare that they have no conflicts of interest.
[1] |
S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu, et al., An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics, 8 (2020), 1–22. https://doi.org/10.3390/math8040558 doi: 10.3390/math8040558
![]() |
[2] |
M. O. Oke, O. M. Ogunmiloro, C. T. Akinwumi, R. A. Raji, Mathematical modeling and stability analysis of a SIRV epidemic model with non-linear force of infection and treatment, Commun. Math. Appl., 10 (2019), 717–731. https://doi.org/10.26713/cma.v10i4.1172 doi: 10.26713/cma.v10i4.1172
![]() |
[3] |
W. F. Alfwzan, D. Baleanu, A. Raza, M. Rafiq, N. Ahmed, Dynamical analysis of a class of SEIR models through delayed strategies, AIP Adv., 13 (2023), 075115. https://doi.org/10.1063/5.0159942 doi: 10.1063/5.0159942
![]() |
[4] |
K. Umapathy, B. Palanivelu, R. Jayaraj, D. Baleanu, P. B. Dhandapani, On the decomposition and analysis of novel simultaneous SEIQR epidemic model, AIMS Math., 8 (2023), 5918–5933. https://doi.org/10.3934/math.2023298 doi: 10.3934/math.2023298
![]() |
[5] |
K. S. Mathur, P. Narayan, Dynamics of an SVEIRS epidemic model with vaccination and saturated incidence rate, Int. J. Appl. Comput. Math., 4 (2018), 118. https://doi.org/10.1007/s40819-018-0548-0 doi: 10.1007/s40819-018-0548-0
![]() |
[6] |
D. Kumar, J. Singh, M. Al Qurashi, D. Baleanu, A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Adv. Differ. Equ., 2019 (2019), 1–19. https://doi.org/10.1186/s13662-019-2199-9 doi: 10.1186/s13662-019-2199-9
![]() |
[7] |
J. Amador, The SEIQS stochastic epidemic model with external source of infection, Appl. Math. Model., 40 (2016), 8352–8365. https://doi.org/10.1016/j.apm.2016.04.023 doi: 10.1016/j.apm.2016.04.023
![]() |
[8] |
A. Atangana, D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., 143 (2017), D4016005. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001091 doi: 10.1061/(ASCE)EM.1943-7889.0001091
![]() |
[9] |
D. Baleanu, A. Mousalou, S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Bound. Value Probl., 2017 (2017), 1–9. https://doi.org/10.1186/s13661-017-0867-9 doi: 10.1186/s13661-017-0867-9
![]() |
[10] |
P. Kumar, S. Kumar, B. S. T. Alkahtani, S. S. Alzaid, The complex dynamical behaviour of fractal-fractional forestry biomass system, Appl. Math. Sci. Eng., 32 (2024), 2375542. https://doi.org/10.1080/27690911.2024.2375542 doi: 10.1080/27690911.2024.2375542
![]() |
[11] |
P. Kumar, S. Kumar, B. S. T. Alkahtani, S. S. Alzaid, A robust numerical study on modified Lumpy skin disease model, AIMS Math., 9 (2024), 22941–22985. https://doi.org/10.3934/math.20241116 doi: 10.3934/math.20241116
![]() |
[12] |
P. Kumar, A. Kumar, S. Kumar, A study on fractional order infectious chronic wasting disease model in deers, Arab J. Basic Appl. Sci., 30 (2023), 601–625. https://doi.org/10.1080/25765299.2023.2270229 doi: 10.1080/25765299.2023.2270229
![]() |
[13] |
A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Thermal Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
![]() |
[14] |
F. A. Rihan, Sensitivity analysis for dynamic systems with time-lags, J. Comput. Appl. Math., 151 (2003), 445–462. https://doi.org/10.1016/S0377-0427(02)00659-3 doi: 10.1016/S0377-0427(02)00659-3
![]() |
[15] |
I. A. Baba, F. A. Rihan, E. Hincal, A fractional order model that studies terrorism and corruption codynamics as epidemic disease, Chaos Solitons Fract., 169 (2023), 113292. https://doi.org/10.1016/j.chaos.2023.113292 doi: 10.1016/j.chaos.2023.113292
![]() |
[16] |
F. A. Rihan, U. Kandasamy, H. J. Alsakaji, N. Sottocornola, Dynamics of a fractional-order delayed model of COVID-19 with vaccination efficacy, Vaccines, 11 (2023), 1–26. https://doi.org/10.3390/vaccines11040758 doi: 10.3390/vaccines11040758
![]() |
[17] |
P. Kumar, A. Kumar, S. Kumar, D. Baleanu, A fractional order co-infection model between malaria and filariasis epidemic, Arab J. Basic Appl. Sci., 31 (2024), 132–153. https://doi.org/10.1080/25765299.2024.2314376 doi: 10.1080/25765299.2024.2314376
![]() |
[18] |
D. Baleanu, H. Mohammadi, S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative, Adv. Differ. Equ., 2020 (2020), 299. https://doi.org/10.1186/s13662-020-02762-2 doi: 10.1186/s13662-020-02762-2
![]() |
[19] |
S. Kumar, A. Kumar, M. Jleli, A numerical analysis for fractional model of the spread of pests in tea plants, Numer. Methods Partial Differ. Equ., 38 (2022), 540–565. https://doi.org/10.1002/num.22663 doi: 10.1002/num.22663
![]() |
[20] |
F. Evirgen, E. Uçar, S. Uçar, N. Özdemir, Modelling influenza a disease dynamics under {Caputo-Fabrizio} fractional derivative with distinct contact rates, Math. Model. Numer. Simul. Appl., 3 (2023), 58–73. https://doi.org/10.53391/mmnsa.1274004 doi: 10.53391/mmnsa.1274004
![]() |
[21] |
H. Joshi, M. Yavuz, Transition dynamics between a novel coinfection model of fractional-order for COVID-19 and tuberculosis via a treatment mechanism, Eur. Phys. J. Plus, 138 (2023), 468. https://doi.org/10.1140/epjp/s13360-023-04095-x doi: 10.1140/epjp/s13360-023-04095-x
![]() |
[22] |
F. Evirgen, E. Ucar, N. Özdemir, E. Altun, T. Abdeljawad, The impact of nonsingular memory on the mathematical model of Hepatitis C virus, Fractals, 31 (2023), 2340065. https://doi.org/10.1142/S0218348X23400650 doi: 10.1142/S0218348X23400650
![]() |
[23] |
M. ur Rahman, M. Arfan, D. Baleanu, Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions, Bull. Biomath., 1 (2023), 1–23. https://doi.org/10.59292/bulletinbiomath.2023001 doi: 10.59292/bulletinbiomath.2023001
![]() |
[24] |
H. Joshi, M. Yavuz, S. Townley, B. K. Jha, Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate, Phys. Scr., 98 (2023), 045216. https://doi.org/10.1088/1402-4896/acbe7a doi: 10.1088/1402-4896/acbe7a
![]() |
[25] |
B. Fatima, M. Yavuz, M. ur Rahman, F. S. Al-Duais, Modeling the epidemic trend of middle eastern respiratory syndrome coronavirus with optimal control, Math. Biosci. Eng., 20 (2023), 11847–11874. https://doi.org/10.3934/mbe.2023527 doi: 10.3934/mbe.2023527
![]() |
[26] |
M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models, Eur. Phys. J. Plus, 132 (2017), 1–16. https://doi.org/10.1140/epjp/i2017-11717-0 doi: 10.1140/epjp/i2017-11717-0
![]() |
[27] |
M. A. Khan, A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alex. Eng. J., 59 (2020), 2379–2389. https://doi.org/10.1016/j.aej.2020.02.033 doi: 10.1016/j.aej.2020.02.033
![]() |
[28] | S. M. Ulam, Problems in modern mathematics, Courier Corporation, 2004. |
[29] |
T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23–130. https://doi.org/10.1023/A:1006499223572 doi: 10.1023/A:1006499223572
![]() |
[30] |
S. Uçar, E. Uçar, N. Özdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solitons Fract., 118 (2019), 300–306. https://doi.org/10.1016/j.chaos.2018.12.003 doi: 10.1016/j.chaos.2018.12.003
![]() |
[31] | J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92. |
[32] |
A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fract., 89 (2016), 447–454. https://doi.org/10.1016/j.chaos.2016.02.012 doi: 10.1016/j.chaos.2016.02.012
![]() |
[33] |
S. Riaz, A. Ali, M. Munir, Sensitivity analysis of an infectious disease model under fuzzy impreciseness, Partial Differ. Equ. Appl. Math., 9 (2024), 100638. https://doi.org/10.1016/j.padiff.2024.100638 doi: 10.1016/j.padiff.2024.100638
![]() |
[34] |
I. Ahmed, A. Akgül, F. Jarad, P. Kumam, K. Nonlaopon, A Caputo-Fabrizio fractional-order cholera model and its sensitivity analysis, Math. Model. Numer. Simul. Appl., 3 (2023), 170–187. https://doi.org/10.53391/mmnsa.1293162 doi: 10.53391/mmnsa.1293162
![]() |
[35] |
N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272–1296. https://doi.org/10.1007/s11538-008-9299-0 doi: 10.1007/s11538-008-9299-0
![]() |
[36] |
D. Baleanu, H. Mohammadi, S. Rezapour, A mathematical theoretical study of a particular system of Caputo-Fabrizio fractional differential equations for the Rubella disease model, Adv. Differ. Equ., 2020 (2020), 1–19. https://doi.org/10.1186/s13662-020-02614-z doi: 10.1186/s13662-020-02614-z
![]() |
[37] |
A. Chavada, N. Pathak, Transmission dynamics of breast cancer through Caputo Fabrizio fractional derivative operator with real data, Math. Model. Control, 4 (2024), 119–132. https://doi.org/10.3934/mmc.2024011 doi: 10.3934/mmc.2024011
![]() |
[38] |
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
![]() |
[39] |
J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57 (1989), 268–273. https://doi.org/10.1016/0021-9045(89)90041-5 doi: 10.1016/0021-9045(89)90041-5
![]() |
[40] |
S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135–1140. https://doi.org/10.1016/j.aml.2003.11.004 doi: 10.1016/j.aml.2003.11.004
![]() |
[41] |
S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, II, Appl. Math. Lett., 19 (2006), 854–858. https://doi.org/10.1016/j.aml.2005.11.004 doi: 10.1016/j.aml.2005.11.004
![]() |
[42] |
S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, III, J. Math. Anal. Appl., 311 (2005), 139–146. https://doi.org/10.1016/j.jmaa.2005.02.025 doi: 10.1016/j.jmaa.2005.02.025
![]() |
[43] | M. N. Qarawani, On Hyers-Ulam stability for nonlinear differential equations of nth order, Int. J. Anal. Appl., 2 (2013), 71–78. |
[44] |
Y. J. Li, Y. Shen, Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett., 23 (2010), 306–309. https://doi.org/10.1016/j.aml.2009.09.020 doi: 10.1016/j.aml.2009.09.020
![]() |
[45] |
K. M. Owolabi, A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos Solitons Fract., 105 (2017), 111–119. https://doi.org/10.1016/j.chaos.2017.10.020 doi: 10.1016/j.chaos.2017.10.020
![]() |