Global dynamics of two-strain epidemic model with single-strain vaccination in complex networks.

Contagious pathogens, such as influenza and COVID-19, are known to be represented by multiple genetic strains. Different genetic strains may have different characteristics, such as spreading more easily, causing more severe diseases, or even evading the immune response of the host. These facts complicate our ability to combat these diseases. There are many ways to prevent the spread of infectious diseases, and vaccination is the most effective. Thus, studying the impact of vaccines on the dynamics of a multi-strain model is crucial. Moreover, the notion of complex networks is commonly used to describe the social contacts that should be of particular concern in epidemic dynamics. In this paper, we investigate a two-strain epidemic model using a single-strain vaccine in complex networks. We first derive two threshold quantities, R 1 and R 2 , for each strain. Then, by using the basic tools for stability analysis in dynamical systems (i.e., Lyapunov function method and LaSalle's invariance principle), we prove that if R 1 < 1 and R 2 < 1 , then the disease-free equilibrium is globally asymptotically stable in the two-strain model. This means that the disease will die out. Furthermore, the global stability of each strain dominance equilibrium is established by introducing further critical values. Under these stability conditions, we can determine which strain will survive. Particularly, we find that the two strains can coexist under certain condition; thus, a coexistence equilibrium exists. Moreover, as long as the equilibrium exists, it is globally stable. Numerical simulations are conducted to validate the theoretical results.

Introduction

It is well known that many infectious diseases are caused by multi-strain pathogens. For example, cholera is caused by an intestinal infection of the bacterium, Vibrio cholerae, in serogroups O1 and/or O139. Dengue fever is caused by the dengue virus, which has four distinct serotypes. Influenza is caused by four virus serotypes. Although COVID-19 is caused by the SARS-CoV-2 virus, several strains have now been discovered. The presence of multiple strains of a pathogen complicates our ability to combat these diseases because different genetic strains may have different characteristics. For example, it may spread more easily, may cause more severe disease, or may evade the immune response of the host. Vaccination is a commonly used control measure for reducing the spread of infectious diseases. Thus, investigating the impact of vaccines on the dynamics of multi-strain diseases is crucial.

In recent years, multi-strain models have been intensively used to study epidemic dynamics. In [1], the authors proposed a two-strain model with a single-strain vaccine to study flu epidemics. The global dynamics of the model were completely determined by using the Lyapunov function method. Later, a two-strain model with vaccine for each strain was investigated in [2]. Stability analysis of the equilibrium solutions was carried out by constructing suitable Lyapunov functions. In [3], a novel multi-strain susceptible–infected–recovered model for emerging viral strains was studied. It was shown that the original and the emergent strain can coexist in an endemic equilibrium if the system is initially in endemic equilibrium for the original strain. In [4], a two-strain model with different susceptibility and strain mutation was formulated and analyzed. The stability of disease-free equilibrium as well as strain dominance equilibria were investigated. In [5], an influenza model with a virus mutation was studied. The authors proved the global stability of disease-free and mutant-dominance equilibria. Furthermore, it was also shown that the influenza is permanent if and only if endemic equilibrium exists. In [6], the impacts of vaccinations and mutations on the dynamics of a two-strain model were analyzed. The existence and stability conditions of equilibria were examined. Additionally, the existence of Hopf bifurcation from the endemic equilibrium was discussed. The influence of cross-immunity, quarantine, and isolation on the epidemic dynamics were studied in [7], [8], and existence conditions of periodic solutions of two-strain influenza model were established therein.

Some studies on multi-strain modeling have focused on the influence of age structures. The ability of vaccinations to generate subthreshold persistence of a disease was analyzed in [9]. An age-structured two-strain model with super-infection was considered in [10]. Stability conditions of strain exclusive equilibria and the existence condition of coexistence equilibria were obtained. Thus far, we can observe that most existing studies were primarily concerned about the stability of equilibria. Identifying the stability of each equilibrium may help us determine the long-term behaviors of disease dynamics. Moreover, the aforementioned studies mainly made homogeneous mixing assumptions in which each individual in the population was assumed to have the same probability of contact with an infected individual. In reality, however, contacts among individuals in a population are heterogeneous. Therefore, the notion of complex networks needs to be incorporated into epidemic modeling. Many network-based single-strain epidemic models have been proposed [11], [12], [13], [14], [15], [16], [17], and they have been heavily cited in other works. However, there are relatively few studies about network-based multi-strain epidemic models. In [18], the authors studied a two-strain model for complex networks with saturated infectivity and mutations. A study of the superinfection process in a networked two-strain epidemic model was presented in [19]. In [20], the authors presented a stability analysis for a networked two-strain model. The latency dynamics of a two-strain epidemic model with a saturation infection force was investigated in [21]. The effect of demographics on the dynamics of a two-strain model was considered in [22].

To the best of our knowledge, no complex-network research has been conducted considering the impact of vaccines on the dynamics of multi-strain epidemic models. To address this deficit, we present a study on the two-strain epidemic model with a single-strain vaccine in complex networks. Existing conditions for strain dominance equilibria and the coexistence equilibrium are obtained. Moreover, the global stability for each equilibrium is determined by using the Lyapunov function method. Numerical simulations are conducted to illustrate our findings.

The remainder of this paper is organized as follows. In Section 2, we introduce the proposed two-strain model with a single-strain vaccine in complex networks and present the existence analysis of the equilibrium solutions. In Section 3, we study the dynamical behaviors of single-strain models. In Section 4, we study the global dynamics of the two-strain model. In Section 5, we describe the numerical simulations to validate the theoretical results. Finally, conclusions and discussions are presented in Section 6.

Model formulation and equilibrium solutions

Model formulation

In [1], a two-strain model was described using the following system of ordinary differential equations: where the state variable, , denotes the number of susceptible individuals, is the number of the strain-1 infected individuals, is the number of individuals who receive the vaccine for strain-1, denotes the number of strain-2 infected individuals, and is the number of recovered individuals. The vaccine is assumed to be effective only for strain-1; hence, vaccinated individuals could be infected by strain-2. The parameters for system (1) show that is the recruitment rate. is the natural death rate, is the vaccination rate, and and are the transmission rates of susceptible individuals to strains 1 and 2, respectively. is the transmission rate of vaccinated individuals to strain-2, and are the recovery rates for strains 1 and 2, respectively, and and are the infection-induced death rates of strains 1 and 2, respectively. In system (1), we set , , and . The transmission diagram for system (1) is described in Fig. 1 (left).

Fig. 1

The transmission diagrams for system (1) (left) and for system (2) (right).

To consider the effect of contact heterogeneity on the epidemic dynamics, we form the entire population as a social network, and a node of the network corresponds to an individual. Besides, an edge connecting two nodes represents the potential contact between two individuals, and the number of edges of a node is the degree of that node. All nodes of the network are assigned to groups according to their degree. More specifically, the nodes in the th group have degree for . Let be the total number of individuals who have degree , , and is the maximal degree. Then the total population size is . To incorporate the nature of contact heterogeneity, we consider a network characterized by the degree distribution , which is defined as the probability that a randomly selected node has degree , i.e., . Motivated by system (1), we consider the following networked two-strain model consisting of subsystems for : where represents the relative density of susceptible nodes with degree , and denotes the relative density of the vaccinated nodes for strain-1 with degree . and are the relative densities of infected nodes for strains 1 and 2 with degree , respectively. is the relative density of the recovered nodes with degree . The transmission diagram for system (2) is described in Fig. 1 (right). Parameters in system (2) have the same meanings as those given in system (1). Nevertheless, it should be noted that the quantities in systems (1), (2) have slightly different units, especially , , , and in system (2) are dimensionless (see, Table 1).

Table 1

Units of quantities in systems (1) and (2).

Quantities of system (1)	Unit	Quantities of system (2)	Unit
S, V_1, I_1, I_2, R	number of individuals	S_k, V_k, I_k, J_k, R_k	dimensionless
Λ	number of individuals/time	Λ	1/time
β_1, β_2, σ	1/(time×number of individuals)	β_1, β_2, σ	1/(time×number of contacts)
r, μ, γ_1, γ_2, ν_1, ν_2	1/time	r, μ, γ_1, γ_2, ν_1, ν_2	1/time

The dynamics of all subsystems in system (2) are coupled via the functions and , which are defined as where denotes the conditional probability that a node with degree is connected to a node with degree . For a degree uncorrelated network, the probability that a link connects to a node of degree is proportional to , and hence, , where is the mean degree of the network [16]. In this study, we study the epidemic dynamics in uncorrelated networks such that the terms, and , can be formed as

In the following lemma, we study the evolution of and . The results can be obtained directly from the - and -equations of system (2) and the expression (3).

Let be a solution to system (2) . Then we have

Now, we can show that a solution with a non-negative initial value remains non-negative on the existing interval.

Let be a solution to system (2) with initial conditions Then for , we have

We first claim that and for all . Indeed, if and for all , then and . It follows from Lemma 2.1 that Thus, we obtain and for all . Given that . Using the first equation of system (2) and the continuity of , we can find a small such that for . Now we prove that for all . Suppose not. Then there exists such that and for . At , we have . This implies for some , which is apparently a contradiction. Thus, for all . Then we show that for all . Applying the method of integrating method on the second equation of system (2), we obtain where is an integrating factor. Thus, we conclude that for all . Similarly, using the nonnegativity of , , , and , we can show that and for all . Subsequently, for all can be made, and this completes the proof. □

Let ; we can derive that , which implies that . Consequently, the solution is eventually uniformly bounded, and then the solution globally exists, denoted by the abbreviation . Hence, throughout this paper, we focus on the dynamics of the solutions of system (2) in the following bounded region: One can easily verify that the set is positively invariant for system (2). In the following, we assume the condition , which ensures that for all . We notice that does not participate in the first four equations in system (2) for each . Thus, we consider the following system: When the convergence dynamics of (6) is achieved, the convergence dynamics of can be demonstrated by applying the theory of asymptotically autonomous systems in [23]. Hence, in this paper, we focus on studying system (6).

Existence of equilibrium solutions

First, we can determine the boundary equilibria of system (6). To find the equilibrium of system (6), we set the derivatives to zero; subsequently, the following relations can be made: and Substituting relation (8) into expression (3), we obtain Define Then, solving system (9) is equivalent to solving It should be noted that after we have determined and from system (11), we can find an associated equilibrium from relations (7), (8).

Obviously, is always a solution to system (11). It follows from relation (8) that , which implies that there is a disease-free equilibrium, for system (6). In addition to the disease-free equilibrium, there are two possible single-strain dominance equilibria. Now, we find the strain-1 dominance equilibrium, , which corresponds to a solution pair, , of system (9) with and . To obtain this solution pair, we need to obtain a positive solution of . From system (10), we have Given that and we can conclude that there is a unique such that if which is equivalent to where . By substituting and into relations (7), (8), the equilibrium, , can be obtained.

Using a similar argument to the one above, we can obtain the strain-2 dominance equilibrium, , which corresponds to a solution pair, , of system (11) with and . To see this, we must find a positive solution for . From system (10), it follows that Given that and we can obtain a unique such that if which is equivalent to After substituting and into relations (7), (8), the equilibrium, , can be obtained. To summarize, we have the following theorem:

For system (6) ,

there is always a disease-free equilibrium, ;

there exists a unique strain-1 dominance equilibrium, , if and only if ;

there exists a unique strain-2 dominance equilibrium, , if and only if .

To explain the biological meaning of , notice that strain-1-infected individuals can only infect susceptible individuals, and is the relative density for the population that consists of only susceptible individuals in the long run without infection, is the average time spent by a strain-1-infected individual, is the transmission rate of susceptible individuals to strain 1, and is the effect from network structure. Thus, represents the basic reproduction number for strain 1 in the absence of strain 2. To explain the meaning of , it should be noted that not only the susceptible individuals, but also vaccinated ones could be infected from strain-2-infected individuals with transmission rate , is the relative density of the vaccinated individuals in the long run without infection, is the average time spent by a strain-2-infected individual, is the transmission rate of susceptible individuals to strain 2. Thus, represents the basic reproduction number for strain 2 in the absence of strain 1, where the first term denotes the new cases from susceptible individuals, while the second term denotes the new cases from vaccinated individuals.

In addition to the above boundary equilibria, the following theorem shows that system (6) can have a coexistence equilibrium: , which corresponds to the solution pair, , of system (11) with and . Furthermore, we define It should be noted that if and if . The following lemma shows the relation between and . More precisely, we show that and cannot occur simultaneously.

Assume that and .

If , then .

If , then .

Given that , we have Thus, if we can deduce that for all . Consequently, we have According to , we have Besides, we can check that which implies that Therefore, the first assertion holds. Similarly, if then we can conclude that for all . Hence, and the second assertion holds. This completes the proof. □

Next, we can state the existence result for the coexistence equilibrium, .

If and , then there exists a coexistence equilibrium, , for system (6) .

To find the coexistence equilibrium, , we must find a positive solution pair, , for system (11). This solution pair, if exists, satisfies Given and , it follows from Theorem 2.1 that there exist and such that and . First, we focus on the curve defined by . We now show that this curve is a straight line. To see this, we compute the slope, , by implicitly differentiating the equation, , with respect to . Thus, It follows from Eq. (13) that , which yields . This means that the slope is always a constant; hence, the curve defined by is indeed a straight line. Because , we conclude that the equation of this line is . Now, we see that there exists a function, , such that . Define . Then, is a continuous function of , and . Owing to , we have Now, we consider the equation, . Clearly, we can obtain . Define It can be claimed that is a continuous function of , , and Because we conclude that if and only if . Consequently, we derive that Owing to the identity we obtain Combining the Intermediate Value Theorem with the relations (14), (15), we see that there exists such that , whenever and , or and . However, from Lemma 2.3, the latter condition does not hold. Hence, exists under and , and accordingly we obtain and . Hence, we find a solution pair that solves system (12). This completes the proof. □

Instability of the boundary equilibria

In this subsection, we will study the instability of the boundary equilibria, namely, , , and . By linearizing system (6) around each boundary equilibrium, we can deduce the instability condition of each boundary equilibrium. To see this, we consider the following system by rearranging the equations in system (6): where and . with , , , , , , , and for . The Jacobian matrix for system (16) has the following block form: where represents the zero matrix, and the entries for the matrices, and , are given by

The following theorems present the instability of boundary equilibria.

If or , then is unstable.

The Jacobian matrix (17) evaluated at is given by Thus, the eigenvalues of are determined by four submatrices: , , , and . By direct computation, we can verify that is an eigenvalue of , and is an eigenvalue of . Hence, if or , then is unstable because the Jacobian matrix, , has a positive eigenvalue. This completes the proof. □

If , then is unstable.

To see this, we compute the Jacobian matrix (17) at : By direct computation, we can check that is an eigenvalue of the submatrix, , and it is also and eigenvalue of the Jacobian matrix, . Hence, we can conclude that is unstable if . This completes the proof. □

If , then is unstable.

To demonstrate this, we calculate the Jacobian matrix (17) at : By direct verification, it can be shown that is an eigenvalue of the submatrix, , which is also an eigenvalue of the Jacobian matrix, . Hence, if , then is unstable because has a positive eigenvalue. This completes the proof. □

Dynamics of single-strain models

In this section, we will study the dynamics of single-strain models. That is, we consider only the strain-1 in system (6), called strain- system, and only the strain-2 in system (6), called strain- system, respectively. As aforementioned in Theorem 2.1, we can see that system (18) always admits the disease-free equilibrium , and the endemic equilibrium exists if and only if . Similarly, system (19) always admits the disease-free equilibrium , and the endemic equilibrium exists if and only if . Furthermore, We focus on the dynamics of solutions of system (18) and solutions of system (19) in the following bounded regions and respectively. The following lemma shows positivity of solutions of systems (18), (19), respectively. We will postpone all proofs of this and further results in this section to Appendix.

Consider system (18) in (resp., system (19) in ), (i) there exist positive constants and such that and ; (ii) if there exist and such that (resp., ), then (resp., ) for , .

We demonstrate the convergence dynamics to the disease-free equilibrium in systems (18), (19), respectively.

If , then the equilibrium attracts all solutions of system (18) in .

If , then the equilibrium attracts all solutions of system (19) in .

Next, we show that the disease uniformly persists in systems (18), (19) when and , respectively. It reveals not only the existence of disease, but also provides us a priority to construct Lyapunov functions used to prove the global convergence to equilibrium and equilibrium , respectively.

If (resp., ), then system (18) (resp., system (19) ), is uniformly persistent, that is, there exists a (resp., ) such that

We end up this section by exploring the convergence dynamics to the endemic equilibrium in system (18), (19) respectively.

If , the equilibrium of system (18) attracts all solutions in .

If , the equilibrium of system (19) attracts all solutions in .

Dynamics of the two-strain model

Persistence of system (6)

This subsection is devoted to show the persistence of strain-1 and 2 disease respectively. First, applying the method in Lemma 3.1, we derive the positivity of and when infection is initiated in one node. We omit the proof to save space.

Consider system (6) in , (i) there exist positive constants and such that and ; (ii) if there exist and such that (resp., ), then (resp., ) for , .

Suppose (resp., ), then strain-1 (resp., strain-2) in system (6) is uniformly persistent, that is, there exists a (resp., ) such that if , or and (resp., , or and ).

We demonstrate the case for strain-1, and that for strain-2 can be shown similarly. Define It is easy to see that both and are positively invariant under the solution flow of system (6), and is relatively closed in . From the boundedness of the solutions, there exists a global attractor in . We further define In fact, we have . Obviously, . Suppose there exists an initial value , then there is some such that , and by Lemma 4.1 each for , and then . Therefore, .

To apply the persistence theory in [24], [25], we divide the supposed parameter range into two cases according to the existence of equilibria in the space .

(i) When and , there exist equilibria . Since (which implies ), there exist constants and such that

The set (resp., ) is a uniformly weak repeller for in the sense that there exist constants (resp., ) such that , (resp., ).

To show the first assertion, we assume by the contrary that there exists such that , . From -equation, we have for . Consider the following auxiliary equation Denoting , it holds that , where Similar to the matrix in Theorem 2.5, one can see that is an eigenvalue of the matrix . It leads to . By the comparison principle, it also concludes that , which is a contradiction, and then the assertion holds true.

To show the second assertion, we again assume by the contrary, there exists such that and , . From -equation, we have for . Consider the following auxiliary equation Denoting , it holds that , where Similar to the matrix in Theorem 2.5, one can see that is an eigenvalue of the matrix . It leads to . By the comparison principle, it also concludes that , which is a contradiction, and then the assertion holds true.

From Theorem 3.5, we see that is globally asymptotically stable in , and then the set is an acyclic invariant set in under the solution flow of system (6). In addition, the above claim indicates that . By the persistence theory in [24], [25], it concludes that system (6) is uniformly persistent with respect to , that is, the strain-1, , uniformly persists in system (6).

(ii) When , there exists unique equilibrium . As in previous case, we can further verify the following result by using the assumption .

The set is a uniformly weak repeller for in the sense that there exist such that

From Theorem 3.2, we see that is globally asymptotically stable in , and then the set is an isolated invariant set and acyclic in under the solution flow of system (6). In addition, the above claim indicates that . Again, by the persistence theory in [24], [25], it concludes that system (6) is uniformly persistent with respect to , that is, the strain-1, , uniformly persists in system (6). This completes the proof. □

Global stability of equilibria

In this subsection, we study the global stability of the equilibrium of system (6). Using the Lyapunov function method, the stability result for each equilibrium is established under appropriate conditions. For later use, let . Then, we can claim that for all and . Moreover, achieves a global minimum at so that . The following theorem shows the global stability of the disease-free equilibrium, .

If , then the disease-free equilibrium, , is globally asymptotically stable.

Consider the Lyapunov function, , which is defined along a given solution of system (6): Then, the derivative of with respect to is given by Applying system (6) and identities (4)–(5) into Eq. (20), we obtain: From the identities, and , we deduce that: Combining with the relation of arithmetic and geometric means, we conclude that with equality holds only at the equilibrium, . Hence, by LaSalle’s well-known invariance principle [26], the disease-free equilibrium, , is globally asymptotically stable. This completes the proof. □

Next, we prove the global stability of two single-strain dominance equilibria.

If one of the following conditions holds:

and ,

and ,

then the strain-1 dominance equilibrium, , is globally asymptotically stable.

In either case, we know from Theorem 2.1 that exists, and from Lemma 4.1 and Theorem 4.1 that it suffices to explore the dynamics of system (6) in . To demonstrate the global stability of , we consider the Lyapunov function as follows: Note that is continuous and bounded along a solution in . Then, the derivative of with respect to is given by Using systems (4), (5), and (6), we deduce that: Taking advantage of the equilibrium equations of , we have . This implies that and Therefore, we obtain Substituting , , and into the above equation, we derive that Note that it satisfies in both conditions (i) and (ii). Hence, by employing the relation of arithmetic and geometric means to the above equation, we can conclude that , where equality holds only at the equilibrium, . Again, according to LaSalle’s invariance principle, the strain-1 dominance equilibrium, , is globally asymptotically stable. This completes the proof. □

If one of the following conditions holds:

and ,

and ,

then the strain-2 dominance equilibrium, , is globally asymptotically stable.

In either case, we know from Theorem 2.1 that exists, and from Lemma 4.1 and Theorem 4.1 that it suffices to explore the dynamics of system (6) in . We define the Lyapunov function: Note that is continuous and bounded along a solution in . Then, the derivative of with respect to is given by Applying system (6) and identities (4)–(5) to the above equation, we obtain From the equilibrium equations of , we have . This yields Thus, we get Substituting , , and into the above equation, we derive that Note that it satisfies in both conditions (i) and (ii). By further employing the relation between arithmetic and geometric means, we can conclude that , where equality holds only at the equilibrium, . It follows from LaSalle’s invariance that the strain-2 dominance equilibrium, , is globally asymptotically stable. This completes the proof. □

Uniform persistence of both strains is a necessary condition for the coexistence equilibrium, , to be globally asymptotically stable. From Theorem 4.1, both strains uniformly persist in system (6) only when and . The following theorem shows the convergence dynamics to under this assumption.

The coexistence equilibrium, , is globally asymptotically stable, if and .

Since and , we see from Theorem 2.2 that exists, and from Theorem 4.1 that both strains uniformly persist in system (6). Hence, it suffices to explore the dynamics of system (6) in .

Consider the following Lyapunov function: Note that is continuous and bounded along a solution in . Differentiating along the trajectories of system (6), we obtain Applying the equilibrium equations for to the above equation, we have From the equilibrium equations, and . Hence, we derive that and Thus, we have Combining the equilibrium equation, , with the identity, , the following equality can be obtained: Because the arithmetic mean is larger than the geometric mean, we can conclude that with equality holding only at the equilibrium, . According to LaSalle’s invariance principle, the coexistence equilibrium, , is globally asymptotically stable. This completes the proof. □

Numerical simulations

In this section, we present a number of numerical simulations to validate our theoretical results. In Example 5.1, Example 5.2, Example 5.3, Example 5.4, we consider scale-free networks whose degree distribution follows a power law, i.e., . The constant indicates the power-law exponent whose value is typically in the range . We select a degree distribution of for , and a constant is given to maintain . Consequently, we can obtain and . In Example 5.5, we shall vary the value of to explore its effect on the dynamics, as well as consider another network with Poisson degree distribution to enrich our observation.

We first consider the stability of the disease-free equilibrium, . In this example, we choose , , , , , , , , and . Then, we can obtain and . This implies that . It follows from Theorem 4.2 that the disease-free equilibrium, , is globally asymptotically stable, which means that the disease will die out. We use 20 different initial conditions to plot the trajectories of in the -plane (see Fig. 2). One can observe that all trajectories approach . For the same set of initial conditions, the solution curves of , , , and for and with respect to time are shown in Fig. 3. It can be seen that , , , and for a sufficiently large . This supports the stability of the disease-free equilibrium, .

Fig. 2

Trajectories of in the -plane with 20 different initial conditions.

Fig. 3

Time evolutions of , , , and for and in Example 5.1.

In the second example, we choose , , , , , , , , and . Thus, one can obtain that and . Thus, from Theorem 2.1, there exist a unique strain-1 dominance equilibrium, , and a unique strain-2 dominance equilibrium, . Besides, for this set of parameters, we can compute and . Consequently, we can obtain and . According to Theorem 4.3(ii), we can conclude that the strain-1 dominance equilibrium, , is globally asymptotically stable. The global stability of reveals that the disease of strain-1 will converge to a positive stationary state, but the disease of strain-2 will eventually become extinct. The trajectories of in the -plane with 20 different initial conditions are shown in Fig. 4. As can be seen, all trajectories converge to . For the same set of initial conditions, the solution curves of , , , and for and with respect to time are shown in Fig. 5. We can observe that and for sufficiently large . This illustrates the stability of the strain-1 dominance equilibrium, .

Fig. 4

Trajectories of in the -plane with 20 different initial conditions.

Fig. 5

Time evolutions of , , , and for and in Example 5.2.

In the third example, we take , , , , , , , , and . Subsequently, we have and . Thus, there exists a unique strain-2 dominance equilibrium, . Furthermore, from Theorem 4.4(i), the global stability of can be established. The stability result of indicates that the disease of strain-2 will tend to a positive stationary state, but the disease of strain-1 will become extinct ultimately. For this set of parameters, we can obtain . The trajectories of in the -plane with 20 different initial conditions are plotted in Fig. 6 from which we can see that all trajectories converge to . For the same set of initial conditions, the solution curves of , , , and for and with respect to time , are shown in Fig. 7. We can observe that and for sufficiently large . This verifies the stability of the strain-2 dominance equilibrium, .

Fig. 6

Trajectories of in the -plane with 20 different initial conditions.

Fig. 7

Time evolutions of , , , and for and in Example 5.3.

In this example, the parameters are selected as , , , , , , , , and . Then, we can and . For this information, we can compute that and . Subsequently, we have and . Therefore, it follows from Theorem 2.2 that there exists a coexistence equilibrium, . According to Theorem 4.5, we can see that is globally asymptotically stable. In other words, the two strains will persist and approach a stationary state. The trajectories of in the -plane with 20 different initial conditions are shown in Fig. 8. It can be observed that all trajectories converge to the interior equilibrium. We also plot the solution curves of , , , and for and with respect to time , as shown in Fig. 9. A stable phenomenon can be observed therein.

Fig. 8

Trajectories of in the -plane with 20 different initial conditions.

Fig. 9

Time evolutions of , , , and for and in Example 5.4.

We summarize Example 5.1, Example 5.2, Example 5.3, Example 5.4 with depicting regions in the -plane to determine the existence and stability of equilibria in Fig. 10, where parameters are chosen as , , , , , , and . Regions therein satisfy : , : and , : and , and we further divide the top-right region ( and ) into : and , : and , : and .

Fig. 10

Diagram of regions in the plane to determine the existence of equilibria in system (6). Regions satisfy (i) : , : , : ; (ii) the top-right region () satisfies : , : , : .

In the final example, we consider the impact of the network structure on the dynamics. The following parameters are fixed: , , , , , , , , and . Then we will consider two different types of networks to explore the dynamical behaviors. The first one is the scale-free network that we have chosen in the previous examples. Here, the degree distribution is given by for . The time evolutions of and with different values of power-law exponent are given in Fig. 11. We can observe that when the value of is small, i.e., , and 2.5, both and tends to positive values as . This indicates that the two strains will coexist. On the contrary, the disease will die out when the value of becomes large. The second type of network we selected is the Poisson network with degree distribution for , where indicates the expected value of . It can be observed in Fig. 12 that if is large, i.e., , and , then coexistence occurs. Besides, it is interesting to note that strain 2 will dominate when . In summary, the network structure does affect the spreading dynamics.

Fig. 11

Time evolutions of and in scale-free networks with degree distribution for , and .

Fig. 12

Time evolutions of and in Poisson networks with degree distribution for , and .

Conclusions and discussions

In this paper, we proposed and analyzed a two-strain epidemic model with a single-strain vaccine using complex networks. The vaccine was assumed to be effective only for one strain. Hence, vaccinated individuals could be infected by another strain. The existence criteria for strain-dominance equilibria and a coexistence equilibrium were derived. Furthermore, with the help of the well-known Lyapunov function method and LaSalle’s invariance principle, we derived the sufficient conditions that ensure the global stability of each equilibrium. More precisely, we obtained four threshold quantities, , , , and . We have proven the following results (see summary in Table 2):

Table 2

Existence and stability of equilibria for system (4).

Equilibrium	Existence	Condition for GAS
E_0=(Sk0,Vk0,Ik0,Jk0)	always exists	R_0≔max{R_1,R_2}<1
E_1=(S¯_k,V¯_k,I¯_k,0)	R_1>1	(i) R_1>1 and R_2<1, or (ii) R¯_1>1 and R¯_2<1<R_2
E_2=(Sˆ_k,Vˆ_k,0,Jˆ_k)	R_2>1	(i) R_2>1 and R_1<1, or (ii) R¯_2>1 and R¯_1<1<R_1
E^∗	R¯_1>1 and R¯_2>1	R¯_1>1 and R¯_2>1

If , then the disease-free equilibrium, , is globally asymptotically stable. That is, the disease will die out.

If , then there exists a unique strain-1 dominance equilibrium, , which is globally asymptotically stable provided that either (i) and , or (ii) and , see Theorem 4.3. Biological meaning of the stability of is that the disease of strain-1 will tend toward a positive stationary state while the disease of strain-2 will become extinct. The strain-1 has this advantage due to two mechanisms: the first one is since the strain-2 intrinsically goes extinction (, and see Theorem 3.2), and the second one is its strong infectivity, compared to the strain-2 ( and ).

If , then there exists a unique strain-2 dominance equilibrium, , which is globally asymptotically stable provided that either (i) and , or (ii) and , see Theorem 4.4. Biological meaning of the stability of is that the disease of strain-2 will tend toward a positive stationary state while the disease of strain-1 will become extinct. The strain-2 has this advantage due to two mechanisms: the first one is since the strain-1 intrinsically goes extinction (, and see Theorem 3.3), and the second one is its strong infectivity, compared to the strain-1 ( and ).

Let and . There exists a coexistence equilibrium, , which is globally asymptotically stable. In this condition, both strains will persist and approach a positive stationary state.

To discuss the impact of vaccination on disease dynamics, let us consider and as functions of the vaccination rate . We see that is decreasing in . Thus, the vaccination is always beneficial for controlling strain-1. However, given we derive that Therefore, if , the vaccination plays a positive role, whereas if , it has a negative impact in controlling strain-2.

In this work, we only considered a single-strain vaccine that was completely effective. However, there could be more than one vaccine for a disease represented by multiple strains. Additionally, vaccine efficacy may be lower than 100%. For example, several COVID-19 vaccines are available globally. However, it is well-known that some of the vaccines such as the Moderna and Pfizer vaccines, reported about efficacy. Thus, studying the impacts of imperfect vaccines on the spreading dynamics of multi-strain epidemic models in complex networks is necessary as future work.