Lyapunov Functionals for Virus-Immune Models with Infinite Delay — One-Strain and Multistrain Models

We have studied the asymptotical behavior of the solutions of the delay differential equation models. To analyze the global stability of the equilibria of those models, Lyapunov functions or Lyapunov functionals are useful and effective. In many papers various techniques to construct Lyapunov functions or functionals and various techniques to prove the nonpositivity of their time derivative have been introduced. For delay differential equations, McCluskey showed the useful integration term. To prove the nonpositivity of the time derivative of the Lyapunov functionals the extension of arithmetic-geometric mean inequality is effective. By these methods the analysis of the global stability of equilibria has progressed. However the general method of the construction of the Lyapunov functions or functionals are unknown. In this thesis we present a systematic method to construct Lyapunov functionals of several delay differential models of infection disease in vivo. We start with the simple model without delay and finally we consider the complexed multistrain model. The Lyapunov functionals are constructed systematically. In the analysis of the stability of the equilibria, there exist some mathematical difficulties. We have made rigorous solutions for those difficulties. The thesis is organized as follows. At first we introduce the simple model and describe the notations and definitions. Secondary we analyze a single strain models and prepare the analysis of successive multistrain models. Finally we analyze some multistrain models such as the model with absorption effect and the model with immune variables. The global stability of the equilibria with the competitive exclusion principle and the coexistence of strains are explained rigorously mathematical method.


Introduction
There often exist multiple strains of pathogens such as HIV, hepatitis C, Epstein-Barr virus, dengue fever, tuberculosis and malaria.For many diseases, the effect of multiplestrain infections remains still unclear, but theoretical works and experimental results from animal models are clarifying it gradually.
Mathematical modeling and analysis of virus dynamics have been subjects of extensive research.It is needless to say that the stability of equilibria is important.For the analysis of the stability, Lyapunov functions and Lyapunov functionals are useful methods [11].But the main difficulty we face is the lack of a general method of constructing a Lyapunov function or a Lyapunov functional.For delay differential equations (DDE), McCluskey [12] showed the useful technique using the integration term to construct Lyapunov functionals for DDE.But in many papers after this the calculations for the Lyapunov functionals are often complicated.
In this paper we propose a systematic method with small calculation to construct Lyapunov functionals.At first we start with a simple model and a simple Lyapunov function.Adding several appropriate terms to the known Lyapunov function or functional we can construct a new Lyapunov functional for the complicated model.Also we can construct Lyapunov functionals for the multistrain models by combining the Lyapunov functionals of the single strain models.Those methods in constructing Lyapunov functionals are simple.In the case of multistrain models, the competitive exclusion principle is found for the model with absorption effect without immune variables, and the condition for coexistence of multiple strains is shown for the model with absorption effect and immune variables.
Some recent papers [9], [10] referred to the construction of the Lyapunov functions or functionals.And the infinitely distributed delay was taken into account in several papers [12], [17], [19].We try to make an accurate description on the mathematical aspect.We describe the definition of the fading memory space, the asymptotic smoothness of the system, the uniform persistence of the virus under R 0 > 1, the well-definedness of the Lyapunov functionals and so on.
The contents of this paper is as follows.In section 3, we consider a model with infinitely distributed delay with absorption effect without an immune variable.In section 4, we consider models with an immune variable.For the model with absorption effect without an immune variable, we describe the detail of the mathematical argument.In the model with immune variable, we describe only outline for several repeated mathematical argument.

Preliminary model
At first we start with a fundamental one-strain model of infection [15] that contains the uninfected cells x, infected cells y and viruses v such as (2.1) We introduce a generalized simple model, which includes (2.1) as a special case.By using the exponential delay kernel we can reconstruct the model which has three variables from the model with two variables x and v.For example we define the function y(t) as follows : where g(s) = exp{−as}/a.Then the time derivative of y(t) becomes as follows : and the third equation of (2.1) can be rewritten as follows : Generally the model with the distributional delay should be considered.It includes the case of finite delay or discrete delay as a particular case of it.In this section we introduce some simple models.
Recently, the construction of the Lyapunov functional of the age-structured model is considered concerned with the delay differential equation model.
The function x(t) denotes the population of uninfected cells, v(t) the population of viruses and y(a, t) the age-specific concentration of infected cells with infection age a and time t.The parameter λ denotes the recruitment rate, δ and b the natural death rate of uninfected cells and viruses respectively, β the contact rate, η(a) the death rate of infected cells with infection age a, k(a) the virus production rate of infected cells at age a. Then  (2. 2) The second equation can be solved by the integral along the characteristic curve as follows : y(a, t) = βσ and The asymptotic behavior of this model is the same as the following model [13] by lim t→∞ F (t) = 0 :

Model with distributed delay with absorption effect without an immune variable
When pathogens infect uninfected cells, the number of pathogens decreases.It is called the effect of absorption.Iggidr et al. [7] described the global stability of the model with absorption effect.We add the effect of absorption expressed by ρ, which is less than r, and consider the nonlinear incidence µ(x) where the function µ(x) satisfies µ(0) = 0, µ(x) is strictly increasing and µ(x)/x is monotone nonincreasing for x > 0 as in [10].
Then we have (3.1) The parameter λ denotes the recruitment rate of uninfected cells, δ the natural death rate for uninfected cells, β the contact rate between uninfected cells and viruses, b the natural death rate and k(a) the viral production rate of an infected cell with infection-age a and the function σ(a) is the probability that an infected cell survives to infection-age a.It is determined by the death rate η(a) of infected cells as The total number of viral particles produced by an infected cell in its life span is called the burst size r and it is defined by where 0 ≤ k(a) ≤ K for some K and σ(a) is integrable.The delay kernel is defined by g(a) = k(a)σ(a)/r, and the nonnegative function g(a) satisfies ∫ ∞ 0 g(a) da = 1.The model is rewritten to the differential equations with infinite delay as follows with the following initial condition Due to the infinite delay, an appropriate phase space is required.We use the phase space of fading memory type [1,5].For ∆ > 0 , let is bounded and uniformly continuous}, (3.6)We define ∆ 0 = lim inf a→∞ ∫ a 0 η(s)ds/(2a) .It is positive because we can assume, in a biological point of view, the death rate η(s) of infection-age s does not decrease to zero.For any ∆ ∈ (0, ∆ 0 ) we can take ∆ 1 which satisfies ∆ < ∆ 1 < ∆ 0 and there exists a sufficiently large A such that for all a ≥ A it holds that 1 2a (3.9) Then the following inequality holds : Thus if the initial functions ϕ 0 and ϕ 1 belong to the phase space C ∆ then the integral of the right hand of the second equation of (3.4) converges.We consider the solution of system (3.4),(x t , v t ) with the initial condition

Positivity and boundedness
The positivity and boundedness of the solution are shown as follows.
Proposition 3.1.Let x(t) and v(t) be the solution of system (3.4) with the initial condition (3.11), then x(t) and v(t) are positive for t > 0.
Proof.Suppose that t 1 is the least positive time such that x(t 1 ) = 0, then x ′ (t 1 ) = λ > 0. Therefore there exists small ϵ > 0 such that x(t) < 0 for t = t 1 − ϵ > 0. It is a contradiction and it follows that x(t) > 0 for all t > 0. Similarly suppose that t 1 is the least positive time such that v(t 1 ) = 0, then v(t) > 0 for 0 ≤ t < t 1 and Then by the positivity of µ(x(t 1 − a))v(t 1 − a) for t 1 − a ∈ (0, t 1 ] , the integral of (3.12) is positive for t = t 1 .Therefore there exists a small ϵ > 0 such that v(t) < 0 for t = t 1 − ϵ > 0. It is a contradiction and it follows that v(t) > 0 for all t > 0.
It is important that x(t) and v(t) are bounded for sufficient large time t.Moreover it is also important that those are bounded until that time.From the first equation in (3.4) we have Then Thus we have the following proposition.

Proposition 3.2.
For each initial condition it holds that and there exists T > 0 such that x(t) ≤ λ/δ + 1 for t ≥ T .
To see the boundedness of v(t) we define W 1 (t) for t > 0 as follows For each initial condition it holds that and there exists Proof.At first we confirm that the integral defining W 1 (t) converges for fixed each t.By using ϕ i ∈ Y ∆ , it holds that ϕ i (θ) ≤ ∥ϕ i ∥e −∆θ for θ ∈ (−∞, 0] (i = 0, 1).Therefore the integrand of the second term of W 1 (t) is dominated by an integrable function as follows.
For any Then by the property of µ(x) it holds that And Then the time derivative of W 1 (t) becomes as follows : ) .
and the statement of this lemma follows.
The initial value W 1 (0) satisfies the following inequality : Therefore v(t) is bounded for t ≥ 0. On the other hand there exists T 1 > 0 such that W 1 (t) ≤ λ/ν 1 + 1 for t ≥ T 1 by Lemma 3.3.By the positivity of x(t) it holds that for ) . (3.28) where M 1 and M 2 are independent of t .
Therefore the solution (x, v) of (3.4) remains in the phase space Y ∆ × Y ∆ for all time t ≥ 0, thus x t (s) ≤ ∥x t ∥e −∆s and v t (s) ≤ ∥v t ∥e −∆s for s ≤ 0.
Let X = Y ∆ × Y ∆ , according to (3.4).Denote by T (t), t ≥ 0 the family of solution operators corresponding to (3.4) such that T (t)u(0) = u(t) where u(0), u(t) ∈ X.We introduce some notations and terminology: the positive orbit γ + (u) through u ∈ X is defined as γ + (u) = ∪ t≥0 {T (t)u}.The ω-limit set ω(u) of u consists of y ∈ X such that there is a sequence t n → ∞ as n → ∞ with T (t n )u → y as n → ∞.The semigroup T (t) is said to be asymptotically smooth, if for any bounded forward invariant subset U of X, there exists a compact set M such that d(T (t)U, M) → 0 as Proposition 3.6.For any bounded forward invariant subset U of X, define M 0 and M as follows : Proof.We know that M is compact in C ∆ × C ∆ from Lemma 3.2 of [2].We write T (t)(ϕ 0 , ϕ 1 ) = (x t , v t ).We note that x(t) ≤ M and v(t) ≤ M for all t ≥ 0. We define the function φ t (s) ∈ M 0 such that Therefore we have d( Similarly we define the function ψ t (s) ∈ M 0 such that By the same argument we can also get that sup Therefore we have d(v t , M 0 ) ≤ (∥ϕ 1 ∥ + x(0)) e −∆t .Thus we get that lim t→∞ d((x t , v t ), M) = 0 uniformly with respect to (ϕ 0 , ϕ 1 ) ∈ U .
Therefore T (t) is asymptotically smooth.The following result of the theory of persistence is taken from [6]: Theorem 3.7.Suppose that we have the following: (i) X 0 is open and dense in X with X 0 ∪ X 0 = X and X 0 ∩ X 0 = ∅ ; (ii) the solution operators T (t) satisfy T (t) : is isolated and has an acyclic covering N , where A b is the global attractor of T (t) restricted to X 0 and N = ∪ k i=1 N i ; (vii) for each N i ∈ N, W s (N i ) ∩ X 0 = ∅ ; where W s refers to the stable set.Then T (t) is a uniform repeller with respect to X 0 , i.e. there is an η > 0 such that for any x ∈ X 0 , lim inf t→∞ d(T (t)x, X 0 ) ≥ η.
It remains to show that W s (E 0 ) ∩ X 0 = ∅.Suppose the contrary, there exists a solution u t ∈ X 0 such that lim We define the function V (t) such that where α(a) = ∫ ∞ a g(τ )dτ .By the assumption v t → 0 we have V (t) → 0 as in the proof of Lemma 3.3.On the other hand differentiating with respect to time gives Now we take advantage of R 0 = rµ(x) ρµ(x) + b > 1 : there exists an ϵ and T > 0 such that for all t > T .Therefore, V (t) goes to infinity or approaches a positive limit as t → ∞.
It is a contradiction.Thus we confirmed (vii) and we can apply Theorem 3.7 to obtain that there exists an η such that lim inf We can apply the following Lemma to x t and v t .
Proof.There exists T 0 > 0 such that For arbitrary positive T 1 larger than Choosing t large enough makes these two values smaller than η ′ /2.Then the value of ∥y t ∥ is attained at θ 0 in the third interval T 1 − t < θ 0 ≤ 0. Therefore Let We can choose T 2 larger than u 1 and by the same argument we can obtain Therefore the assertion follows.Proof.We use Proposition 3.6 and Theorem 2.2 in [18] with ρ = x or ρ = v , where ρ : X → (0, ∞) is a continuous strictly positive functional on X.
Now we construct the Lyapunov functional for the model (3.4).We define the functionals for our construction of Lyapunov functionals, where and c is a positive constant.To guarantee the well-definedness of the integration we require that there exist positive ε and M such that ε ≤ ϕ(t − a) ≤ M for all 0 ≤ a < ∞.
Suppose R 0 > 1.Let (x, ṽ) be a solution of the equation (3.4) with (x 0 , ṽ0 ) ∈ X 0 .Then combined with Proposition 3.2, Proposition 3.4 and Proposition 3.10, it follows that the ω-limit set Ω of (x, ṽ) is non-empty, compact and invariant.It follows that Ω is the union of entire orbits of the equation (3.4).That is, if (ϕ 0 , ϕ 1 ) ∈ Y ∆ × Y ∆ is a point in Ω, then there exists an entire solution through (ϕ 0 , ϕ 1 ) such that every point on the solution is in Ω.For the solution (x, v) that lies in Ω, combined with Proposition 3.2, Proposition 3.4 and Proposition 3.10, there exist ϵ > 0 and M > 0 such that Then the functional (3.39) is well defined for every solution that lies in Ω.

Stability of the equilibria
Proposition 3.11.Suppose R 0 > 1 and (x, v) is a solution of (3.4) that lies in Ω, then the time derivative of is nonpositive under the condition : Proof.When we assume R 0 > 1, there exists an infected equilibrium (x * , v * ).We can rewrite the first equation of (3.1) as follows and at the equilibrium (3.44) The monotone nonincreasing of µ(x)/x leads and it follows that then the derivative is nonpositive and U 1 becomes a Lyapunov functional for the equilibrium (x * , v * ).
Theorem 3.12.If R 0 > 1, then all solutions of equation (3.4) for which the disease is initially present converge to (x * , v * ) under the condition (3.42).
Proof.We can show that the maximal invariant set M in is the singleton {(x * , v * )}.Let (x, v) be a solution of (3.4) in Ω.Then the ω-limit set and α-limit set are contained in M .They are equal to {(x * , v * )}.Since U 1 (x t , v t ) is nonincreasing along the solution (x, v), (x, v) must be equal to (x * , v * ) identically.Then the ω-limit set Ω is equal to {(x * , v * )}.It follows that all solutions of equation (3.4) for which disease is initially present converge to (x * , v * ).
If there does not exist absorption effect, that is ρ = 0, then the condition (3.42) is satisfied.
Assume R 0 ≤ 1 and define the following functional : The integral W ∞ 0 can be defined for every bounded solution.Let (x, ṽ) be an arbitrary solution.As in the case R 0 > 1, the ω-limit set Ω of (x, ṽ) is non-empty.Then there exists an entire solution (x, v) through an element (ϕ 0 , ϕ 1 ) ∈ Ω. Proposition 3.13.Suppose R 0 ≤ 1.Let (x, v) be a solution of (3.4) that lies in Ω.Then the time derivative of We can rewrite the model (3.4) as follows ) . (3.52) The nonpositivity of the time derivative of U 2 (x t , v t ) along (3.52) is shown by the monotonous increase of µ(x) and R 0 ≤ 1 as follows : ) is a Lyapunov functional for the disease free equilibrium (x, 0).Theorem 3.14.If R 0 ≤ 1, then all solutions converge to the infection-free equilibrium.
Proof.We can show that the maximal invariant set M in is the singleton {(λ/δ, 0)}.As in Theorem 3.12, we can show that all solutions converge to the infection-free equilibrium.

Model with an immune variable
We consider a model with the immune variable z.The model is incorporated with humoral immunity.The system of equations is where vq(z) represents the activation of immunity by infected cells and m represents the death rate of immune variable.Kajiwara et al. [10] studied a method of construction of Lyapunov functions for an ODE model containing those terms.Several types of q(z) are considered.Nowak and Bangham [15] and Murase et al. [14] used q(z) = qz.Inoue et al. [8] used q(z) = q.Gomez-Acevedo and Li [4] used q(z) = qz/(z + K) , where q and K are constants.In this paper we examine two types of q(z).The first type is q(z) = q, and the second type is q(z) = qz, where q is constant.Using an argument in Section 3 we can choose Y ∆ × Y ∆ × R ≥0 as a phase space for system (4.1), and the integral in (4.1) converges.The basic reproduction number of (4.1 The disease free equilibrium of the model (4.1) is E 0 = (λ/δ, 0, 0).The infected equilibrium is (x * , v * , z * ) that satisfies following equations : where v * > 0.

Positivity and boundedness
By a similar argument in Proposition 3.1 we have the positivity of x, v and z.We will show the boundedness of them.By (4.1) we obtain Then by a similar argument in Proposition 3.2 and Proposition 3.4 we can see the boundedness of x(t) and v(t).
In the case q(z) = q, let V be the upper bound of v.By the third equation of (4.1) we obtain Then we have the followings : therefore it holds that and there exists a T such that z ≤ qV /m + 1 for all t ≥ T .
In the case q(z) = qz, we define the following function : Then the time derivative of W 2 (t) becomes as follows : and it holds that Therefore there exists T 2 (> T ) such that W 2 ≤ C 2 /ν 2 + 1 for all t ≥ T 2 .The positivity of x and v leads the boundedness of z.

Stability of the infected equilibrium : q(z) = qz
If R 0 > 1 then the infected equilibrium exists.At first we consider the case q(z) = qz.If z(0) = 0 then z(t) ≡ 0 for all t ≥ 0 by the third equation of (4.1).Then the model is reduced to the model (3.4) without an immune variable.At most two infected equilibria are obtained from (4.2),(4.3)and (4.4).The one equilibrium ) − b ≤ 0 then there does not exist an equilibrium with positive immune variable and the infected equilibrium is only E † .Else if it holds that (r − ρ)µ(x ‡ ) − b > 0 then there exists the equilibrium ) with positive immune variable where x ‡ > x † .The condition (r − ρ)µ(x ‡ ) − b > 0 is equivalent to the following inequality : .12) Figure 1 shows the relationship of these two infected equilibria and the disease free equilibrium. x Suppose (4.12) , we define Then the attractor A restricted to X 0 is {E 0 }.A similar argument as Theorem 3.8, Lemma 3.9 and Proposition 3.10 leads the following Proposition : Proposition 4.1.Assume R 0 > 1.Let (x, v, z) be a solution of (4.1) with (x 0 , v 0 , z(0)) ∈ X 0 , then there exists a positive η ′′ such that Suppose R 0 > 1.Let (x, ṽ, z) be a solution of the equation (4.1) with (x 0 , ṽ0 , z(0)) ∈ X 0 .Then the ω-limit set Ω of (x, ṽ, z) is non-empty, compact and invariant.The set Ω is the union of entire orbits of the equation (4.1).That is , if (ϕ 0 , ϕ 1 , α) ∈ Y ∆ × Y ∆ × R + is an omega limit point of (x, ṽ, z), then there exists a solution through (ϕ 0 , ϕ 1 , α) such that every point on the solution is in Ω.As in the model without immune variable, For the solution that lies in Ω, there exist ϵ > 0 and M > 0 such that Then the functional defined by (3.39) is well defined for every solution that lies in Ω.
Proposition 4.2.Suppose that R 0 > 1 and that there exists the infected equilibrium E ‡ with positive immune variable.Let (x, v, z) be a solution of (4.1) that lies in Ω, then the time derivative of is nonpositive under the condition : Proof.If there exists the interior equilibrium (x ‡ , v ‡ , z ‡ ), we define the functional U 3 as follows : We consider the following modified system with absorption effect : This model has the same equilibrium as (4.1) and a Lyapunov functional similar to U 1 .We rewrite the system (4.1) as follows : The time derivative of The second term is nonpositive by the extension of arithmetic-geometric mean inequality [9].The last two terms become p r − ρ ) This is zero because v ‡ = m/q.Therefore when r > ρ x ‡ the derivative of U 3 holds nonpositive and U 3 is a Lyapunov functional for the equilibrium E ‡ = (x ‡ , m/q, z ‡ ).Theorem 4.3.If R 0 > 1 and there exists the equilibrium E ‡ = (x ‡ , q/m, z ‡ ) , then all solutions of (4.1) for which the disease is initially present converge to the infected equilibrium (x ‡ , m/q, z ‡ ) under the condition (4.16).
Proof.We can show that the maximal invariant set M in is the singleton {(x ‡ , v ‡ , z ‡ )}.Let (x, v, z) be a solution of (4.1) in Ω.Then the ω-limit set and α-limit set are contained in M .They are equal to {(x ‡ , v ‡ , z ‡ )}.Since U 3 (x t , v t , z) is nonincreasing along the solution (x, v, z), (x, v, z) must be equal to (x ‡ , v ‡ , z ‡ ) identically.
Then the ω-limit set Ω is equal to {(x ‡ , v ‡ , z ‡ )}.It follows that all solutions of (4.1) for which disease is initially present converge to (x ‡ , v ‡ , z ‡ ).
When there does not exist absorption effect, that is ρ = 0, then the condition (4.16) is obviously satisfied.
Let (x, v, z) be a solution that lies in Ω, then the time derivative of is nonpositive under the condition : Proof.The functional U 4 is defined as follows : The time derivative of U 4 along (4.19) is The second term is nonpositive by the extension of arithmetic-geometric mean inequality.The last two terms become p r − ρ Proof.We can show that the maximal invariant set M in is the singleton {(x † , v † , 0)}.Let (x, v, z) be a solution of (4.1) in Ω.Then the ω-limit set and α-limit set are contained in M .They are equal to {(x † , v † , 0)}.Since U 4 (x t , v t , z) is nonincreasing along the solution (x, v, z), (x, v, z) must be equal to (x † , v † , 0) identically.Then the ω-limit set Ω is equal to {(x † , v † , 0)}.It follows that all solutions of (4.1) for which disease is initially present converge to (x † , v † , 0).

Stability of the infected equilibrium : q(z) = q
Now we consider the case q(z) = q.The infected equilibrium derived from (4.2),(4.3)and (4.4) is only Let Ω be the ω-limit set of the solution of (4.1).Then by the same argument for Proposition 3.10, the functional (3.39) is well defined for every solution that lies in Ω.
Proposition 4.6.Suppose R 0 > 1.Let (x, v, z) be a solution of (4.1) that lies in Ω, then the time derivative of is nonpositive under the condition : Proof.If R 0 > 1 then there exists the interior equilibrium (x * , v * , z * ).The functional U 5 is

.30)
We consider the following modified system with absorption effect: This model has the same equilibrium as (4.1) and a Lyapunov functional similar to U 1 .We rewrite the system (4.1) as follows : If we note m = v * q/z * , the time derivative of U 5 along (4.32) is The second term is nonpositive by the extension of arithmetic-geometric mean inequality.Since q(z) = q then the last two terms become This is nonpositive by arithmetic-geometric mean inequality.
Therefore when r > ρ x * the derivative of U 5 holds nonpositive and U 5 is a Lyapunov functional for (x * , v * , z * ).Theorem 4.7.If R 0 > 1 , then all solutions of (4.1) for which the disease is initially present converge to the infected equilibrium (x * , v * , z * ) under the condition (4.29).
Proof.We can show that the maximal invariant set M in is the singleton {(x * , v * , z * )}.Let (x, v, z) be a solution of (4.1) in Ω.Then the ω-limit set and α-limit set are contained in M .They are equal to {(x * , v * , z * )}.Since U 5 (x t , v t , z) is nonincreasing along the solution (x, v, z), (x, v, z) must be equal to (x * , v * , z * ) identically.
Then the ω-limit set Ω is equal to {(x * , v * , z * )}.It follows that all solutions of (4.1) for which disease is initially present converge to (x * , v * , z * ).

Stability of the infection-free equilibrium
We consider the case R 0 ≤ 1.Let (x, v, z) be an arbitrary solution.As in the case R 0 > 1, the ω-limit set Ω of (x, v, z) is non-empty and the Lyapunov functional is well-defined for every solution.
Then the time derivative of is nonpositive.
Proof.If R 0 ≤ 1 then there exists no interior equilibrium and the only equilibrium is (x, 0, 0).The functional U 6 is Then the time derivative of U 6 along (4.1) is as follows: , then the derivative becomes nonpositive and U 6 is a Lyapunov functional for the equilibrium (x, 0, 0).Theorem 4.9.If R 0 ≤ 1, then all solutions converge to the infection-free equilibrium.
Proof.We can show that the maximal invariant set M in is the singleton {(λ/δ, 0, 0)}.As in Theorem 3.14 , we can show that all solutions converge to the infection-free equilibrium.

Multistrain model with absorption effect without immune variables
In many infection processes, there often exist multiple strains in pathogens.We consider whether the competitive exclusion principle holds or the coexistence of multiple strains holds in multistrain models.In this section we will show that the competitive exclusion principle holds in the model with absorption effect without immune variables.
We consider the multistrain model with absorption extended from the single strain model (3.1).If the variable v i denotes the virus population of i-th strain, we have with the initial condition When the i-th strain satisfies the condition ϕ 0 (θ)ϕ i (θ) > 0 for some θ ≤ 0 we say that the strain is present.All parameters concerned with i-th strain are expressed with i.The delay kernel g i (τ ) is defined by The phase space of (5.1) is Y ∆ ×Y n ∆ .We assume ρ i < r i and define Ri 0 = (r i −ρ i )β i µ(x)/b i for every i-th strain, where x = λ/δ.The parameter Ri 0 quantifies the strength of infection.We may assume without loss of generality.The exceptional case Ri 0 = Ri+1 0 for some i is not considered in this paper.For j ̸ = i it holds that We define an order relation by In our paper we assume (5.4), thus for the subset J ⊂ N n the max ▹ J is uniquely defined by ▹ max J = i ⇔ i ∈ J and j i for all j ∈ J.
If the initial condition of the i-th strain is ϕ 0 (θ)ϕ i (θ) = 0 for all θ ≤ 0 then v i (t) ≡ 0 for all t ≥ 0. Else if the initial condition of the i-th strain is ϕ 0 (θ)ϕ i (θ) > 0 for some θ ≤ 0 then the positivity and boundedness of x(t) and v i (t) are shown in the same way as in Section 3.
We define xi by the equation (5.7) as follows : and (v) j = 0 for all j ̸ = i.Else if it holds that x * ̸ = xi for any i then the equilibrium is

The competitive exclusion principle
For the variable x(t) Proposition 3.2 also holds in this model (5.1).For each initial condition it holds that and there exists T > 0 such that x(t) ≤ λ/δ + 1 for t ≥ T .
Proposition 5.1.We can choose positive C i 's and C i 's that satisfy the following : for each initial condition, v i (t) ≤ max{v i (0), C i /b i } for t ≥ 0, and there exists Proof.We define the functional W i (t) by the same way as Lemma 3.3 as follows : where ∆ < ∆ 1 < ∆ 0 .Then the time derivative of W i (t) becomes as follows : ) . ( Let It follows that and Therefore there exists Using this result we can show the boundedness of v i (t).It holds that Assume that k i (a) is bounded and continuous.Let ∥k i ∥ ∞ = sup a≥0 {k i (a)} and assume that σ i (a) is continuous.For a ≥ A it holds that σ i (a) ≤ e −2∆1a and let Then by using (5.15) it holds that for t ≥ 0 ) (5.17) Then (5.18) . Then by the same argument of Proposition 3.4 we can confirm that there exist We will state that the competitive exclusion principle holds in the multistrain model with the absorption effect without immune variables under the condition with some parameters.That is, the only i-th strain survives which is present and has the largest parameter Ri 0 larger than unity.Let us define the set of strains that can potentially survive as S as follows : For J ⊂ S let us consider the phase space X S\J as follows: Let us consider the semiflow {U S\J (t)} t≥0 : X S\J → X S\J such that for u 0 ∈ X , U S\J (t)u 0 is a solution of (5.1).Let s = max ▹ J for J ̸ = ∅.We also consider the sets N and ∂N as follows : (5.21) For the initial condition (ϕ 0 , ϕ 1 , . . ., ϕ n ) ∈ X let us define J ⊂ S by When the i-th strain is not present, then v i (t) ≡ 0 for all t ≥ 0. Let ũ = (x, ṽ) be a solution of (5.1).We can confirm that the semigroup associated with this model is asymptotically smooth as in Proposition 3.6.The ω-limit set Ω of ũ is non-empty.Then there exists an entire solution u through an element (ϕ 0 , ϕ) ∈ Ω, where ϕ Theorem 5.2.Suppose J = ∅.Let u = (x, v) be a solution of (5.1) that lies in Ω.Then the time derivative of ) The nonpositivity of the time derivative of U M 1 (u) along (5.1) is shown by v i ≡ 0 for i ∈ S, Ri 0 ≤ 1 for i / ∈ S and the monotonous increase of µ(x) .
Theorem 5.3.If J = ∅, then all solutions converge to the infection free equilibrium E 0 .
Proof.We can show that the maximal invariant set M in is the singleton {(λ/δ, 0, • • • , 0)}.Let u be a solution of (5.1) in Ω.Then the ω-limit set and α-limit set are contained in M .They are equal to {E 0 }.Since U M 1 (u) is nonincreasing along the solution u, u must be equal to E 0 identically.Then the ω-limit set Ω is equal to {E 0 }.It follows that all solutions of equation (5.1) for which no strain in S is present converges to E 0 .Theorem 5.4.If J ̸ = ∅ then every solution on X S\J converges to the equilibrium E s under the condition : where s = max ▹ J .
We will show this Theorem 5.4 by mathematical induction on #J .For mathematical induction of Theorem 5.4 we assume that (5.26) This inequality (5.26) holds under the condition that the absorption effect ρ i is sufficiently small.If ρ i is zero then the inequality holds unconditionally .

Case #J = 1
When #J = 1, it can be shown in the same way as the case of single strain.Let J = {s} and define (N, ∂N ) as (5.21).Every strain except s-th is not present or will be extinct by the condition R0 ≤ 1.
When the s-th strain is present then we will show that the solution will not converge to the equilibrium E 0 .Let define the function V s (t) as follows : where α s (a) = ∫ ∞ a g s (τ )dτ .Suppose that the solution converges to the disease free equilibrium E 0 , then v s (t) → 0, x(t) → λ/δ as t → ∞.Therefore it holds that V s (t) → 0 as t → ∞.On the other hand it holds that (5.28) The inequality xs < λ/δ leads µ(x s ) < µ(λ/δ).Thus for some ϵ > 0 there exists a T > 0 such that µ(x(t)) − µ(x s ) ≥ ϵ for all t ≥ T .Therefore it holds that d dt V s (t) > 0 for all t ≥ T and it is a contradiction.Now we can apply Theorem 4.2 in [6] and the infection is persistent.We define the following functional for our construction of Lyapunov functionals, ) da, (5.29)where H(u) = u − 1 − log u and c is a positive constant for ϕ ∈ Y ∆ .To guarantee the well-definedness of the integration we require that there exist positive ε and M such that ε ≤ ϕ(t − a) ≤ M for all 0 ≤ a < ∞.A similar argument as Theorem 3.8 , Lemma 3.9 and Proposition 3.10 leads the following Proposition : Proposition 5.5.Assume Ri 0 > 1.Let (x, v) be an entire solution of (5.1) with (x 0 , v 0 ) ∈ N , then there exists a positive η ′′ such that lim inf t→∞ x(t) > η ′′ and lim inf t→∞ v i (t) > η ′′ , ( for i = 1, 2, . . ., n).
(5.30) Let ũ = (x, ṽ) be a solution of equation (5.1) with (x 0 , ṽ0 ) ∈ N .Then the ω-limit set Ω of (x, ṽ) is non-empty, compact and invariant.It follows that Ω is the union of the entire orbits of the equation ( 5 ∆ is a point in Ω, then there exists an entire solution through (ϕ 0 , ϕ) such that every point on the solution is in Ω.For the solution u = (x, v) that lies in Ω, combined with Proposition 5.1 and Proposition 3.10, there exist ϵ > 0 and M > 0 such that ϵ ≤ x(t) ≤ M, ϵ ≤ v s (t) ≤ M for all t ∈ R. (5.31) Then the functional (5.29) is well-defined for every entire solution that lies in Ω. (5.35) The monotone nonincrease of µ(x)/x leads and it follows that then dU 2 /dt is nonpositive.
Proposition 5.7.All solutions of equation (5.1) for which the s-th strain is present converge to E s under the condition (5.38),where s = max ▹ J .
Proof.We can show that the maximal invariant set M in is the singleton {E s }.Let u = (x t , v t ) be a solution of (5.1) in Ω.Then the ω-limit set and α-limit set are contained in M .They are equal to {E s }.Since U M 2 (x t , v t ) is nonincreasing along the solution u, u must be equal to E s identically.Then the ω-limit set Ω is equal to {E s }.It follows that all solutions of equation (5.1) for which disease is present converge to E s .

Case #S ≥ 2 and 2 ≤ #J ≤ #S
In this section we assume that #S ≥ 2 and #J ≥ 2 for a subset J ⊂ S. Our induction hypothesis is concerned with the validity of Theorem 5.4 for each subset J ′ ⊂ S such that 1 ≤ #J ′ < #J .
Let s = max ▹ J .We define N and ∂N by (5.21).Let J ′ ⊂ J \{s} then by #J ′ < #J we can use induction hypothesis when #J ′ ≥ 1.That is if J ′ ̸ = ∅ then there

Multistrain model with absorption effect and immune variables
Inoue et al. [8] analyed the dynamics of ODE models of humoral immunity.For all models, they proved the global stability of the disease steady state.Our purpose is to analyze a system with delay.In this section we consider the multistrain model with absorption effect and immune variables extended from the single strain model (4.1) such that We consider the case the immune activity increases at a rate proportional to the number of virus.The phase space of (6.1) is without loss of generality.The exceptional case Ri 0 = Ri+1 0 for some i is not considered in this paper.For i ▹ j it holds that Ri We define the set of strains which can potentially survive as S defined by Let J be a subset of S. We introduce a phase space X S\J = {(ψ 0 , ψ 0 , . . ., ψ 0 , α 1 , . . ., α n ) ∈ X ψ 0 (θ)ψ i (θ) = 0 for all θ ≤ 0 ∀i ∈ (S\J)} and consider the semiflow {U S\J (t)} t≥0 : X S\J → X S\J such that for u 0 ∈ X , U S\J (t)u 0 is a solution of (6.1).The positivity and boundedness of x(t) and v i (t) are shown in the same way as Proposition 5.1, where ϕ 0 (θ)ϕ i (θ) > 0 for some θ ≤ 0. And it holds that x(t) ≤ max{x(0), λ/δ}, lim sup t→∞ x(t) ≤ λ/δ.
When the initial condition of immune variable is z i (0) > 0, the positivity of z i is shown as follows.If t 0 > 0 is the least positive time such that z i (t 0 ) = 0 then the third equation of (6.1) leads Because of the positivity of v i it holds that z i (t) < 0 for some positive t such that t < t 0 .It is a contradiction.Therefore it holds that z i (t) > 0 for all t ≥ 0. The boundedness of z i is shown as follows.Let V i be the upper bound of v i .By the third equation of (6.1) we obtain Then we have the followings : therefore it holds that and there exists a T such that z i ≤ q i V i /m i + 1 for all t ≥ T .We denote by (x * , v * , z * ) the candidate of the equilibrium.At this equilibrium the right hand side of (6.1) becomes Thus where Then by (6.11) The values of xi are defined by (5.8).And we define the function h 2,i (x) as follows:
For the initial condition (ϕ 0 , ϕ 1 , . . ., ϕ n , α 1 , . . ., α n ) ∈ X let us define the set J as follows : When J ̸ = ∅ let x * be the solution of h 1 (x) = h 2,J (x) and define The equilibrium is represented by E KJ .If J = ∅ then K J = ∅ and the equilibrium is where We define U M 4 (u; K) as follows

.22)
Let ũ = (x, ṽ, z) be a solution of (6.1).We can confirm that the semigroup associated with this model is asymptotically smooth as in Proposition 3.6.The ω-limit set Ω of ũ is non-empty.Then there exists an entire solution u through an element (ϕ 0 , ϕ, z(0)) ∈ Ω. Proposition 6.1.Suppose J = ∅.Let u = (x, v, z) be a solution of (6.1) that lies in Ω.Then the time derivative of is nonpositive, where α i (a) = ∫ ∞ a g i (τ )dτ .Proof.By J = ∅ it holds that K J = ∅.The time derivative of U M 4 (u; ∅) along (6.1) Theorem 6.3.If J ̸ = ∅ then every solution on X S\J converges to the equilibrium E KJ under the condition : where J ̸ = ∅ and the equilibrium is (x * , v * , z * ).
We will show this Theorem 6.3 by mathematical induction on #J .

Case #J = 1
Let J = {s}.Other strains are not present or it holds that Ri 0 ≤ 1 for them.If the s-th strain is present, then the solution converges to the equilibrium E KJ = E {s} .Else if s-th strain is not present, then the solution converges to the equilibrium E ∅ .
When the s-th strain is present then we will show that the solution will not converge to the equilibrium E ∅ same as in 5.1.1.Let define the function V s (t) by (5.27).If the solution converges to E ∅ then v s (t) → 0, x(t) → λ/δ and z s → 0 as t → ∞.Therefore it holds that V s (t) → 0 as t → ∞.On the other hand the time derivative of V s (t) becomes as follows : By s ∈ S it holds that xs < λ/δ, that is µ(x s ) < µ(λ/δ).Then there exist ϵ > 0 and T > 0 such that µ(x(t)) − µ(x s ) ≥ ϵ and z s is sufficiently small for all t ≥ T .Thus it holds that d dt V s (t) > 0 for all t ≥ T , it is a contradiction.Therefore we can apply Theorem 4.2 in [6], and the infection is persistent.The similar argument in 5.1.1 will show that the ω-limit set Ω of (x, ṽ1 , • • • ṽn , z) is nonempty, compact and invariant.It follows that Ω is the union of the entire orbits of the equation (6.1).That is, if (ϕ 0 , ϕ, z(0)) ∈ Y ∆ × Y n ∆ × R n + is a point in Ω, then there exists an entire solution through (ϕ 0 , ϕ, z(0)) such that every point on the solution is in Ω.For the solution u = (x, v, z) that lies in Ω, combined with Proposition 5.1 and Proposition 3.10, there exist ϵ > 0 and M > 0 such that ϵ ≤ x(t) ≤ M, ϵ ≤ v s (t) ≤ M for all t ∈ R. (6.28) Then the functional W ∞ 1 defined by (5.29) is well defined for every solution of (6.1) that lies in Ω. Proposition 6.4.Let u = (x, v, z) be an entire solution of (6.1) that lies in Ω.Then Proof.We consider the functional U M 4 (u; K) for K ∈ K defined by (6.22).
The integral of the second line is well defined by the hypothesis of induction because K = K J ′ and #J ′ < #J .The time derivative of U M 4 (u; K) becomes When the inequality (6.36) holds, it holds that dU M 4 /dt ≤ 0 for all t.The assumption (6.35) leads u(t) ≡ E K .
Therefore we can apply Theorem 4.2 in [6] and the flow U S\J is persistent on (N, ∂N ).Let ũ = (x, ṽ, z) be a solution of equation (6.1) with (x 0 , ṽ0 , z(0)) ∈ N .Then same as in Section 3, it follows that the ω-limit set Ω of (x, ṽ, z) is non-empty, compact and invariant.It follows that Ω is the union of orbits of equation (6.1).That is, if (ϕ 0 , ϕ, α) ∈ Y ∆ × Y n ∆ × R n + is an omega limit point in Ω, then there exists an entire solution through (ϕ 0 , ϕ, α) such that every point on the solution is in Ω.For the solution u = (x, v, z) that lies in Ω, the functional U M 4 (u; K J ) is well defined.Proposition 6.8.Let Ω be the ω-limit set of the solution of (6.1) and let u be a solution that lies in Ω.Then the time derivative of

Conclusion
We used constructive method to formulate Lyapunov functionals from the simple onestrain models to the multistrain models.We have made mathematically rigorous description.These methods and techniques stated in this paper are applicable to the other various models.
The conditions that we can construct Lyapunov functionals for the models are slightly strong sufficient conditions.More appropriate conditions are desirable.In this thesis we does not consider the multistrain model with immune variable such that the activation of immunity is represented by q i v i z i for every i-th strain.This model includes much mathematical difficulty compared to other models.The analysis of this model will be an important study.

Theorem 4 . 5 .
If R 0 > 1 and the infected equilibrium is only E † , then all solutions of equation (4.1) for which the disease is initially present converge to the infected equilibrium (x † , v † , 0) under the condition (4.23).

. 8 )
By the equation (5.5) it holds that if i ̸ = j then xi ̸ = xj and if j ▹ i then xi < xj .By the equation (5.7) it holds that x * = xi or v Then there exist M 1 and M 2 such that ∥x t ∥ ≤ M 1 and ∥v t ∥ ≤ M 2 for t ≥ 0 where M 1 and M 2 are independent of t .