Effect of density dependence on coinfection dynamics: part 2

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in the 1st part of the paper. We look for coexistence equilibrium points, their stability and dependence on the carrying capacity $K$. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by $K$. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a section of coexistence points starting at a bifurcation equilibrium point with zero second single infections and finishing at another bifurcation point with zero first single infections. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of $K$ and the rate $\bar \gamma$ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.


Introduction
In this paper we continue on the work of [2] where we studied the equilibrium dynamics for a continuous compartmental model of two infectious diseases with the ability to co-infect individuals. In the model we assume that only the susceptibles can give birth and that the reproductive rate depends on the density of the susceptibles. This dependence is modelled with a parameter K > 0 which represent the carrying capacity of the population. Recall that by an (equilibrium) branch we understand any continuous in K ≥ 0 family of equilibrium points of a dynamic system which are locally stable for all but finitely many threshold values of K.
In [2] it was discovered that for a certain set of parameters excluding K there exists an equilibrium branch with respect to K of locally stable equilibrium. For this branch all of the equilibrium points where expressed explicitly and K for which a compartment changed from being zero to non-zero or vice versa where pointed out.
In this paper we will show the same holds for the rest of the parametric choices. The main difficulty compared to [2] is that for our parameters the equilibrium branch consists of coexistence equilibrium where single infection of each disease and coinfection both occurs. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems.
Following [1,3,10], we assume limited population growth by making the per capita reproduction rate depend on the density of population. We also consider the recovery of each infected class (see the last equation in (1.1)). The fundamental parameters of the system are: • r = b − d 0 is the intrinsic rate of natural increase, where b is the birthrate and d 0 is the death rate of S-class, • K is the carrying capacity (see also the next section), • ρ i is the recovery rate from each infected class (i = 1, 2, 3), • d i is the death rate of each class, (i = 1, 2, 3, 4), where d 3 and d 4 correspond I 12 and R respectively, • µ i = ρ i + d i , i = 1, 2, 3.
• α 1 , α 2 , α 3 are the rates of transmission of strain 1, strain 2 and both strains (in the case of coinfection), • γ i is the rate at which infected with one strain get infected with the other strain and move to a coinfected class (i = 1, 2), • η i is the rate at which infected from one strain getting infection from a co-infected class (i = 1, 2); We only consider the case when the reproduction rate of susceptibles is not less than their death rate since we know that the population will go extinct in that case. The system is considered under the natural initial conditions S(0) > 0, I 1 (0) ≥ 0, I 2 (0) ≥ 0, I 12 (0) ≥ 0 and by N = S + I 1 + I 2 + I 12 + R. we denote the total population.
Since the variable R is not present in the first four equations, without loss of generality, we may consider only the first four equations of system (1.1). It is convenient to introduce the notation And similarly to the first part [2] we make the following assumption We shall also assume that (1.1) satisfies the non-degenerate condition This condition have a natural biological explanation: the virus strains 1 and 2 have different (co)infections rates. We use the notation γ = γ 1 + γ 2 , γ = (γ 1 , γ 2 ), and A 1 = α1α3 r (σ 3 − σ 1 ), η * 1 := Notice that by (2) one has A 1 , A 2 , A 3 > 0. We have The determinants ∆ α and ∆ µ = η 1 µ 2 − η 2 µ 1 are related to each other by This implies an inequality which will be useful in the further analysis: We shall also make use of the following relations: A consequence of (7) and (5) is that for η * 1 > η * 2 we have ∆ α , ∆ µ > 0. On the other hand, one has

The main result
It is elementary to see that except for the trivial equilibrium state G 1 = (0, 0, 0, 0) and the disease free equilibrium G 2 = (K, 0, 0, 0), there exist only 6 possible types of equilibrium points G 3 , G 4 , G 5 , G 6 , G 7 , G 8 determined by their non-zero compartments (see Table 1 and Proposition 1 below for explicit representations). More precisely, the equilibrium points G 3 , G 4 , G 5 have two non-zero components and represent points where only one of the diseases are present or where the diseases only exist together as coinfection. At the points G 6 , G 7 one of the diseases are only present in coinfected individuals while the other disease also occurs as single infections. The point G 8 is the coexistence equilibrium were both types of single infection is present as well as coinfection. Our main results extends the results of [6] on the case of small values of γ i . More precisely, we have only four possible scenarios of developing of a locally stable equilibrium point as a continuous function of increasing carrying capacity K: Table 1 The types of equilibrium states of (1.1), where ⋆ denotes a non-zero coordinate Theorem 1 Let all parameters α i , µ i , η i , γ i of (1.1) be fixed withγ sufficiently small. Then one has exactly one locally stable nonnegative equilibrium point depending on K > 0. Furthermore, changing the carrying capacity K from zero to infinity, the type of this locally stable equilibrium point changes according to one of the following alternative scenarios: The first two scenarios are considered in our paper [2]. In this paper we consider the remained two scenarios, (iii) and (vi). These cases require a more nontrivial bifurcation analysis with application of methods similar to the principle of the exchange of stability developed in [8], see also [9] and [2] for recent applications in population analysis. In our context, this require a delicate analysis of the inner equilibrium state G 8 , as well as a new bifurcation technique.

Equilibrium points
We note that the last equation in (1.1) can be solved explicitly with respect to R: therefore it suffices to study the dynamics of the first four equations in (1.1). If I 1 , I 2 and I 12 have limitsÎ 1 ,Î 2 andÎ 12 respectively as t → ∞ then R will have the limitR Let us turn to the first four equations in (1.1). The equilibrium points satisfy the following system and in [2] we had the following proposition Proposition 1 Except for the trivial equilibrium G 1 = (0, 0, 0, 0) and the disease free equilibrium G 2 = (K, 0, 0, 0) there exist only the following equilibrium states: , . All required information about equilibrium points G 1 -G 6 can be found in [2]. Here we take only points G 8 and G 7 . To highlight the dependence of the equilibrium points on K we will write sometimes G j (K).

Coexistence equilibria
The coordinates of coexistence equilibrium points satisfy Furthermore, as it is shown in [2], Sect. 3.2, the S coordinate of an inner equilibrium point (coexistence equilibrium) satisfies P (S) = 0 where P (S) := P γ,K (S) :=

One can verify that
Here ρ := If the S component is known the other components can easily be solved from the linear system of equations that results from the first three equations of (9).
Let us introduce the Jacobian matrix of the right hand side of (1.1), with the redundant last row removed, evaluated at an inner equilibrium point G 8 = (S, I 1 , I 2 , I 12 ): Adding the first three rows of J 8 to its last row one obtains applying systematically (9) that det B = det(J 8 ) The last equality is verified directly by using the definition of P (S). This implies an important property We assume that ∂P γ,K (S) ∂S > 0 for any coexistence equilibrium point.
Remark 1 Inequality (13) together with (12) implies, in particular, that the Jacobian matrix is invertible at every coexistence equilibrium and so there exist a curve G(K) through this point, parameterized by K and consisting of equilibrium points satisfying (13). Moreover (13) implies that the product of all eigenvalues of the Jacobian matrix at a coexistence eq. point is positive which is in agreement with the local stability of the corresponding equilibrium point. By Lemma 4 and (7) we have that ∆ α > 0 if the condition (13) is valid and the set of coexistence equilibria is non empty. Then since inequality (13) is true for smallγ.
Lemma 1 Let G(K) = (S(K), I 1 (K), I 2 (K), I 12 (K)) be a curve consisting of coexistence equilibrium points satisfying (13). Let also (K 1 , K 2 ) be the maximal interval of existence of such curve. Then (i) ∂S ∂K < 0 and ∂I12 ∂K < 0 for K ∈ (K 1 , K 2 ). (ii) K 1 ≥ σ 1 and there exists the limit lim K→K1 G(K) which is an equilibrium point with at least one zero component.
(iii) if K 2 < ∞ then there exists the limit lim K→K2 G(K) which is an equilibrium point with at least one zero component.
(iv) if K = ∞ then there is a limit lim K→∞ G(K) which is an equilibrium point of the limit system (K = ∞).
To prove (ii) we note first that the equilibrium point G 2 is globally stable for K ∈ (0, σ 1 ) according to [2], Proposition 2, and therefore K 1 ≥ σ 1 . Next, since S and I 12 components are monotone according to (i), and bounded there is a limit S (1) = lim The I 1 and I 2 components satisfying equations which implies convergence of these components to I and I (1) 2 respectively as K → K 1 .
12 ) is an equilibrium point which must be on the boundary of the positive octant, otherwise one can continue the branch G(K) outside the maximal interval of existence. This argument proves (ii). Proof of (iii) and (iv) are the same up to some small changes as the proof of (ii).
To exclude from our analysis the equilibrium point G 4 we will require in this text that γ * < 1.
Under this condition G 4 is always unstable. Since we are interested only in locally stable equilibrium point the point G 4 will not appear in our forthcoming analysis.

The equilibrium state G 7
Let us consider the equilibrium point G 7 . The components are given by proposition 1 as G 7 = (S * , 0, I * 2 , I * 12 ), , This point has type G 7 (i.e. three positive components) if and only if where the first relation is equivalent to Similarly to above we find the Jacobian matrix evaluated at G 7 as where S, I 2 , I 12 are given by Proposition 1. Since the submatrix is stable by Routh-Hurwitz criteria, we conclude that the matrix J 7 is stable whenever Using Proposition 1, we can rewrite (16) as If ∆ α − γ 1 α 3 = 0 then the linear stability holds whenever It can be verified also that This readily yields the (local) stability criterion: The equilibrium point G 7 is nonnegative and locally stable if and only if η * 2 > 1 and exactly one of the following conditions holds: By (18) we get that for small values of γ 1 one has the following refinement of the above proposition.
Then the equilibrium point G 7 is nonnegative and linearly stable if and only if Therefore the bifurcation pointK 2 appears here only in the case (i).
Proof By the made assumption, the case (i) in Proposition 2 is impossible. So it is sufficient to prove (i). It is thus required that η * where we used equation (4) in the last equality. Furthermore, since (19) and ∆ α − γ 1 α 3 > 0 holds for this case we also obtain from (18) thatŜ 2 − σ 2 > 0, therefore max(Ŝ 2 , σ 2 ) =Ŝ 2 , and we arrive at the desired conclusion.
In what follows we will assume that We note that the first inequality guarantees (19) since the equilibrium point G 7 exists only if η * 2 > 1.

Bifurcation of G 6
From [2] we know that the equilibrium point G 6 with the only zero component I 2 has the form where We also know that it has positive components (except I 2 ) when η * 1 > 1 and The bifurcation point (the point where the Jacobian is zero) corresponds to and is denotedĜ 6 = G(K 1 ). Stability analysis of G 6 is given in the next proposition Proposition 3 The equilibrium point G 6 is nonnegative if η * 1 > 1 and it is stable if the following conditions hold: The case η * 2 > η * 1 (when we have no bifurcation) is considered in our paper [2]. Here we will assume that By (7) the last inequality implies that (5), one verifies straightforward that Notice a useful identity (the last equality is by (8)) As a result of Corollary 1 and proposition 3 we get thatŜ 1 ,Ŝ 2 only exist as parts of equilibrium points when ∆ α − γ 1 α 3 > 0 and η * 1 > η * 2 > 1. In this case we see from (24) thatŜ 1 >Ŝ 2 .
We will now prove the following lemma Lemma 2 Let (23) be valid and ∂P ∂S | S=Ŝ1 > 0. Then there exists a smooth branch of equilibrium points G 8 (K) = (S, I 1 , I 2 , I 12 )(K) defined for small |K − K 1 | with the asymptotics Furthermore this equilibrium point is locally stable forK 1 < K ≤K 1 + ε, where ε is a small positive number. Moreover all equilibrium points in a small neighborhood of G 6 (K 1 ) are exhausted by two branches G 6 (K) and G 8 (K).

Remark 2
The constant ε does not depend on γ i but it does depend on α i , µ i and η i .
Proof In order to use results of Appendix B we write system (9) in the following form where By the definition of the bifurcation point (22) and (21), we have that solves the equation F (x * , 0; 0) = 0 and f (x * ) = 0. Futhermore, the vector With the help of Hurwitz stability criterion we can deduce that A(y * , 0) is stable and invertible. Since Let us evaluate Θ and check that Θ = 0. First we observe the equality By (12) det(left-hand side of (36)) = 1 I 12 Let us show that det(right-hand side of (36)) = Θ detÂ .
For this purpose consider the equation We denote byB the matrix in the left-hand side of (30) and using the expression for the matrix inverse, we get Solving (30) as a linear system by finding first X and then x from the last equation, we obtain −Θx = 1. The last relation together with (31) gives (29). Now the relations (28) and (29) imply which along with det(Â(K 1 )) = 1 Next, since f (x * , 0; 0) = 0, we have Now applying (71) in the appendix we get which is equivalent to (25).
To prove local stability let us consider the matrix Since the matrix A is stable the matrix B has three eigenvalues with negative real part and one eigenvalue zero. The eigenvalues of the Jacobian matrix are small perturbation of the eigenvalue of B = J (s). Therefore three of them have negative real part for small s and the last one λ(x(s)), which is perturbation of zero eigenvalue of B, has the following asymptotics (see (72) in the appendix) and hence it is negative for small positive s. This proves the local stability of the coexistence equilibrium point.

Bifurcation of G 7
We will assume in this section that From [2] we know that the equilibrium point G 7 with only zero component I 1 has the form . We also know that it has positive components (except I 1 ) when η * 2 > 1 and The bifurcation point (the point where the Jacobian is zero) corresponds to and is denotedĜ 7 = G 7 (K 2 ).
For γ * 1 < η * 2 we have thatŜ 2 > σ 2 and that G 7 is stable for K in the We will now prove the following lemma Lemma 3 Let (32) be valid and ∂P ∂S S=Ŝ2 > 0. Then there exist a smooth branch of equilibrium points G 8 = (S, I 1 , I 2 , I 12 )(K) defined for small |K −K 2 | with the asymptotics These equilibrium points are locally stable forK 2 − ε ≤ K <K 2 , where ε is a small positive number. Moreover all equilibrium points in a small neighborhood of G 7 (K 2 ) are exhausted by two branches G 7 (K) and G 8 (K).

Remark 3
The constant ε does not depend on γ but it does depend on α, µ and η.
Proof We write system (9) in the form By the definition of the bifurcation point (34) and (33), we have that solves the equation F (x * , 0; 0) = 0 and f (x * ) = 0. Furthermore, the vector solves the equation F (ξ(s), 0, s) = 0. The matrix A = (∂ xj F k (x * , 0, 0)) 3 j,k=1 is evaluated as with the help of Hurwitz stability criterion we can deduce that A is stable and invertible. Since Let us evaluate Θ and check that Θ = 0. First we observe the equality In the same way as in section 3.1 we get Next, since f (x * , 0; 0) = 0, we have Now applying (71) we get which is equivalent to (35). To prove local stability let us consider the matrix Since the matrix A is stable the matrix B has three eigenvalues with negative real part and one eigenvalue zero. The eigenvalues of the Jacobian matrix are small perturbation of the eigenvalue of B = J (s). Therefore three of them have negative real part for small s and the last one λ(x(s)), which is a perturbation of the zero eigenvalue of B, has the following asymptotics (see (72) in appendix B) and hence it is negative for small positive s. This proves the local stability of the coexistence equilibrium point.

Equilibrium transition for coexistence equilibrium points
Lemma 4 Let the assumption (13) be valid. If there exist a coexistence equilibrium point then (i) η * 1 > η * 2 and η * 1 > 1 and this point lies on the branch of coexistence eq. points which starts at K = K 1 at the bifurcation pointĜ 6 . Moreover (ii) if additionally η * 1 > η * 2 > 1 then the above branch is finished at K =K 2 at the pointĜ 7 .
Proof Let us assume that there is a coexistence eq. point G * 8 for K = K * . Let (K 1 , K 2 ) be the maximal existence interval for existence of the branch G 8 (K) of coexistence equilibrium points containing K * and G 8 (K * ) = G * 8 . According to Lemma 1 there exists the limit G * = lim K→K1 G 8 (K) and this limit is an equilibrium with at least one zero component. According Lemma 2 in [2] the only possible scenarios are either that G * = G 6 and α 2 S * −η 2 I 12 −γ 2 I 1 −µ 2 = 0 or that G * = G 7 and α 1 S − η 1 I 12 − γ 1 I 2 − µ 1 = 0. This happens only if G * =Ĝ 6 or G * =Ĝ 7 . The case G * = G 4 is disregarded due to assumption (14). According to (24) with its associated comment we have thatŜ 1 >Ŝ 2 and η * 1 > η * 2 . SinceŜ 1 > S 2 and ∂S ∂K < 0 according to Lemma 1 deduce that G * =Ĝ 6 . From existence ofĜ 6 it follows that η * 1 > 1 and we obtain (i). IfK 2 is finite then there is a limit of G 8 (K) as K →K 2 and this limit lies on the boundary. Simple modification of the above arguments shows that this limit isĜ 7 which gives (ii).
In the case (iii) there are noĜ 6 orĜ 7 and hence the branch can be continued for all K >K 1 .
Then there is a branch of coexistence equilibrium points starting atĜ 6 , K =K 1 , and ending atĜ 7 , K =K 2 . All possible coexistence equilibrium points lies on this branch.
(ii) Let η * 1 > 1 > η * 2 and ∂ S P (Ŝ 1 ) > 0 when K =K 1 . There is a branch of coexistence equilibrium points starting atĜ 6 , K =K 1 , and defined for all K >K 1 . All possible coexistence equilibrium points lies on this branch. Proof (i) By Lemma 3 there is a branch of coexistence equilibrium points ending atĜ 7 , K =K 2 and defined for smallK 2 − K > 0. By Lemma 4 it can be continued to the interval (K 1 ,K 2 ) and the limit when K →K 1 is equal toĜ 6 . If we take any coexistence equilibrium then by Lemma 4 it must lie on an equilibrium curve starting atĜ 6 . Then by uniqueness in Lemma 2 this curve must coincide with the coexistence equilibrium branch constructed in the beginning.
(ii) In this case there is no bifurcation pointĜ 7 and the proof repeats with some simplifications the proof of (i).

Auxiliary assertion
Let Q and q be two positive constants. We introduce the set of Consider the matrix depending on the parameters Y and K: Let also λ k = λ k (Y, K), k = 1, 2, 3, 4, be their eigenvalues numerated according to the order |λ 1 | ≥ |λ 2 | ≥ |λ 3 | ≥ |λ 4 |. In the next lemma we give some more information about the first three eigenvalues.
Proof First assume that all components of Y are non-zero. Let λ ∈ C be an eigenvalue of M, i.e. MX = λX, X = (X 1 , X 2 , X 3 , X 4 ) T ∈ C 4 , X = 0.
This implies and (·, ·) is the inner product in C 4 . Therefore This gives ℜλ ≤ 0. Assume now that λ = iτ , τ ∈ R, which implies X 1 = 0. Then (38) implies If λ = 0 then X 4 = 0 and from the first and last equations in (39) we get that X 2 = X 3 = 0. If X 4 = 0 and λ = 0 then from the middle equations in (39) we obtain X 2 = X 3 = 0. Consider the case when λ = 0 and X 4 = 0. Expressing X 2 and X 3 through X 4 from the middle equations in (39) and putting them in the first equation, we get which implies X 4 = 0. Thus we have shown that there are no eigenvalues of M on the imaginary line, i.e. ℜλ j < 0, j = 1, 2, 3, 4, provided all Y j ar positive. Next consider the case Y 2 = 0. Then one eigenvalue of M is zero and the remaining three can be found from the eigenvalue problem Similar to the eigenvalue problem (38) one can show that ℜλ < 0 for (40). The argument in the case Y 3 = 0 is the same as in the case Y 2 = 0. Thus we have shown that for all (Y, K) ∈ Y, ℜλ j (Y, K) < 0, j = 1, 2, 3. Since the eigenvalues continuously depend on (Y, K) and the set Y is compact, we arrive at (37).

Local stability in the case
The main stability result for the equilibrium points branch in Lemma 6 is the following Proposition 4 Let η * 1 > η * 2 > 1 and G 8 (K),K 1 ≤ K ≤K 2 be the branch constructed in Lemma 5 (i). Then there exists a constant δ depending only on α j , j = 1, 2, 3, and η 1 , η 2 such that ifγ ≤ δ then all points on this branch for K 1 < K <K 2 are locally stable.
Proof Consider equilibrium points G 8 (K) = (S(K), I 1 (K), I 2 (K), I 12 (K)), K ∈ [K 1 ,K 2 ]. By Corollary 2 of [2] and Lemma 5 all of the components are non-negative and bounded by a certain constant independent of K and γ 1 , γ 2 . Solving for S and I 12 in the second and third of (10) gives Furthermore, from the last equation in (10) we get which implies due to (41) We will keep the notation Y for our case (Y = (S, I 1 , I 2 , I 12 )), where andγ is kept sufficiently small. Then we can use Lemma 6, where K j =K j , j = 1, 2. We introduce two subsets of Y × [K 1 ,K 2 ]. The first oneŶ 1 consists of all (Y ; K) ∈ Y × [K 1 ,K 2 ] such that ℜλ 1 ≥ Ξ/2, where Ξ is the constant from Lemma 6 (in our case it depends only on α j , j = 1, 2, 3, and η 1 , η 2 . The second setŶ 2 consists of all (Y ; K) ∈ Y × [K 1 ,K 2 ] such that ℜλ 1 ≤ Ξ/2. Introduce the contours where C is sufficiently large. Put By Lemma 6 there are at least 3 eigenvalues of M with ℜλ ≤ Ξ Consider two cases (i) the remaining eigenvalue satisfies ℜλ < 5 8 Ξ or (ii) it satisfies ℜλ ≥ 5 8 Ξ. Since the norm of the matrix is estimated by C 1γ with C 1 independent on γ and K we conclude that by Rouche's theorem the number of eigenvalues inside Γ 2 of the matrix M and M + N is the same for smallγ in the case (i). Similarly we have that in the case (ii) the number of eigenvalues of M and M + N is the same inside the contour Γ 1 for smallγ and this number is equal to 4. This implies that for smallγ there are at least three eigenvalues of the Jacobian matrix J 8 with negative real part on the branch G 8 (K). Since we conclude that all eigenvalues of J 8 (G 8 (K)) must have negative real part. This proves the proposition.

Instability for large K
In this section we assume that According to Lemma 5 there exists a branch G 8 (K), K ∈ [K 1 , ∞), of coexistence equilibrium points starting from G 8 (K 1 ) =Ĝ 6 . For K = ∞ and γ = 0 the interior point has the coordinates All eigenvalues of the corresponding Jacobian matrix lie on the imaginary axis. For K = ∞ and small γ > 0 the interior point has the coordinates G ∞ (γ) = G ∞ (0) + O(|γ|), where γ = (γ 1 , γ 2 ). Our goal is to analyze the location of eigenvalues of the Jacobian matrix when γ is small.
The characteristic polynomial of the Jacobian matrix of the interior point is (up to a positive factor) where r i are defined in (11). It is clear that the polynomial p is monic. The necessary condition for stability of the polynomial p is the positivity of all its coefficients. Let us evaluate the coefficient p 1 of the λ term and show that it can be negative for certain choice of parameters (we note that for γ = 0 this coefficient is zero). This will imply that some of all eigenvalues must have positive real part. We have Plugging in the values of S, I 12 , r 1 , r 2 , for K = ∞ and γ = 0 we continue the above equalities

Hopf bifurcation
In this section we assume that (42) is satisfied. Thus there exists a branch of coexistence equilibrium points G 8 (K) defined for K >K 1 . We assume also that the parameters α 1 , α 2 , α 3 and η 1 , η 2 are chosen such that the stability is lost when Kγ is large. Since the point G 8 (K) is stable when K is close toK 1 there exist a point K = K c where the local stability of G 8 is lost. Since the trace of the Jacobian matrix is always negative the eigenvalues can only reach the imaginary axis in pairs. If we assume that the derivative of their real part at K = K c is positive then there is a simple Hopf bifurcation so for K close to K c there are periodic oscillations, see [9].
Proof In what follows in the proof we will denote by c and C, possibly with indexes, various positive constants depending on α 1 , α 2 , α 3 and η 1 , η 2 . The Jacobian matrix is equal to J 8 (K) = D(A+K+Γ ), where D = diag(S, I 1 , I 2 , I 12 ), Consider the eigenvalue problem If γ = 0 then the eigenvalue of this problem lie in the half-plane ℜλ < 0. If we show that the are no eigenvalues of (43) with λ = iτ , τ ∈ R, for all γ and K satisfying Kγ ≤ ω then by continuity argument for eigenvalues we obtain the required result. Therefore let us assume that one of eigenvalues has the form λ = iτ, τ ∈ R and that no eigenvalue has positive real part. We will now show that the problem (43) has only the trivial solution. For large K and small γ (say K ≥ K * andγ ≤γ * ) we have where C and c are positive constants depending on α and η. Furthermore, the Jacobian matrix at the point (44) is and therefore, det J 8 (K) = SI 1 I 2 I 12 ∆ 2 α + O(γ + K −1 ). Thus we may assume that det J 8 (K) ≥ c 1 > 0. This fact together with (45) gives where λ j (K), j = 1, 2, 3, 4, are eigenvalues of J 8 (K). Assume that λ = iτ, c 2 ≤ τ ≤ c 3 is an eigenvalue to J 8 . We will now show that this leads to a contradiction. Multiplying both sides of (43) by D −1ū and taking the real part and using that ℜ(Au, u) = 0 we get = ℜ((K + Γ )u, u) = 0 or (46) Let us derive some relations between u 1 , u 2 , u 3 and u 4 . From the first three equations in (43) we obtain We rewrite the last two equations as iτ u 2 + γ 1 I 1 u 3 = α 1 I 1 u 1 − η 1 I 1 u 4 iτ u 3 + γ 2 I 2 u 2 = α 2 I 2 u 1 − η 2 I 2 u 4 and solving them we obtain u 2 = (−iτ α 1 I 1 + α 2 γ 1 I 1 I 2 )u 1 − (η 2 γ 1 I 1 I 2 − iτ η 1 I 1 )u 4 τ 2 + γ 1 γ 2 I 1 I 2 (48) Inserting these relations in (47) we get This leads to The relations (48) and (49) together with (50) gives This is impossible if C 1 is sufficiently small. Thus the local stability of G 8 (K), K ≥ K * , is proved. The local stability of G 8 (K) for K ∈ (K 1 , K * ] is proved in the same manner as in the proof of Proposition 4.

Equilibrium transition with increasing K
In this section we finalize our results in two theorems describing the equilibrium branch for the sets of parameter η * 1 > η * 2 > 1 and η * 1 > 1 > η * 2 .

Equilibrium transition when
In this section we will prove that there exist an equilibrium branch By Corollary 5 in [2] we know that for these parameters there is an equilibrium branch Furthermore from section (3.1) we know the this branch continues onto G 8 at K = K 1 . From section 2.2 and section 3.2 in this paper as well as Theorem 1 from [2], we get that there exist an equilibrium branch One could suspect that these two equilibrium branches are the two parts of a complete equilibrium branch. We shall now prove that indeed that is the case.
Theorem 3 Let (13), (20) and Assumption II hold and let η * 1 > η * 2 > 1. Then there exist a unique branch of equilibrium points G * (K) parameterised by K ∈ (0, ∞): we display this schematically as (see figure 1) The point G * (K) is locally stable whenever it is not a coexistence point. It is also locally stable near the end on the intervalK 1 < K <K 2 and it is locally stable on the whole interval ifγ is small Theorem 4 Let (13), (20) and Assumption II hold and let η * 1 > 1 > η * 2 . Then there exists a unique branch of equilibrium points G * (K) parameterised by K ∈ (0, ∞): We display this schematically as (see figure 2) The point G * (K) is locally stable whenever it is not a coexistence point. It is also locally stable near the left end on the intervalK 1 < K < ∞ and it is locally stable if Kγ is small.
Proof This theorem follow from Lemma 4 and Theorem 2 in section 4.5.

Some concluding remarks
Below we briefly comment on our results from the biological point of view. We start from K = 0 and reason how the dynamics changes as K increases. For small carrying capacity K the susceptible population will be kept so low that the likelihood of an infected individual spreading its disease will be too low (below 50% ) for any disease to spread. As K increases the stable susceptible population increase.
When the stable susceptible population reaches σ 1 , any increase in S * due to increased K will result in the disease 1 with highest transmission rate to be able to spread. But it can only spread until the susceptible population is equal to σ 1 . So from now on S * = σ 1 and an increases in K gives an increase of I * 1 . Disease 2, with lower transmission rates then disease 1, can not spread since it is outcompeted by disease 1.
The disease 2 can however spread through the population of infected with disease 1. Under the condition the sum of susceptibles and infected of disease 1 will be so high that disease 2 can spread. However disease 2 will only occur as a coinfection in the stable state. This is a result of the fact that we assume that coinfected individuals can only spread both disease simultaneously. The single infections of disease 2 are either outcompeted by disease 1 or they become part of the coinfected compartment. For these K the compartment of single infected of disease 1 will decrease with K. This does however not mean that disease 1 becomes less prevalent, only that it occurs more as a coinfection. The susceptibles increase for these K. This is a consequence of the assumption of the coinfection being less transmissible then single infection. When the coinfection rises the average transmission rate of the diseases decrease allowing the susceptible population to increase.
For the parameters dealt with in this paper (η * 1 > 1) it will happen that as the average transmission rate of the disease decrease eventually single infection of disease 2 will be more transmissible then disease 1 and the coinfection and will thus be able to spread as a single infection giving rise to a stable coexistence point. From there either the equilibrium point stays as a coexistence point for all large K or the single infections starts to only occur in coinfections, with disease 1 being the first to stop occurring as a single infection. The susceptible population can only increase to σ 3 at which point any increase in susceptibles would even be absorbed be the least transmittable compartment (coinfection). The sick population can by assumption not reproduce and so it must also have an upper bound. If this upper bound is large compared to σ 3 we will have a situation where a large proportion of the population is sick making coinfection far more likely to occur then single infections resulting in the diseases only occuring as coinfection. while the overall sick population can increase indefinitely. So when K is large enough the number of sick individuals will be far more then the susceptibles making coinfections far more likely to occur then single infection leading to a stable state of coinfection with no single infections.

A Implicit function theorem
Let F : R n × R m → R n be a C 2 mapping. Let us consider the equation We assume that F (0, 0) = 0 and that the matrix A := DxF (0, 0) is invertible.
Our aim is to find a solution to (63) x = x(y) such that x(0) = 0 and estimate the region where such solution exists. We fix positive numbers a and b and put Let also Ba = {x : |x| ≤ a}.
We introduce the quantities Here the above norms are understood in the following sense Here | · | is the usual euclidian norm. We also introduce where ||DyF (x, y)|| = max |ξ|=1 |DyF (x, y)ξ|.
The following result is a well known implicit function theorem. We supply it with a short proof since we want to include in the formulation a quantitative information about the solution.
Theorem 5 If the constants a and b satisfies where q < 1 and ||A −1 || is the usual operator-norm of A −1 . Then there exist a C 2 -function x = x(y) defined for |y| ≤ b which delivers all solutions to (53) from Λ.
Proof We write (53) as a fixed point problem Let us check that F maps Ba into itself and that it is a contraction operator there.
To show the first property we note that ∂y k F (tx, ty)y k dt. Since ∂y k ∂x i F (τ tx, τ y)y k x i dτ dt which guarantees that F maps Ba on to itself. For checking the contraction property we write so F is a contraction and by the Banach fixed point theorem we can conclude that there exist a unique c 1 -function x = x(y) defined for |y| ≤ b. Since F ∈ C 2 the same is true for x(y).
In the next assertion we present estimates of the derivatives of the solution x(y).

Theorem 6
The matrix DxF (x, y) is invertible for all (x, y) ∈ Λ and and where

Since
Fx k x k y i + Fy i = 0, we arrive at (57) by using (56) and definition of L.
Derivating once again (59) with respect to y we obtain with Einsteins summation index ∂ 2 ∂y i ∂y j F = Fx k x l x k y i x l y j + Fy i x l x l y j + Fx p y j x p y i + Fx p x p y i y j + Fy i y j = 0.
Solving for xy i y j we get xy i y j = −(Fx) −1 (Fx k x l x k y i x l y j + Fy i x l x l y j + Fx pyj x p y i + Fy i y j ).
Using the definitions of norms we obtain (58). If where cn,m is a positive constant depending only on n and m, then there exist a C 2 -function x = x(y) defined for |y| ≤ b and such that |x| ≤ √ b, which delivers all solution to (53) from Λ b . Moreover, the matrix DxF (x, y) is invertible for all (x, y) ∈ Λ and |DxF (x, y) −1 | ≤ C||A −1 ||, |Dyx(y)| ≤ C||A −1 ||M, where C depends only on n and m.

B Bifurcation from a degenerate bifurcation point
The results of the following section can not be considered as new. They can be deduced from the classical results from [4] and [5], see also [8] for more complete presentation and related references. Here we give another more direct presentation which are more suitable for application to to models appearing in biological applications. First, systems here are finite dimensional and have a special structure, which essentially simplifies the proofs. Second the bifurcating parameter is fixed from the begining and we are interesting in bifurcation with respect to this parameter. Therefore we present here proofs which are more addapted to our situation.
It is assumed that there exists x * ∈ R n−1 such that F (x * , 0; 0) = 0 and that the (n − 1) × (n − 1)-matrix is invertible. This implies, in particular, that the equation We assume that f (x * , 0; 0) = 0 and our goal is to construct a solution to equations (63), (64) different fromx(s). This will be achieved if we solve the problem F (x; s) = 0, f (x; s) = 0.
instead of (63), (64). We denote the Jacobian matrix of the right-hand side at the point (x * , 0; 0) by A. Direct calculations show that To find the inverse of the matrix consider the equation A(X ′ , Xn) T = (Y ′ , Yn). Then where and in what follows we assume that Θ := ∇ x ′ f A −1 ∂x n F (x * , 0; 0) − ∂x n f (x * , 0; 0) = 0. Therefore Without lost of generality we can put Xn = 1. Solving this system with respect to X ′ and putting the result in the last equation, we get −xn∇ x ′ f · D x ′ F + xn∂ x ′ G(x; s) − λ Comparing (72) and (73), we see that the first derivative of smallest eigenvalue corresponding to solutionsx andx has opposite sign.