A time scales approach to coinfection by opportunistic diseases

Traditional biomedical approaches treat diseases in isolation, but the importance of synergistic disease interactions is now recognized. As a first step we present and analyze a simple coinfection model for two diseases affecting simultaneously a population. The host population is affected by the \emph{primary disease}, a long-term infection whose dynamics is described by a SIS model with demography, which facilitates individuals acquiring a second disease, \emph{secondary (or \emph{opportunistic}) disease}. The secondary disease is instead a short-term infection affecting only the primary-infected individuals. Its dynamics is also represented by a SIS model with no demography. To distinguish between short and long-term infection the complete model is written as a two time scales system. The primary disease acts at the slow time scale while the secondary disease does at the fast one, allowing a dimension reduction of the system and making its analysis tractable. We show that an opportunistic disease outbreak might change drastically the outcome of the primary epidemic process, although it does among the outcomes allowed by the primary disease. We have found situations in which either acting on the opportunistic disease transmission or recovery rates or controlling the susceptible and infected population size allow to eradicate/promote disease endemicity.

We consider a population where a fraction of its individuals is infected by the socalled primary disease. We assume that mediated by this illness they can also be infected by an opportunistic or secondary disease. We focus on the impact of the opportunistic disease on the epidemiological behavior of the primary infection. We are mainly interested in the interrelation between these two infectious processes: possible feedback phenomena, strengthening effect, effect of the secondary disease on persistence primary infection thresholds, and so on, [16], [17].
This class of epidemiological problems concerns syndemics [16], an emerging conception of epidemics and public health, [16], [13]. Instead of considering each disease as an isolated fact, syndemics focuses on the consequences of possible synergistic interaction between epidemics. We note that the term syndemic is referred not only to the temporal or locational co-occurrence of two or more diseases or health problems, but also to the health consequences of the biological interactions among the current health conditions [1], [6]. Our study is also related to superinfection and coinfection problems. Superinfection means that one disease invades an organism already invaded by another disease. Many times the second invasion is carried out by different strain of the established disease. Coinfection is used when individuals are simultaneously infected by two pathogens regardless of whether both invaders establish at the same time or one of them allows the other to invade. Anyway, both diseases may compete and this competition may lead to disease coexistence or to the (competitive) exclusion of one of them. Coexisting diseases are, for instance, bacterial superinfection in viral respiratory disease or infection of a chronic hepatitis B carrier with hepatitis D virus. Superinfection by different strains of HIV which result in coinfection have been reported in [18] whereas one strain exclusion by the arrival of a different one is described in [10] or in [11].
There are many examples of syndemics: a globally common coinfection problem involves HIV and tuberculosis. In some countries, up to 80% of tuberculosis patients are also HIV-positive [19]. Other common examples are AIDS, which involves coinfection of end-stage HIV with opportunistic parasites [9] and polymicrobial infections like Lyme disease (note that this is a vector borne disease) with other diseases [14]. Other syndemics (even if not all of the related diseases are infectious ones) are the hookworm, malaria and HIV/AIDS syndemic, the chagas disease, rheumatic heart disease and congestive heart failure syndemics, the whooping cough, influenza and tuberculosis syndemic or the HIV and sexually transmitted diseases syndemic. And those are just a few examples for human population. Domestic and wild animals and plants suffer their own syndemics, for an example see [5].
Our interest relies in understanding the effect of an opportunistic disease, which may suddenly appear, on the outcome of long-term illness, which is the primary disease. Due to this long illness period, the immune system of the infected individual is weakened thus enabling the secondary invasion. This feature strongly suggests using time scales models. We distinguish a slow and a fast processes, namely, the primary and the secondary infections, respectively. For instance, this is the case for tuberculosis and influenza or AIDS and many opportunistic diseases (being the later the actual cause of death).
In spite of the impact of opportunistic diseases in public health [4], [7], [8], [20], there is a lack of mathematical models devoted to understand this phenomenon. A major difficulty in the study of such kind of models is that the resulting system consist of a large number of coupled equations.
Approximate aggregation techniques (see [2], [3] take advantage on the time scales separation to get a reduced system of lower dimension attaining asymptotic information on the complete model. Roughly, this is obtained by considering the events occurring at the fastest time scale as being instantaneous when compared to the slower ones. This consideration entails a reduction of the number of state variables and parameters needed to describe the dynamics of the system at the slower time scale.
Using such a technique we deal with syndemic problems. We analyze a model consisting of a three compartments: susceptible individuals in front of the primary disease, infected individuals by the primary disease and, among them, those infected also by the opportunistic disease.
Among other results, we find that, on the one hand, an outbreak of the opportunistic disease may allow the primary disease to persist. On the other hand, there are ranges of parameter values for which the opportunistic disease may drive infected individuals to extinction, which is a brutal way of avoiding epidemics. These values are related not only to biological considerations but also to socioeconomic conditions as prevention and education, salubrity conditions, prophylaxis, access to medicines or medical advisement. It will be apparent that, under certain conditions, the outcome of the model is more sensitive to slight changes in the opportunistic disease recovery rate than in the transmission rate, which suggests a way to face coinfection problems.
This work is organized as follows. In section 2 we state the main assumptions and build up a two time scales ordinary differential equations model describing the dynamics of a syndemic problem caused by an opportunistic disease. Furthermore, we derive the aforementioned reduced model via aggregation techniques. Section 3 is devoted to the analysis of the reduced model. It is written according to the slow time scale and incorporates the asymptotic features of the fast dynamics. In section 4 we interpret the results achieved on the aggregated model in terms of the original system. 2 The model, main assumption and notation.
We consider a population whose individuals are classified as susceptible, S, or infected, I, by a primary disease. Parameters β and γ stand for the transmission and recovery rates, respectively. We also consider demography with no vertical transmission on newborns, although infected individuals have an extra mortality rate and lower reproduction rate when compared with susceptible individuals. As mentioned before, these processes, demography and the primary disease, evolve at the slower time scale and the primary disease rates are comparable with demographic rates. Thus, we assume that individuals are homogeneously mixed. Thus all of them have the same probability of meeting an infected individual, so that we consider mass action transmission law [12]. Summing up, the dynamics of the primary disease model with demography are given by the following system where r > 0 and m > 0 stand for the growth and death rates, respectively. Parameters a ∈ (0, 1) and µ > 0 describe, respectively, the decline of reproduction rate and the increase of mortality rate due to the primary disease. Besides, k, c, f, g measure the intra-specific pressure due to population on each compartment and between compartments. We stress once again that the disease is assumed not to be vertically transmitted.
We assume that those infected by the primary disease are also susceptible or infected by a secondary, opportunistic disease. Then, infected individuals I are classified as susceptible to the secondary disease U (that is, the number of individuals infected only by the primary disease) and coinfected V , i.e., those individuals infected by both primary and secondary diseases. We stress that the important modelling assumption we make here describes the contacts of the opportunistic disease as fast processes, for which individuals cannot meet the whole population. Therefore, the standard transmission law is appropriated to model [12]. The equations describing the behavior of the opportunistic disease constitute the well known SIS model where δ and λ stand, respectively, for the recovery and transmission rates.
Finally, incorporating the opportunistic disease dynamics (2) into the primary disease equations (1) along with the time scales assumption yields the so-called complete two time scales system where ε > 0 is a parameter measuring the difference of time scales. Note that coinfected individuals are also reproductive individuals although their reproduction rate is even smaller than that of individuals infected by the primary disease, 0 ≤ b < a < 1, i.e. b = 0 if coinfected individuals become non-reproductive. The parameter ε is small enough so that we can use approximate aggregation techniques to analyze the system. First, we need to write (3) in an appropriate slow-fast form. Let us recall that I = U +V is the total amount of infected individuals regardless of whether they are infected by one or two diseases. Then, we can study system (3) by means of a two dimensional one. The interested reader can find the technical details and examples, for instance, in [2]. In order to get the mentioned two dimensional model, we change variables (S, U, V ) → (S, I, V ), which yields and set ε = 0, bringing system (4) into the form Note that, indeed, the last equation in system (5) corresponds to the classical SIS epidemic model following the standard transmission law, in this case relative to infectedcoinfected individuals U = I −V and V , respectively. It is well known (see, for instance, [12]) that the proportion of infected and coinfected individuals is, respectively, Getting back to system (5), it is clear that the variables S and I are constant, and that for each value of I it follows that where µ v is defined in (6). We point out that µ v I is referred to as the fast equilibrium (see, for instance, [2]). Summing up, the solutions (S, I, V ) of system (5) tend to (S, I, µ v I). Furthermore, thanks to the approximate aggregation techniques, for ε > 0 small enough the solution (S, U, V ) of system (3) tends to (S, µ u I, µ v I) .
Then, taking into account that (U, V ) rapidly approaches (µ u I, µ v I), the reduced system is obtained from the equations for S and I in system (4) replacing U and V by their respective limits µ u I and µ v I along with a change on the independent variable t = ετ . This procedure, the quick derivation method (whose mathematical foundations can be found in [2], [3], [15]) gives rise to the aggregated system where 3 Analysis of the model Thanks to the approximate aggregation technique the original system (3) can be analyzed by means of the aggregated system (8). Note that expression (6) implies that the coefficients (9) change depending on whether λ/δ < 1 or λ/δ > 1. This fact is related to whether there is coinfection or not, which is discussed in section 4 using the results achieved in the present section.

Equilibria and stability.
We begin this section with some preliminary comments. It is straightforward to see that E * 0 = (0, 0) and E * 1 = (Ŝ 1 , 0) withŜ 1 = (r − m)/k, are equilibrium points for the system (8). We name them respectively the trivial and the semi trivial equilibrium,.
in view of the value attained byŜ 1 . As for the endemic equilibrium of (8), direct calculations show that The roots of Φ(S) are S = 0 and S =Ŝ 1 , while S = A/B is asymptotic to Φ(S). Figure 1 qualitatively displays the function Φ(S) depending on the sign ofŜ 1 , and when positive, whether it is larger or smaller than A/B. It is immediate that D < 0 implies that there is no feasible equilibrium point. On the contrary, when D > 0, in cases (a) and (b) there can be either zero, one or two feasible equilibrium points. In case (c) instead there is one or no feasible equilibrium point. In the following we analyze these situations describing the dynamics of system (8); namely, Theorem 1 Assume that D < 0. Then, there is no coexistence equilibrium point for system (8). Furthermore, 1. If r < m then E * 0 is globally asymptotically stable (GAS) for system (8).
2. If r > m then E * 0 is unstable and E * 1 , with nonegative population levels, is GAS for system (8).
Thus, there is a transcritical bifurcation for r = m.
Proof : The fact that D < 0 prevents the existence of strictly positive equilibrium point for system system (8) follows in a straightforward way from the second equation of system (8).
We denote by F (S, I) the flow of system (8) at (S, I). Given an equilibrium point E = ( S, I) of system (8), we note JF ( E) the jacobian matrix of F at E, It is straightforward to see that Then, from the signs of the eigenvalues of JF (E * 0 ), E * 0 is locally asymptotically stable (unstable) if r < m (if r > m). In the same way, arguing on the signs of the eigenvalues of JF (E * 1 ) we find that E * 1 is locally asymptotically stable if r < m (since D < 0) and unstable if r > m.
Next, we prove the global stability of E * 0 when D < 0 and r < m, which, recalling (10), impliesŜ 1 < 0, (case (a)). First, we show that the solutions of the system (8) are bounded from above.
In figure 3.1 (a), the non negative cone is divided in three regions: region I lies in between the vertical axes and the positive branch of Φ(S), region II is the region between the positive branch of Φ(S) and the vertical line S = A/B and region III lies on the right of the vertical line S = A/B.
We note that trajectories cannot enter region III. In fact, for S = A/B, because (10) does not hold, we find Indeed, trajectories starting in region III leave this region after a transient time. Consider the first equation in system (8). If S ≥ A/B so that A − BS ≤ 0 we have i.e. S(t) is strictly decreasing and its derivative is negative and bounded away from zero as long as S ≥ A/B. Therefore, for any solution S(t) of system (8) starting at S 0 ≥ A/B it follows that S(t) ≤ S 0 e ((r−m)A/B−kA 2 /B 2 )t → 0 as long as A/B ≤ S(t). Thus, there exists t 0 > 0 (which depends on S 0 ) such that 0 < S(t) < A/B, ∀t > t 0 . Then, any trajectory with initial value in region III will leave this region (entering into region II) and will never again get back.
The function Φ(S) is a nullcline for S (t). Then, S (t) > 0 in region I while S (t) < 0 in region II. On the other hand, since D < 0, from the second equation of system (8) we get Then, solutions are strictly monotone in regions I and II. Consider a solution (S(t), I(t)) with initial value in region I. Then either the omega limit of (S(t), I(t)) consists of an equilibrium point (and it is (0, 0)) or the trajectory (S(t), I(t)) leaves region I, reaching the curve defined by Φ(S). If (S(t), I(t)) meets Φ(S) at t * , then (S (t * ), I (t * )) = (0, −ξ), ξ > 0 which means that the trajectory enters into region II. Furthermore, trajectories starting in region II cannot leave region II. Summing up, any trajectory starting in regions I or III enters into region II or reaches the origin. Furthermore, trajectories are strictly monotone in region II and thus, arguing as before, any trajectory in region II reaches the origin.
Proceeding similarly, it is also easy to prove the global stability of E * 1 when (10) is satisfied.
We now continue assuming that D > 0 and the examine cases (a), (b) and (c) illustrated in figure 3.1. In case (a) we have the following result.
Theorem 2 Assume that D > 0 and r < m. Then, 1. E * 0 is locally asymptotically stable and E * 1 is infeasible.
(b) If R = 1 there is a single interior equilibrium point E * = (S * , I * ).
(c) If R < 1 there is no interior equilibrium point and E * 0 is GAS stable.
Remark: The equilibrium point arising in case 2 (b) is not hyperbolic. We skip its analysis because it is not relevant for the general analysis. Istead, it would require a considerable amount of work involving singular perturbation theory. This underlaying mathematical tool does not directly apply to our considerations here.
Proof : Statement 1 follows easily by linearizing the flow of the system at E * 0 as in the proof of theorem 1.
As for statement 2, the equilibrium points lying in the interior of the first quadrant are the solutions of Φ(S) = Ψ(S).
Assuming S = B/A, this is equivalent to solve the following equation Using Decartes Rule the existence of either no or two positive equilibrium points (counting multiplicities) follows immediately. Furthermore, the fact that R is larger, equal or smaller than 1 is equivalent to the discriminant of equation (13)  We analyze the stability by means of the well known trace-determinant criterion. Recalling expression (11) for the Jacobian for the flow of system (8) at the equilibrium point E * = (S * , I * ), taking into account that Φ(S * ) = I * = Ψ(S * ), (11) simplifies to This immediately yields trJF (E * ) < 0. Furthermore, a direct calculation lead to Using again the fact that I * = Ψ(S * ) if and only if EI * = DS * − C, we have detĴ so that detJF (E * ) > 0 is equivalent to which entails local stability. On the contrary, E * is instable if The equilibrium points E * uns and E * 2 can be explicitly calculated from (13) and direct calculations show that E * uns (respectively, E * 2 ) fulfills condition (15) (respectively, condition (14)).
Proceeding as for the proof of theorem 1, it is easy to prove statement 2 (c).
We next analyze case (b) in figure 3.1.
(b) If R = 1 then there is a single interior equilibrium point E * = (S * , I * ).
(c) If R < 1 then there is no interior equilibrium points and E * 1 is GAS stable.
Proof : It can be easily accomplished proceeding as in the proofs of Theorems 1 and 2.
And, finally, we examine case (c) in figure 3.1.
Proof : Statement 1 follows in a straightforward way analyzing the eigenvalues of JF (E * 0 ). As for statement 2 (a), we get also the local stability analyzing the eigenvalues of JF (E * 1 ). Again, the global stability follows from an argument based on the regions of the positive cone where the sign of the derivatives S (t) and I (t) remains constant (as we did in the proof of theorem 1.
The local stability of E * 2 follows from the proof of theorem 2. The proof is completed by reasoning as we sketched in statement 2.

Conclusions
This work aims at ascertaining the impact of an opportunistic disease outbreak in a population already affected by a primary disease. From our results in section 3 the following relevant quantities have been identified: • The relative size of the nonzero interceptsŜ 1 andŜ 2 with the horizontal axis of the nullclines I = Ψ(S) and I = Φ(S) relative to the S and I population dynamics, respectively.
• The reproductive number λ/δ of the opportunistic disease whose dynamics is expressed by system (2).
• The parameter D defined in (9) arising for the aggregated system (8), which represents the recruitment rate of new infected individuals I due to their interaction with susceptible individuals, including competition and recruitment via primary disease infection.
• The parameter R defined in (12) by both epidemiological and demographic parameters. It therefore represents a combination of the reproduction number for the primary disease and a demographic threshold. The primary disease persists or gets eradicated depending on whether R is larger or smaller that 1, but in this latter case the whole population is wiped out.
It is apparent that all the aforementioned quantities depend on λ and δ, the epidemiological parameters that drive the opportunistic disease. Indeed, as shown next, the opportunistic disease may change the overall outcome of the primary epidemic process. In order to interpret the results achieved in the previous section 3, let us recall that E * 0 stands for the population extinction state, E * 1 is the disease-free equilibrium, while E * 2 notes the endemic disease state. In terms of the complete model, E * 2 gives rise to the equilibrium point (S * , U * , V * ) = (S * , µ u I * , µ v I * ) where (µ u , µ v ) are given by (6). Thus, assuming that E * 2 exists, the opportunistic disease is able to invade the infected population if and only if δ/λ > 1.
We have found that the population can persists even if (10) does not hold, r < m i.e. the natural death rate is larger than the growth rate, via coinfection. It is apparent that r < m implies that the semitrivial disease-free equilibrium E * 1 is not feasible and the trivial equilibrium corresponding to population extinction is locally asymptotically stable. In this case, and according to theorem 2, there exists a locally asymptotically stable interior equilibium whenever D > 0 and R > 1. This entails population persistence. This apparent paradox is possible because there is no vertical transmission, so that newborns from infected individuals enter into the class of susceptible individuals. At the same time condition D > 0 is required. This means that in general the competition effect on infected individuals with susceptible individuals is negligible when compared with recruitment from susceptible individuals due to infection by the primary disease. Summing up, there is a kind of double feedback allowing population to persist.
We analyze now the effect of the opportunistic disease on the epidemic outcome when the population persists, meaning that E * 0 is asymptotically unstable, i.e. when (10) is satisfied. We also assume D > 0, a necessary (but no sufficient) condition for the existence of positive equilibrium points. We already know that condition 1 >Ŝ 1 /Ŝ 2 makes the disease-free equilibrium E * 1 at least a local attractor, which otherwise is unstable: • When 1 >Ŝ 1 /Ŝ 2 , the value of R determines whether E * 1 is GAS or there exists also an asymptotically stable endemic disease equilibrium. When the primary disease is established, coinfection insurgence depends on the relative size of δ and λ (see equation (6)).
• When 1 <Ŝ 1 /Ŝ 2 , there is a unique asymptotically stable interior equilibrium point, meaning that at least the primary disease becomes endemic. Again, the opportunistic disease may also invade the population giving rise to coinfection depending on the relative size of δ and λ.
So we analyze the effect on R andŜ 1 /Ŝ 2 of varying the parameters λ and δ. Observe thatŜ So it is clear thatŜ 2 depends on δ/λ since µ u = δ/λ and µ v = 1 − δ/λ (see expression (6)). We claim thatŜ 1 /Ŝ 2 decreases as λ/δ = 1/µ u increases. Let us note ξ(µ u ) := S 1 /Ŝ 2 (µ u ); direct calculations show that Taking into account that we are assuming m < r and D = cµ u + eµ v + βµ u > 0, so that from (9) the latter implies β > f , we get that ξ(µ u ) is an increasing function. Therefore, ξ(µ u ) decreases as λ/δ = 1/µ u increases (assuming λ/δ > 1), as we wanted to show. Then, ifŜ 1 /Ŝ 2 < 1 in the region λ/δ < 1, this inequality holds everywhere in the δ − λ plane. For instance, under the hypotheses of theorem 3, it means that if E * 1 is AS then it is so regardless of the values of δ and λ. Now, depending on the values of R, the disease-free equilibrium can be GAS if R < 1, or an endemic AS equilibrium exists when R > 1. Figure 4 displays R as function of δ and λ, where parameter values satisfy the assumptions (3a) and (3c) of theorem 3. From figure 4 gather the following information 1. Whenever R < 1 the equilibrium E * 1 is GAS. Note that this region overlaps with the set λ/δ > 1. That is, if we were considering only the fast dynamics given by (2), the opportunistic disease would be able to invade the population. Anyhow, there is no primary disease so that neither the opportunistic disease is present, because it can survive only in the presence of the primary disease. In other words, the synergistic behavior between the two diseases is not strong enough to lift the system to the epidemic state.
2. On the other hand, there is a region where R > 1 is included in the set λ/δ > 1.
In this case equilibrium E * 2 exists and it is locally AS. A separatrix curve exists partitioning the domain into the basins of attraction of equilibrium E * 1 and In this case, and in order to control the epidemics, i.e. switching the epidemiological state, acting on the recovery rate δ rather than on the transmission rate λ seems to be more effective. This remark could constitute a relevant clue to manage epidemic outbreaks. IfŜ 1 /Ŝ 2 > 1 in the region λ/δ < 1, this inequality may be reversed in the region λ/δ > 1. This is interpreted as follows: the primary disease is endemic in the region λ/δ < 1, where the opportunistic disease cannot invade the population. As λ/δ crosses the threshold value 1, coinfection arises. At the same time,Ŝ 1 /Ŝ 2 decreases as λ/δ > 1 increases. ThenŜ 1 /Ŝ 2 may fall below 1, so that E * 1 becomes AS. Again, the value of R determines whether the disease-free equilibrium is GAS or an asymptotically stable endemic equilibrium may exist. Figure 3 displays the above mentioned situation. The apparently contradictory behavior of a strengthening opportunistic disease that implies diseases disappearance is explained as follows: as λ/δ crosses the threshold value 1 the coefficient µ v turns positive and D decreases towards 0 if λ/δ grows. Keeping in mind expression (16), we see that the coefficient n, the effect on coinfected individuals of competition with susceptible individuals, enters into the scene with increasing weight. It means that coinfected individuals become weaker because of the coinfection and, if λ/δ is large enough, i.e. when there is a large fraction of coinfection among infected individuals, competition with susceptible individuals wipes out the infected individuals.