Periodically Forced Discrete-time Sis Epidemic Model with Disease Induced Mortality

We use a periodically forced SIS epidemic model with dieases induced mortality to study the combined effects of seasonal trends and death on the extinction and persistence of discretely reproducing populations. We introduce the epidemic threshold parameter, R 0 , for predicting disease dynamics in periodic environments. Typically, R 0 < 1 implies disease extinction. However, in the presence of disease induced mortality, we extend the results of Franke and Yakubu to periodic environments and show that a small number of infec-tives can drive an otherwise persistent population with R 0 > 1 to extinction. Furthermore, we obtain conditions for the persistence of the total population. In addition, we use the Beverton-Holt recruitment function to show that the infective population exhibits period-doubling bifurcations route to chaos where the disease-free susceptible population lives on a 2-cycle (non-chaotic) attrac-tor.

In this paper, we focus on the impact of seasonal factors on a discretetime SIS (susceptible-infected-susceptible) epidemic model with disease induced mortality.When the environment is constant and the disease is not fatal our model reduces to the SIS epidemic model of Castillo-Chavez and Yakubu [9,10,11].However, when the environment is constant and the disease is fatal, our model reduces to that of Franke and Yakubu [25].Hwang and Kuang [34,39] as well as Berezovsky et al [6,7] illustrated surprising dynamics in a simple continuous-time susceptible-infected (SI) model with variable population size and disease-induced mortality.In particular, in the continuous-time SI model, Berezovsky et al [6,7] showed the existence of homoclinic trajectories that allow the possibility of outbreak of the disease at very low levels of infection.In this paper, we use a periodically forced discrete-time SIS model with disease induced mortality to show that a small number of infectives may cause the total population to go extinct although R 0 > 1.
To construct our compartmental model, we assume that a disease invades and subdivides the target population into two classes: susceptibles (noninfectives) and infectives.Prior to the time of disease invasion, the population is assumed to be governed by a periodically forced demographic equation with a periodic recruitment function.Hence, the population is assumed to be either at a demographic "steady state" (an attracting cycle or a chaotic attractor) or at a periodic geometric growth rate.The transition from susceptible to infective is a function of the contact rate α (between individuals) and the proportion of infectives (prevalence) in the population.
We derive the epidemic threshold parameter, R 0 , for predicting disease persistence or extinction in periodic environments.Franke and Yakubu, in [25], showed that in constant environments, when R 0 > 1 a tiny number of infected individuals can drive an otherwise persistent population to extinction.We extend this result to include periodic (non-constant) environments.In particular, we use numerical simulations and a periodically forced Beverton-Holt recruitment function to illustrate this in a specific example [8].In addition, we show that it is possible for the infective population to exhibit period-doubling bifurcations route to chaos while the disease-free (susceptible) demographic dynamics is cyclic but nonchaotic [30].
The paper is organized as follows.In Section 2, we introduce the periodically forced demographic equation with disease induced mortality and the main model, a periodically forced discrete-time SIS epidemic model with disease-induced mortality.We extend, in Section 3, the results of Franke and Yakubu on periodically forced SIS epidemic model without disease-induced mortality.The demographic threshold parameter R Di and the basic reproductive number R 0 are introduced in Section 4. These are used to predict the (uniform) persistence or extinction of the infective population in the SIS model.Section 5 concerns applications of the persistence and extinction results to an SIS model with the Beverton-Holt recruitment function.Also in Section 5, we illustrate period-doubling bifurcations in the epidemic model where the disease-free susceptible dynamics is cyclic and non-chaotic [41].The implications of our results are discussed in Section 6.

Demographic equations with seasonality
In constant environments, theoretical discrete-time epidemic models with diseaseinduced mortality are usually formulated under the assumption that the dynamics of the total population size in generation t, denoted by N (t), is governed by equations of the form where γ 1 and γ 2 ∈ (0, 1) are respectively the constant "probabilities" of surviving of the susceptibles and infectives per generation and f : R + → R + models the birth or recruitment process; where γ 1 ≥ γ 2 [25].
To introduce seasonality into (1), we p-periodically force the recruitment function and the survival rates.This is modeled with the p-periodic demographic equation where ∃ p ∈ N such that and for each i ∈ {1, 2} where γ it ∈ (0, 1).We assume throughout that f (t, In the absence of disease, I(t) = 0, Model (2) reduces to the disease-free demographic equation Franke and Yakubu, in [28], studied model (3) with periodic constant recruitment function with periodic Beverton-Holt recruitment function and with periodic Ricker function [42] where the carrying capacity k t and the survival rate of the susceptibles γ 1t are p-periodic, k t+p = k t and γ 1(t+p) = γ 1t for all t ∈ Z + [16,28].Franke and Yakubu proved that periodically forced recruitment functions can generate globally attracting cycles in Model (3) [28].In the following result, Franke and Yakubu obtained a globally attracting cycle for the periodic constant recruitment function.
Then Model (3) has a globally attracting q -periodic cycle that starts at , where q divides p.
Next, we state the result of Franke and Yakubu on globally attracting cycles for convex monotone recruitment functions [28].
Theorem 2 If the recruitment functions of x for each t, f(t, x), are increasing, concave down, bounded, pioneer functions, then the p− periodic demographic system, Model (3), has a globally attracting positive q -cycle, where q divides p.
By Theorem 2, Model (3) has a globally attracting cycle whenever the recruitment function is the periodically forced Beverton-Holt model.
Theorems 1 and 2 imply that in the absence of the disease, the susceptible population is asymptotically periodic (bounded) and lives on a cyclic attractor when the recruitment function is either a periodic constant or the periodic Beverton-Holt model.Denote this cycle by {S 0 , S 1 , . . ., S q−1 }.
When new recruits arrive at the periodic positive per-capita growth rate λ t , then where λ t+p = λ t for all t ∈ Z + .The solution to the disease-free equation, Model , is and the demographic basic reproductive number is If R D < 1, the total population goes extinct at a geometric rate, and if R D > 1, the total population explodes at a geometric rate.In constant environments, p = 1, λ J = λ, γ 1J = γ 1 and R D reduces to In [25], Franke and Yakubu used R D1 = λ to study the long-term behavior of geometrically growing disease-free state in constant environments.
To introduce our periodically forced SIS epidemic model with disease-induced mortality, we assume that infective individuals recover with constant probability (1 − σ).Furthermore, we assume that is a monotone convex probability function with φ(0) = 1, φ ′ (x) < 0 and φ ′′ (x) ≥ 0 for all x ∈ [0, ∞).Also, we assume that the susceptible individuals become infected with nonlinear probability 1 − φ α I N per generation, where the transmission constant α > 0. When infections are modeled as Poisson processes, for example, then ]).
Our assumptions and notation lead to the following periodically forced SIS epidemic model with disease induced mortality: We assume throughout that ∃ p ∈ N such that and for each i ∈ {1, 2} where γ it , σ ∈ (0, 1), reduces to the SIS epidemic model of Franke and Yakubu when the environment is constant, f (t, N (t)) = f(N (t)) and γ 2t = γ 2 ≤ γ 1t = γ 1 [25].In Model (5), the total population in generation t + 1 (N (t + 1) = S(t + 1) + I(t + 1)), the sum of the two equations of the model, is Equation (2).

Preliminary Results
Here, we obtain some auxiliary results that will be used to study disease persistence and extinction in our periodically forced SIS model.In the following result, we obtain one-variable bounds on the total population of the model.

Lemma 3
In Model (5), Using the substitution S(t) = N (t)−I(t), the I-equation and the N -equation in Model (5) become and respectively.
On the closed interval [0, N ], let When F N,t has a unique positive fixed point and a unique critical point, we denote them by I N,t and C N,t , respectively.The sets of sequences generated by and are the sets of density sequences generated by the infective and the total population equations, respectively.
To study Model (5), we need the following results on the properties of F N,t and G N,t .
Lemma 4 F N,t (I) and G N,t (I) satisfy the following conditions.

Proof. (a)
Since It is easy to check that the equality holds if and only if (N, I) = (0, 0).(b) (c) Since φ ′ < 0 and φ ′′ ≥ 0 on [0, ∞), we have 0)I is the tangent line to the graph of F N,t (I) at 0. Since F N,t is concave down on [0, N ], its graph is below the tangent line at the origin on [0, N ].Hence, the graph of F N,t (I) starts out higher than the diagonal and must cross it before I = N.The concavity property of F N,t (I) (see (c)) implies that there is a unique positive fixed point. (f) (h) Topological conjugacy preserves critical points.The result follows from (f ).(i) ).
Note that I N 1 < I N 0 .Since the graph of F 1,t goes through the origin with positive slope and is concave down, the ray through the origin and I N1 , F 1,t ( I N1 ) has a larger slope than the ray through the origin and I N0 , F 1,t ( I N0 ) .The first ray contains the point I, N 1 F 1,t ( I N1 ) , while the second ray contains . Next, we obtain the invariance of the positive quadrant.

Disease Extinction or Persistence
To study the qualitative dynamics of Model (5), we define the p − periodic dynamical system where H t = H t+p for all t and Ω = {(N, I)|0 ≤ I ≤ N }.Since φ is a decreasing function and 0 ≤ I N ≤ 1, {H t } is a continuous p − periodic dynamical system that is C 1 away from (0, 0).Since φ is a decreasing function and 0 ≤ I N ≤ 1, {H t } is a continuous p − periodic dynamical system that is C 1 away from (0, 0).We will study the behavior of solutions to Model (5) by analyzing H t on the triangular region Ω.
Lemmas (4) and (5) show that the set of iterates of the p−periodic dynamical system {H t } on {(N, I)|0 ≤ I ≤ N } is equivalent to set of density sequences generated by Model (5), where denotes the i − th component of the t − th iterate (under {H t }) of the initial condition (N, I).

Definition 6
The total population is uniformly persistent under {H t } if there exists a constant η > 0 such that The total population is said to be persistent under [48].Consequently, uniform persistence implies the persistence of the total population.

Definition 7
The total population is driven to extinction under for every initial condition.
For each i ∈ {1, 2}, define the p − periodic dynamical system By Lemma 3, Now, we introduce the demographic threshold parameter In the following lemma, we show that R D 1 > 1 implies the uniform persistence of the susceptible population while R D1 < 1 implies local extinction of the total population.
Lemma 8 Let f(t, 0) = 0 for all t.If R D1 > 1, then the disease-free susceptible population described by Model ( 3) is uniformly persistent.However, if R D1 < 1 then {(0, 0)} is locally asymptotically stable in Model (5), and both the susceptible and infected populations go extinct at low values of initial population sizes.
Since R D1 > 1, there is a κ > 0 such that N ∈ (0, κ) implies that for all t.By taking κ smaller if needed, we can assume that each D 1,t is increasing on (0, κ).Note that Let ⌊x⌋ be the greatest integer less than or equal to x.If the j th image of N is N * and if it and the next n images under the p-periodic dynamical system {D 1,t } are in (0, κ), then Thus the orbit must leave (0, κ) and Π p−1 t=0 γ 1t N * is a lower bound for these values.
D 1,t is positive on the compact set A κ , and it has a minimum Hence the disease-free susceptible population is uniformly persistent.We now consider the case where )+γ 1t > 0, hence there is an interval containing 0 on which D 1,t is increasing.So if N is small the continuity of H t gives Inductively, we obtain that if N is small then whenever N ∈ (0, κ).Therefore {(0, 0)} is locally asymptotically stable.N = S + I implies the extinction of the susceptible and infected populations at positive small initial values of S and I.
Lemma 9 If either f (t, 0) > 0 for some t or f(t, 0) = 0 for all t and R D 2 > 1, then the total population is uniformly persistent.
Proof.First, we consider the case f(t * , 0) > 0 for some t * .Since lim Thus a nonzero initial condition gets above m * and stays above ) is larger than m * and stays larger than Π p−1 t=0 γ 2t m * for every t > 0. Hence the total population is uniformly persistent.Now, we consider the case f (t, 0) = 0 for all t and R D2 > 1.Since D 2,t is continuous and Inductively, we obtain that if N is small then H 1 is positive on the compact set A κ , and it has a minimum κ > 0 on A κ .Consequently, for all nonzero initial conditions and the total population is uniformly persistent.
By Lemma (9), the total population is uniformly persistent when the recruitment function is periodically constant, When the recruitment function is the periodic Beverton-Holt function, , and Π p−1 t=0 (µ + γ 2t ) > 1 then the total population is uniformly persistent.When the recruitment function is the periodic Ricker function and Π p−1 t=0 ((1 − γ 1t )e r + γ 2t ) > 1 then the total population is uniformly persistent.
Next, we obtain that all positive initial conditions are attracted to a trapping region of our epidemic model.
then there is a compact subset, W, of cylinder space {(N, attracts all initial conditions under {H t } iterations.That is, there is no population explosion.
There is an N > 0 such that N ≥ N implies is a compact subset of the cylinder space which attracts all initial conditions under {H t } iterations.Hence, all orbits are bounded.
To prevent population explosion in our p − periodic epidemic model, we assume throughout that Next, we prove that R 0 < 1 implies disease extinction whereas R 0 > 1 together with the persistence of the total population implies persistence of the disease.
That is, the disease goes extinct.(b) If R 0 > 1 and the total population is uniformly persistent, then lim That is, the disease is uniformly persistent.
The proof of Theorem 11 is in the Appendix.In most epidemic models, R 0 > 1 implies disease persistence.However, in our SIS epidemic model with disease induced mortality, we obtain sufficient conditions that guarantee total population extinction for some initial conditions, where R 0 > 1.
Theorem 12 Let R 0 > 1, f (t, 0) = 0 and f(t, N ) ≤ f ′ (t, 0)N for all t and N > 0. If γ 1t * > γ 2t * for some t * , then there is a ς > 1, such that if 1 < R D1 < ς then the total population goes extinct under {H t } iteration whenever I(0) > 0. The ς can be chosen so that it is independent of f.The proof of Theorem 12 is in the Appendix.

Application
In this section, we use the periodically forced Beverton-Holt model with its specific recruitment function to illustrate the general results from the previous sections [43].
With our choice of parameters, the disease-free dynamics are governed by the Beverton-Holt model and the susceptible population persists.That is, in the absence of the disease, the susceptible population exhibits a globally attracting positive 2-cycle.Moreover, R 0,t = −αγ 1t φ ′ (0)+γ 2t σ and Hence, all of the hypotheses of Theorem ( 12) are satisfied and our numerical results show that 0.11 < a < 0.15 gives the extinction of the total population predicted by the theorem (see Figure (1)).In Figure 1, a = 0.12.The population without disease exhibits an attracting period 2 cycle from 1.589 to 1.513.However, adding a small number of infectives first increases the number of infectives and then drives the total population to extinction.
Figure 1: A small number of infectives drives an otherwise peristent total population to extinction.
The behavior of the orbit in Figure 2 is a bit surprising.By our simulations, as in Figure 2 most positive orbits seem to move along a line and then turn and crash.Understanding the basin of attraction of the origin is an interesting open problem.

Bifurcation Diagram
Without infectives the total population in the example is governed by the Beverton-Holt model.So under 2-cycle survival rates, the population exhibits an attracting 2-cycle.However, the dynamics of the infective population can be much more complicated.In fact the infective population can exhibits a period doubling route to chaos.To illustrate this, we make the following choice of parameters in the example.To illustrate the structure of the chaotic attractor depicted in Figure 2, we let α = 250 and keep all the other parameters fixed at the same values as in Figure 1. Figure 3 shows a complicated structure for a 4-piece chaotic attractor in the (N, I) − plane [47].To illustrate the region that leads to extinction, we let That is, γ 2 is the average of the γ 2,i .In the (a, γ 2 ) -parameter space of the Beverton-Holt model, we continuously vary the intrinsic growth rate a between 0 and 1, and γ 2 between 0 and 0.9 leaving γ 2,1 − γ 2,2 = 0.1, where all the other parameters are kept fixed at the same values as in Figure 1. Figure 4 shows that in (a, γ 2 )-space, the species goes extinct at low values of the intrinsic growth rate whenever R D2 < 1.

Conclusion
The study of the combined effects of disease induced mortality and seasonal trends on the control of diseases in discretely reproducing populations has received little attention.In this paper, we focus on the joint impact of periodic environments and disease induced mortality on the persistence or extinction of discretely reproducing populations.We formulated and analyzed a periodically forced discrete-time SIS epidemic model with disease induced mortality.For our model, we computed the basic reproduction number, R 0 , and used it to investigate the relationship between disease persistence and extinction.We showed that R 0 < 1 implies the extinction of the infective population.
In constant (non-periodic) environments, Franke and Yakubu, in an earlier work, used a discrete-time SIS epidemic model with disease induced mortality to show that a tiny number of infectious individuals can drive an otherwise persistent (susceptible) population to extinction whenever R 0 > 1.In this paper, we extend this result to include periodic (non-constant) environments.That is, in SIS models with diseases induced mortality, when the environment is either constant or periodic and R 0 > 1, a tiny number of infective population can drive the total population to extinction.In addition, we obtained conditions that guarantee the persistence of the total population whenever R 0 > 1 and the environment is periodic [25].
Periodically forced models are known to exhibit oscillatory and chaotic dynamics [25].In this paper, we used a periodically forced Beverton-Holt model to illustrate period-doubling bifurcations route to chaos in SIS models with disease induced mortality where the disease-free susceptible population exhibits a cyclic and non-chaotic attractor.In addition, we use an example to highlight that the structure of the chaotic attractors can be very complicated.It is also possible that a continuous-time model with an infective period of fixed length might exhibit similar dynamical behaviors.Studies of analogous questions for SIR epidemic models are in the process and will be reported elsewhere.
(a) Since I(0).Thus, and inductively The sequence {I(t)} is dominated by the decreasing sequence which converges to 0. Hence, (b) By Lemma (5), since I(0) > 0 we have I(t) > 0 for all t ∈ Z + .Lemma (4) gives Note that F N,t (N, 0) = 0 for all t and N. The derivative of By uniform persistence of the total population there is a η * > 0 such that eventually and it remains true for larger t values.By Lemma (10) there is a η * > 0 such that eventually That is, the disease is uniformly persistent.
Proof of Theorem 12. Let 0 < β ≤ 1.Now, we investigate the ray through the origin with slope β.If a point (N(t), I(t)) on the orbit of a positive initial condition (N (0), I(0)) is on this ray then I(t) = βN (t).To calculate the slope of the ray that contains the image of this point under H t , we have The new slope is Since f (t, N ) ≤ f ′ (t, 0)N for all t and N > 0, The limit, lim By taking ε 0 even smaller, if necessary, we can assume that for each t Also there is a β 0 > 0 such that for each β ∈ (0, β 0 ] and for each t Hence when β and m N(0),0 (β) ∈ (0, β 0 ], Let j be a nonnegative integer.Inductively, as long as the iterations, m In particular , is independent of f.
The first p-1 iterates could decrease but they stay above  Thus, the slopes eventually get larger than β 0 > 0.
Thus, the slopes grow when they are small until they reach at least β 0 .They can then decrease but not less than β in one step.The next few steps could take a slope lower before it starts increasing geometrically.The smallest it could get is  Thus, the lim inf of the slopes is at least Note β η is independent of f.We now need to show that the total population decreases when β ≥ β η .This will be accomplished by taking 1 < R D 1 < ς 2 for some appropriate ς 2 that is independent of f .Since the t th iterate of (N(0), βN(0)) is < a < 0.15, b t = 1.2 + (−1) t * 0.1, α = 5, γ 1t = 0.9 + (−1) t * 0.05, γ 2t = 0.8 + (−1) t * 0.05, and σ = 0.9.