ON EXACT SOLUTIONS TO EPIDEMIC DYNAMIC MODELS

In this study, we address an SIR (susceptible-infected-recovered) model that is given as a system of first order differential equations and propose the SIR model on time scales which unifies and extends continuous and discrete models. More precisely, we derive the exact solution to the SIR model and discuss the asymptotic behavior of the number of susceptibles and infectives. Next, we introduce an SIS (susceptible-infected-susceptible) model on time scales and find the exact solution. We solve the models by using the Bernoulli equation on time scales which provides an alternative method to the existing methods. Having the models on time scales also leads to new discrete models. We illustrate our results with examples where the number of infectives in the population is obtained on different time scales.


Introduction
Epidemic models are used for understanding infectious disease dynamics where the population dynamics is divided into compartments. In the susceptible-infectedrecovered (SIR) epidemic model, susceptible individuals may become infected, and infected individuals may recover and become immune. No other transitions are considered in this model. The structure of the SIR model dates back to Kermack and McKendrick in 1927 [11] which has provided the basic framework for almost all later epidemic models ever since. In the susceptible-infected-susceptible (SIS) epidemic model, susceptible individuals may become infected, and infected individuals may recover and revert to being susceptible.
The continuous and discrete SIR and SIS models have been investigated in a number of recent works, see [5,10,12]. One of the continuous SIR models is presented in [13] as where S(t) and I(t) are the number of susceptibles and the number of infectives at time t, respectively with constant population N and the average number of adequate contacts of a person per unit time, i.e, the transmission rate β and the recovery rate γ. The authors eliminate the variable S and obtain the second equation of (1.1) in the form of the Bernoulli equation, and by using a suitable substitution the authors find a solution to (1.1). In the recent articles [3] and [2], the authors call attention to the importance of discrete modeling of HIV-1, Pseudomonas putida bacteria and mammary tumor dynamics, respectively. Especially, comparison of discrete and continuous models of HIV-1 dynamics shows that the data collected in a clinical trial is described by the discrete models better than the continuous model, see [3]. According to our knowledge, the discrete case of system (1.1) has not been studied earlier. Therefore, our purpose is to unify and extend the continuous and the discrete systems on time scales T, nonempty closed subset of real numbers. Motivated by system (1.1) and the Bernoulli equation on time scales, see [4], we propose the SIR model on time scales in the following form If T = R, then p = −p, and system (1.2) with positive β and γ constants turns out to be system (1.1). If T = Z, then p = −p 1+p for p = −1, and system (1.2) is equivalent to the system of first order difference equations as follows (1.3) In Section 3, we find the exact number of susceptibles and infectives of system (1.2) and discuss their asymptotic behaviors. Furthermore, we illustrate the behavior of infectives of the continuous and the discrete SIR models by examples. The exact solution of the following SIS model with the initial conditions S(0) = S 0 > 0, I(0) = I 0 > 0 satisfying S 0 + I 0 = N , where β and γ are positive constants is studied in [13] while the discrete model of is studied in [1]. Motivated by system (1.4) and the Bernoulli equation on time scales, see [4], we propose the SIS model on time scales as where β, γ ∈ C rd ([0, ∞) T , R + ), S, I ∈ C 1 rd ([0, ∞) T , R + ). If T = R, then system (1.6) with positive constants β and γ is equivalent to system (1.4). However, the discrete model of (1.6) when T = Z is . Note that a different form of system (1.6) with constant coefficients is studied in [9]. In Section 4, we find the exact number of susceptibles and infectives of (1.6) and demostrate the behavior of the infectives on a quantum calculus with an example. Now let us present some preliminary concepts regarding the calculus on time scales without proofs to help understanding the key points in our main results. We refer readers to books by Bohner and Peterson [7,8] and manuscripts [4,6].

Essentials of Time Scales
There are two important operators in T. The forward jump operator σ : T → T is defined as σ(t) := inf{s ∈ T : s > t} for t ∈ T while the backward jump operator ρ : T → T is defined as ρ(t) := sup{s ∈ T : s < t}. The graininess function If t < sup T and σ(t) = t, then t is called right-dense, and if t > inf T and ρ(t) = t, then t is called left-dense. Besides, if ρ(t) < t, we say that t is left-scattered. If T has a left-scattered maximum m, Assume f : T → R is a function and let t ∈ T κ . Then, the delta (or Hilger) derivative of f , denoted by f ∆ , on T κ is defined to be the number (provided it exists) such that for given any > 0, there is a neighborhood U = (t − δ, t + δ) for some δ > 0 such that for all s ∈ U A function f : T → R is called rd-continuous provided it is continuous at rightdense points in T and its left-sided limit exists (finite) at left dense points in T. The set of rd-continuous f : The set of functions f : T → R that are differentiable and whose derivative rd-continuous is denoted by . Every rd-continuous function has an antiderivative. In particular, if t 0 ∈ T, then for t ∈ T holds for all t ∈ T κ . For any left-dense t 0 ∈ T and any > 0, let The following theorem is one of several L'Hôpital Rules on time scales.
Theorem 2.1 (Theorem 1.120, [7]). Assume f and g are differentiable on T with

The set of all regressive and rd-continuous functions
for all t ∈ T κ , while the circle dot multiplication is defined by to find a simple form of the derivative of p α on time scales.
Theorem 2.2. Suppose p ∈ R and fix t 0 ∈ T. Then the initial value problem has a unique solution e p (·, t 0 ), so called the exponential function on time scales.
and if T = q N0 = {q n : n ∈ N}, where q > 1 and q ∈ R, i.e., the quantum calculus We use the following properties of exponential functions on time scales in our proofs, see Theorems 2.36, 2.39 and 2.44 in [7]. Theorem 2.3. If p, q ∈ R and t 0 , t, s ∈ T, then (i) e 0 (t, s) = 1 and e p (t, t) = 1; One of the Variation of Constants Formulas in [7, Theorem 2.77] is stated as follows.
Theorem 2.4. Suppose p ∈ R and f ∈ C rd . Then the unique solution of the initial value problem is given by where t 0 ∈ T and y 0 ∈ R.
As we mention in the introduction, our main results are based on solutions of the Bernoulli equation on time scales of the form where α ∈ R \ 0, and the proof of the existence of solutions of (2.6) can be found in [4, Theorem 6.1].
for all t ∈ T, then Note that in the case of α = 1 in (2.6), we have Hence, the Bernoulli equation (2.6) is equivalent to where we use (2.2) and (2.7), and the solution of (2.8) with x(t 0 ) = x 0 is by Theorem 2.5.
The following inequalities, see [6,Lemma 2] and [6,Remark 2], are necessary to show the asymptotic behavior of solutions of system (1.2). For nonnegative f if −f ∈ R + , then and if f is rd-continuous, then for all t ≥ s.

An SIR Model on Time Scales
In this section, we find the exact solution to SIR model (1.2) with the initial conditions (S 0 , I 0 ). Then, we discuss the asymptotic behavior of the solutions and illustrate the behavior of infectives on continuous and discrete time scales.
Since −γ ∈ R + , from Theorem 2.4, the solution to (3.3) with D(0) = D 0 is for t ∈ [0, ∞) T , where we use Theorem 2.3 (iv). SIR model (1.2) on time scales can be rewritten as Note that the positivity of β and I implies that 1 + µ(t)β(t)I = 0 for t ∈ [0, ∞) T . By plugging S = D − I into the second equation of (3.5), we have where p is defined as in (3.2). Note that (3.6) is a Bernoulli equation in the form of (2.8). Therefore, by Theorem 2.5 when α = 1, we obtain I as in (3.1). This implies that S = D − I is obtained as in (3.1). Therefore, the proof is completed. We now consider system (1.2) with positive β and γ constants for the following examples.
Example 3.1. Let T = [0, ∞) and D = 1 in system (1.2). Then, since µ = 0, we have p = β − γ from (3.2). From Theorem 3.1, the number of infectives to the continuous SIR model with initial conditions S 0 and I 0 is given by if p = 0, that is β = γ. Moreover, if p = 0 then where we use (2.3), and so S = 1 − I.
The results in Remark 3.1 are important to analyze the asymptotic behavior of infectives and susceptibles to system (1.2) with positive constants β and γ in the following theorem. Proof. Assume that β and γ are positive constants, −γ ∈ R + , and p is as in (3.2) for t ∈ [0, ∞) T . In the proof of Theorem 3.1, we show that p ∈ R + .
The following examples illustrate Theorem 3.2, where the number of infectives is obtained for the continuous and discrete SIR models.  Figure 1 shows the number of infectives for all t ∈ [0, 50] based on the sign of p.

An SIS Model on Time Scales
We now find the exact solution to SIS model (1.6) with the initial conditions (S 0 , I 0 ) where the population size is constant. An example on quantum calculus is presented at the end of this section.

This implies that
Now from the second equation of (4.2), we get where we use (2.1) in the last step. Note that when β and γ are positive constants, (4.3) and (4.4) give SIS model (3.1) in [9].
Remark 4.2. Let β and γ be positive constants and R 0 = βN γ be the reproduction number. If q = 0, i.e., R 0 = 1, then Theorem 4.1 states that the number of susceptibles is S = N − I, where I is given as in (3.7). If q = 0, i.e., R 0 = 1, then the number of infectives is qI 0 e q (t, 0) q − βI 0 + βI 0 e q (t, 0) .  (1 + q2 n ) by (2.5). Hence, (4.5) implies that the number of infectives can be found as   Figure 3 illustrates the behavior of infectives on [0, 1024) 2 N 0 based on the sign of q. Here, the number of infectives is computed from (4.6) and (4.7) in Example 4.1.