Electronic Journal of Qualitative Theory of Differential Equations

Exact solution of the Susceptible–Infectious–Recovered–Deceased (SIRD) epidemic model is established, and various properties of solution are derived directly from the exact solution. The exact solution of an initial value problem for SIRD epidemic model is represented in an explicit form, and it is shown that the parametric form of the exact solution is a solution of some linear differential system.


Introduction
Recently there is an increasing requirement for mathematical approach to epidemic models. It goes without saying that a vast literature and research papers, dealing with epidemic models has been published so far (see, e.g., [2][3][4]7]). It seems that little is known about exact solutions of epidemic models. Exact solutions of the Susceptible-Infectious-Recovered (SIR) epidemic model were studied by Bohner, Streipert and Torres [1], Harko, Lobo and Mak [5], Shabbir, Khan and Sadiq [9] and Yoshida [11]. However there appears to be no known results about exact solutions of the Susceptible-Infectious-Recovered-Deceased (SIRD) epidemic models. The objective of this paper is to obtain an exact solution of SIRD differential system, and to derive various properties of the exact solution. Furthermore we show that the parametric form of the exact solution satisfies some linear differential system.
In Section 2 we show that a positive solution of the SIRD differential system can be represented in a parametric form, and we derive an exact solution of the SIRD differential system (1.1)- (1.4). Section 3 is devoted to the investigation of various properties of the exact solution.

Exact solution of SIRD differential system
First we need the following important lemma. Lemma 2.1. If S(t) > 0 for t > 0, then the following holds: Proof. From (1.1) and (1.3) we see that and integrating the above on [0, t] yields Therefore we obtain It follows from (1.3) and (1.4) that and hence we get for some constant C. The initial condition (1.5) implies Consequently we obtain Taking account of (2.2), (2.3) and which is the desired identity (2.1).
Proof. Since S(t) > 0 and I(t) > 0, we see that S (t) < 0, and therefore S(t) is decreasing on [0, ∞). It is trivial that S(t) is bounded from below because S(t) > 0. Hence, there exists the limit S(∞). We observe that R(t) is increasing on [0, ∞) and bounded from above in view of the fact that R (t) = γI(t) > 0 and R(t) < N. Therefore there exists R(∞). Similarly there exists D(∞). Since I(t) = N − S(t) − R(t) − D(t) and there exist S(∞), R(∞) and D(∞), it follows that there exists I(∞). Theorem 2.3. Let (S(t), I(t), R(t), D(t)) be a solution of the initial value problem (1.1)-(1.5) such that S(t) > 0 and I(t) > 0 for t > 0. Then (S(t), I(t), R(t), D(t)) can be represented in the following parametric form:

Lemma 2.4. Under the hypothesis (A 3 )
, the transcendental equation Proof. First we note that in view of (1.6). We define the sequence {a n } ∞ n=1 by It is easily seen that If a n+1 ≥ a n , then Therefore we find that a n+2 ≥ a n+1 , and hence the sequence {a n } is nondecreasing by the mathematical induction. We observe that the sequence {a n } is bounded because Since {a n } is nondecreasing and bounded, there exists lim n→∞ a n = α. Taking the limit as n → ∞ in (2.15), we have The straight line y = F(N,D,R, γ, µ)−x and the exponential curve y = (γ/(γ+µ))Se (β/γ)R e −(β/γ)x has only one intersecting point in 0 < x < N by virtue of (2.14), and so the uniqueness of α follows. We claim thatR < α < N. Since The inequality α >R follows from the following inequality We assume that the following hypothesis holds in the rest of this paper. We note that (A 5 ) is equivalent to the following where ψ(ξ) is given by (2.12).

Various properties of the exact solution
We can derive various properties of solutions of SIRD epidemic model via the differential system qualitatively, however we obtain more detailed properties directly from the exact solution of the SIRD differential system.
Furthermore, I(t) is increasing in [0, T) and is decreasing in (T, ∞).
Proof. Taking account of (2.16), we easily check that is positive for t > 0. Differentiating both sides of (2.24), we arrive at in view of (2.23). It is obvious that I (t) = 0 holds if and only if We note that in light of the hypotheses (A 1 ) and (A 5 ). Since ϕ −1 (t) < 0 and ϕ −1 (t) is decreasing on [0, ∞), we observe that I (t) > 0 [resp. < 0] if and only if t < T [resp. > T]. Hence, I(t) is increasing in [0, T) and is decreasing in (T, ∞). We find that the maximum of I(t) on [0, ∞) is given by (N,S,R,D, β, γ, µ) , and T satisfies the following inequality

Corollary 3.2. The function I(t) has the maximum at
where
It is obvious that R(t) is increasing on [0, ∞) in view of the fact that ϕ −1 (t) is decreasing [0, ∞).
Theorem 3.5. The following holds: and D(t) is an increasing function on [0, ∞) such that Proof. Taking account of (2.20), we obtain which is the desired identity (3.4).

Theorem 3.6. IfS
then there exists a number T 1 (T < T 1 ) such that I(t) is concave in (0, T 1 ), and is convex in (T 1 , ∞). IfS