In this section, we analyze the local and global stability of the disease-free equilibrium and the global stability of the endemic equilibrium. We can check that the variables R and D do not appear in the first three equations of the system.

Therefore, we can omit the last two equations of dynamical system (4) and focus on the system

$$begin{aligned} textstylebegin{cases} S'(t) = alpha (1-S)-beta I(t)S(t)+eta A(t)-p(M)S(t), \ A'(t)= p(M)S(t)-(eta +alpha )A(t), \ I'(t)= beta I(t)S(t)-(gamma +omega +alpha )I(t), \ M'(t)= ag(I)-aM(t). end{cases}displaystyle end{aligned}$$

(5)

Local stability of the disease-free equilibrium

Let (P_{0}) be the disease-free equilibrium (DFE),

$$begin{aligned} P_{0}= biggl(frac{alpha +eta}{alpha +eta +p(0)}, frac{p(0)}{alpha +eta +p(0)}, 0, 0 biggr). end{aligned}$$

Theorem 4.1

If (mathcal{R}_{0} mathcal{R}_{1} < 1), then the diseasefree equilibrium (P_{0}) is locally stable.

Proof

Linearizing the system at DFE for these infectious variables and substituting the solution of the form (X(t) = e^{lambda t}v), (v in mathrm{R}^{4}), into the matrix equation corresponding to the system, we get the characteristic polynomial

$$begin{aligned} p(lambda ) = det|lambda I-A|, end{aligned}$$

where (A = J|_{P_{0}}). This polynomial has the roots

$$begin{aligned} &lambda _{1} = (omega +gamma +alpha ) (mathcal{R}_{0} mathcal{R}_{1} – 1), \ &lambda _{2}= -a, end{aligned}$$

and the other two are roots of the quadratic polynomial

$$begin{aligned} lambda ^{2} +theta lambda +omega =0, end{aligned}$$

where (theta = p(0)+eta +2alpha ) and (omega = alpha (p(0)+eta +alpha )). The root (lambda _{1} = (omega +gamma +alpha )(mathcal{R}_{0}mathcal{R}_{1}-1) ) is negative if (mathcal{R}_{0}mathcal{R}_{1}< 1), the root (lambda _{2} = -a < 0), and the roots of the polynomial (lambda ^{2} + theta lambda +omega = 0) are always negative. □

Local stability and Hopf bifurcation of the endemic equilibrium

We have proved that the endemic equilibrium

$$begin{aligned} P_{1}= biggl(frac{1}{mathcal{R}_{0}}, frac{p(g(I_{1}))}{(alpha +eta )mathcal{R}_{0}}, I_{1}, frac{gamma}{alpha}I_{1}, D^{*}, g(I_{1}) biggr) end{aligned}$$

exists if (mathcal{R}_{0} mathcal{R}_{1} > 1).

Theorem 4.2

If (mathcal{R}_{0} mathcal{R}_{1} > 1) and the coefficients (B_{i}>0), (i = 2, 1, 0) (independent of the parameter a), then the endemic equilibrium (P_{1}) is locally and asymptotically stable.

Proof

The characteristic equation corresponding to this equilibrium point is of the form

$$begin{aligned} lambda ^{4} + b_{3}lambda ^{3} + b_{2} lambda ^{2} + b_{1}lambda + b_{0} = 0 end{aligned}$$

with coefficients

$$begin{aligned} begin{aligned}& b_{3} = a+alpha +Delta >0, \ &b_{2}= (1+a) (alpha +Delta )+beta I_{1}(eta +omega + gamma + alpha )>0, \ &b_{1} = a(alpha Delta +beta eta I_{1}+Lambda )+beta (eta + alpha )I_{1}(omega +gamma +alpha )>0, \ &b_{0}=alpha bigl(beta (eta +alpha )I_{1}(omega + gamma +alpha )+ alpha Lambda bigr)>0, end{aligned} end{aligned}$$

(6)

where (Delta = eta +alpha +beta I_{1} + p(g(I_{1})) > 0) and (Lambda = (omega +gamma + alpha )I_{1} g'(I_{1})p'(g(I_{1})) > 0). The positivity of the (b_{i}) rules out the possibility of positive real eigenvalues (Descartes’ theorem). Consequently, the loss of stability can only occur via a Hopf bifurcation.

Since the value of the parameter a, which represents the inverse of the average delay of the information collected about the disease, only affects the stability of the endemic point, we will use this a as the bifurcation parameter. Define

$$begin{aligned} &C_{1} = alpha +Delta >0, \ &C_{2}= c_{1}+beta I_{1}(eta +omega +gamma +alpha )>0, \ &C_{3} = alpha Delta +beta eta I_{1}+Lambda >0, \ &C_{4}=beta (eta +alpha )I_{1}(omega +gamma +alpha )>0. end{aligned}$$

By the Routh–Hurwitz theorem, if (b_{3}b_{2}b_{1}-b_{1}^{2}-b_{0} > 0), then the endemic point is locally and asymptotically stable. When observing the coefficients (b_{i}) defined in (6), this last condition is equivalent to the positivity of the cubic polynomial

$$begin{aligned} f(a) = B_{3}a^{3} + B_{2}a^{2} + B_{1}a + B_{0}, end{aligned}$$

(7)

where

$$begin{aligned} &B_{3} = C_{1}C_{2} >0, \ &B_{2}= C_{1}C_{4}+C_{1}^{2}C_{2}-C_{2}^{2} -1, \ &B_{1} = C_{2}C_{4}+C_{1}^{2}C_{4}+C_{1}C_{2}^{2} -2C_{2}C_{4}-2C_{1}b_{0}, \ &B_{0}=C-1C_{2}C_{4}-C_{4}^{2} -C_{1}^{2} b_{0}. end{aligned}$$

Only the first coefficient is guaranteed to be positive, and the others are variable. Therefore, if we require that the remaining (B_{i}), (i = 2, 1, 0), are positive, then the endemic point is locally and asymptotically stable regardless of the delay, whereas if any of them is negative, then we obtain the possible instability of the endemic point. □

Theorem 4.3

If (mathcal{R}_{0} mathcal{R}_{1} > 1) and the coefficients (B_{i}>0), (i = 2, 1, 0), then there exist two values (0< a_{1}< a_{2}) of the delay parameter a such that the endemic equilibrium (P_{1}) is unstable for (ain (a_{1},a_{2})), whereas it is locally asymptotically stable if (ain (0.a_{1})) or (ain (a_{2},infty )). At the values (a_{1}) and (a_{2}), Hopf bifurcations occur.

Proof

As (B_{3}>0), the cubic polynomial (7) has one negative real root, and if (B_{0}>0), then the other two roots can be

  • two real positive distinct roots, or

  • one real positive root of multiplicity 2, or

  • two complex roots.

In the last two cases, the endemic point is always locally asymptotically stable independently of a. In the first case, since (f(0)>0), we have (f(infty )>0); the endemic equilibrium is always locally asymptotically stable for both small and large values of information delay parameter (this means large or small values of parameter a). Then there exist two positive values (a_{1}) and (a_{2}) ((a_{1}< a_{2})) of the delay parameter a for the equation (f(a)=0), which determines two bifurcating values of the delay parameter a. In consequence, (f(a)) is positive for (0< a< a_{1}) and (a>a_{2}), which means that the endemic point is locally asymptotically stable, whereas the endemic point is unstable for (a in [a_{1}, a_{2}]).

Finally, the following transversality condition for a Hopf bifurcation is satisfied:

$$begin{aligned} biggl[frac {d(b_{3}b_{2}b_{1}-b_{1}^{2}-b_{0})}{da} biggr]_{a=a_{i}}= biggl[frac {df(a)}{da} biggr]_{a=a_{i}} neq 0. end{aligned}$$

 □

Global stability of the disease-free equilibrium

Theorem 4.4

If (mathcal{R}_{0} < 1), then the diseasefree equilibrium is globally and asymptotically stable.

Proof

The disease-free equilibrium (P_{0}) always exists. In the reduced system, defining (phi (t) = S(t) + A(t) + I(t)) with derivative (phi ‘(t) = S'(t) + A'(t) + I'(t)), we get

$$begin{aligned} phi ‘(t)leq alpha -alpha phi (t). end{aligned}$$

By a comparison theorem for ODEs (Hale, 1969) we get that (S+A+I leq 1). In consequence, if we replace it in the infected population dynamics, then we obtain

$$begin{aligned} I’leq Ibigl[beta -(omega +gamma +alpha )bigr]. end{aligned}$$

As

$$begin{aligned} mathcal{R}_{0} = frac{beta}{gamma +omega +alpha} < 1, end{aligned}$$

this implies (I(t) rightarrow 0) and (S(t) + A(t) rightarrow 1). Therefore the disease-free equilibrium is globally and asymptotically stable. □

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