Model 3: Zika Virus SEIR Horizontal and Vertical Transmission Model

Imran et al. [60] proposed a deterministic SEIR model to investigate the transmission dynamics of Zika virus including the horizontal and vertical modes of transmission in both humans and vectors. To consider a vertical transmission in the model, the authors made the assumption that a fraction of newborn individuals from parents in Etl and Itl classes will be infected and thus remain in Eh class before becoming infectious [100]. Because of this vertical transmission, a fraction of susceptible individuals will enter the exposed class. Thus, the newborn individuals entering the Eh class are represented by [pYli.Ei, + yFlidi,)- Similarly, the newborn individuals exiting the S(, class are represented by (pBi,Eh + In both cases, 0 < p < 1 and 0 < q < 1; p is the fraction of new babies from

exposed individuals, and q is the fraction of newborn babies from infected individuals. The exposed population E/,(f) is depleted at the natural death rate ph. Additionally, exposed individuals develop symptoms and move into the infected class Ih at a rate The infected population Ii,(t) is decreased due to the natural death rate nh, the disease-induced death rate <5/, and the recovery rate of infected individuals в. The recovered population Rh(t) is decreased due to the natural death rate yU/,. The susceptible human population S;,(f) has a constant recruitment rate П;, = BhKi, where Kh is the carrying capacity and Bt, is the natural birth rate [20]. Susceptible individuals get infected with Zika virus (due to contact with infected vectors) at a rate Xh and thus enter the exposed class Eh. The total human population Ni,(t) at time t is divided into four mutually exclusive classes: (i) susceptible humans Si,{t), (ii) exposed humans Ei,(t), (iii) infected humans £,(f), and (iv) recovered humans R,,(f). It was assumed that individuals who recover from infection with a particular serotype of Zika virus gain lifelong immunity to it. Similarly, the total vector population Nv(t) is divided into three mutually exclusive classes: (i) susceptible vectors S„(f), (ii) exposed vectors E„(f), and (iii) infected vectors /„(f). It is assumed that vectors (mosquitoes) infected with Zika virus never recover. The susceptible vector population Sv(t) has a constant recruitment rate П;1 = BVKV and a natural death rate /uv. A fraction of the offspring in £„ and /„ classes will be infected and thus remain in Е/, class before becoming infectious. Because of this vertical transmission, a fraction of susceptible individuals will enter the exposed class. Thus, the newborn individuals entering the EJ( class are represented by (rB„E„ + sBvIv), where 0 < r < 1 and 0 < s < 1, and similarly, these offsprings are exiting the Sz, class, where r is the fraction of offsprings from exposed individuals and s is the fraction of offsprings from infected individuals. Susceptible vectors are infected with Zika virus (due to effective contact with infected humans) at a rate A„ and thus move to the exposed vector class Ev. The exposed vector class £„(f) is depleted due to the natural death rate /uv. In addition, exposed vectors develop symptoms and move to the infected vector class /„(f) at a rate a. Infected vectors, in addition to the natural death rate //„, die at a disease-induced death rate <5„. The forces of infection are given by, Xh = ChvIv/Nh and Xv = CiwIh/Nh where C(,„ is the effective contact rate, that is, the rate at which mosquitoes/humans acquire infection from infected humans/mosquitoes. The transmission of Zika virus is governed by the following set of non-linear differential equations [60]

where K. = + jUi,, K.o = 6 + <5/ + /Ji,, K.% = о + and К4 = ju;, + 8V.

Analysis of equilibrium points: The model has the DFE, No(n(,/jU;,,0,0,0,n!,/jU!,,0,0). LetN, = (S(‘, E'h, Ij,, Rl, Sl,E'v,I',) bean EE of the model. Denote A/‘ = Q.Jj./Nj,, A,*, = C,wll/N’,:. We obtain the equilibrium point as

where 12KK2Xit + tttifii,, = /С;,Х4Я.. + tn2pv/

For the existence of the EE, we require KK2 > K5, and K2K4 > K6, where K5 = В/, (pK2 + qg), K6 = (rK4 + as)Bv (Problem 5.4, Exercise 5).

Substituting (5.23b) into (5.23a), we obtain the quadratic equation

where

where 7?0 is the basic reproduction number given by (Problem 5.5, Exercise 5)

When p = 0 = <7, and r = 0 = s, vertical transmission is not present in the model and 7?0 reduces to the basic reproduction number rR<) = 'IZt, for an SEIR model with vector population as given by Derouich and Boutayeb [34].

We find that a > 0. We have the following two cases: (i) 'Rq < 1: in this case, c>0. If b>0, then the signs of the coefficients are +, +, +. The equation has no positive root. If b < 0, then the signs of the coefficients are The equation has two positive roots when b2 - 4ac > 0.

In this case, two EE points may exist. Otherwise, there is no positive root, (ii) 'Rq > 1: in this case, c < 0. If b > 0, then the signs of the coefficients are when b< 0, the signs of

the coefficients are Hence, the equation has a positive root always. The unique EE is

obtained from equation (5.22). If 7^o = 1, then c = 0. If b < 0, then a positive solution exists for equation (5.24), which is given by XJ, = -(b/a). In this case, a unique EE point exists.

DFE А/о(П;1//и/,,0,0,0,П!,//и!.,0,0) is locally asymptotically stable if R„ < 1 and unstable if Ro > 1 (Problem 5.6, Exercise 5 [61]).

The result implies that Zika virus can be eliminated from the population when Rq < 1, if the initial sizes of the subpopulations of the model (5.21) are in the basin of attraction of the DFE N0. The effective contact rate and vertical transmission can help to control the disease. The reproduction number R0 can be reduced and maintained at a value below unity, if the initial sizes of the subpopulations of the model are in the neighborhood of attraction of the DFE.

Bifurcation analysis: Denote Si, = x4, Е/, = x2, h, = x3, Ri, = x4, Sv = x5, Ev = x6/ Iv = x7. Let / = [/i, ■ ■ ■, /7 ] denote the vector field of the original model in terms of x's. Then, the model system (5.21) reduces to the system

The DFE of the system is given by e0 = {.v[, 0,0,0, *5,0,0}, where x = П;,///,,, and .vj = П;,/pv. Consider the case Rq = 1. Let Chv = Cf, be a bifurcation parameter

The Jacobian of the matrix at the DFE is given by

where j2 = C'twПг,/1;,/П/,/1г). The Jacobian matrix of the linearized system has zero as a simple eigenvalue and the remaining eigenvalues have negative real parts. Hence, the center manifold theory of Castillo-Chavez and Song [27] can be used to analyze the dynamics of the system. Following their analysis, the right and left eigenvectors of the Jacobian matrix corresponding to zero eigenvalue at Cjw are given by w = (wu...,zv7) and v = (vu...,v7), where

The values of non-zero derivatives are

The values of the parameters a and b defined in the center manifold theory are obtained as

Since b is always positive, backward bifurcation occurs when a > 0. Therefore, 7?(l < 1 is a necessary condition but it is not sufficient to effectively control the spread of Zika virus in the population. In other words, efforts to bring % < 1, may fail to lead to effective control of ZIKV (due to the phenomenon of backward bifurcation). In such a backward bifurcation scenario, effective disease control is dependent on the initial sizes of the populations of the model. The global stability is not investigated in this case because of the existence of a backward bifurcation due to a large vector population. The authors [60] have shown numerically that the model exhibits the phenomenon of backward bifurcation when the stable DFE co-exists with two endemic equilibria (one of which suggested to be stable) when 7?o < 1. To study the impact of the model parameters on the prevalence of Zika infection, uncertainty and sensitivity analyses were also performed. The authors concluded that the effective contact rates, the recovery rate of the infected individuals, and the birth rate of mosquitoes are the most influential parameters. The authors [60] have also studied a modified model system (5.21) by neglecting the disease-induced death rate (<5/ = 8V = 0). In this case, both the total host population and the vector population are asymptotically constant. The endemic state of the modified model (5.21) is locally asymptotically stable for 'Rq > 1. The proof is based on the Krasnoselskii sublinearity trick [54,55]. Rewrite the system as [60]:

Linearizing the system (5.25) about the EE = (SfE, /,, ,R°h, S°, E°, T,), we obtain

The Jacobian of the system evaluated at ег is

___i____т/° r , ,, : ChvlfiCftvIv . ClwSh__j . C/IVSV

where /C2 = e + ft, /. =~NT'1> = =157'and ’< =~йГ

Write the solution of the model (5.26) in the formZ(f) = Ze®', where Z = (Zi,Z2,Z3,Z4,Z5). Substituting Z(f) into (5.26), we obtain

Rearrange the above system of equations as follows: first, move the negative terms in the last four equations of (5.27) to the respective left-hand sides. Secondly, the last four equations are then re-written in terms of Z. We obtain

Substituting the values of Z2,Z3,ZS in the first equation of (5.27), we obtain where F, (a)), i = 1,...,5 are positive functions of parameters and

Note that the matrix M has non-negative entries. The notation (MZ)| denotes the i"‘ coordinate of the vector MZ. Define F(co') = min,|l + F,|. It can be verified that the equilibrium point £[ satisfies the equation e, = Me,. If Z is a solution of (5.28), then it is possible to find a minimal positive real number r such that ||Z|| < re,, [54,55]. To show that Re(co) < 0, assume that Re((o) > 0, and consider the following two cases.

Case 1: ft) = 0: In this case, equation (5.27) is a homogeneous linear system. It is easy to show that the determinant of this system is negative. It follows that the system has a unique solution Z = 0, which corresponds to the disease-free steady state of the modified model system (5.21).

Case 2: ft) 0: By assumption, |l + F, ( 1. Since r is a minimal positive real number, it follows that ||Z||> r£,/F((o), where F(ft)) is minimal of |l + F(®)|- From the second equation of (5.28), we have F(ft))||Z2|| < rll, which contradicts ||Z|| > re,/F(co). Hence, Re(co)< 0. Thus, all eigenvalues of the characteristic equation associated with the linearized system (5.26) have negative real parts. This implies the local asymptotical stability of the endemic state.

Numerical simulations: To study the stability and bifurcation, numerical computations are performed using the set of parameter values as

The initial conditions are taken as (100000,100,3U0,10,1U0U0,50,200). We obtain the DFE as N0 =(2.17391 xlO6,0,0,0,10101,0,0), R0 = 0.65884, and Qw = 1.43. The local asymptotic stability of DFE is shown in Figure 5.6a. We obtain the EE as N, = (1.73284xlO6, 61.4744,100.905,438718,10089.5,1.04632,10.462), Chv = 2.43, and R0 = 1.11957. The local asymptotic stability of EE is shown in Figure 5.6b.

As discussed earlier, the epidemiological implication of the backward bifurcation phenomenon of the model (5.21) is that having Ro < 1 is only a necessary condition to effectively control the spread of Zika virus in the population. In other words, the efforts to bring Ro < 1 may fail to lead to effective control of ZIKV. In such a backward bifurcation scenario, effective disease control (7?o < 1) is dependent on the initial sizes of the populations of the model. This effect is shown in Figure 5.7. One of the reasons for the occurrence of the backward bifurcation phenomenon in the model is a large vector population. If the vector population can be controlled, then backward bifurcation may be eliminated.

FIGURE 5.6

Time series of model (5.21) converging to steady states, (a) DFE with Cj,t, = 1.43, % = 0.65884 and vector (mosquito) population, (b) EE with C,„, = 2.43 and = 1.11957.

FIGURE 5.7

Backward bifurcation in model (5.26). [From Imran, M, Usman, M., Dur-e-Ahmad, M., Khan. A. 2017. Transmission dynamics of Zika fever: A SEIR based model. Diff. Eys. Dyn. Sys. 1-24, [60], Copyright 2017, Springer Nature. Reprinted with kind permission from author and Springer Nature.]

 
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