
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

38548897
58025
10.1038/s41598-024-58025-7
Article
Transmission dynamics of Zika virus with multiple infection routes and a case study in Brazil
Wang Liying 12
Jia Qiaojuan 2
Zhu Guanghu 12
Ou Guanlin 1
Tang Tian ttang@guet.edu.cn

13
1 https://ror.org/00jbpxw47 Key Laboratory of Cognitive Radio and Information Processing, Ministry of Education (Guilin University of Electronic Technology), Guilin, 541004 China
2 https://ror.org/05arjae42 grid.440723.6 0000 0001 0807 124X School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541004 China
3 https://ror.org/05arjae42 grid.440723.6 0000 0001 0807 124X School of Information and Communication, Guilin University of Electronic Technology, Guilin, 541004 China
28 3 2024
28 3 2024
2024
14 74243 12 2023
25 3 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
The Zika virus (ZIKV) is a serious global public health crisis. A major control challenge is its multiple transmission modes. This paper aims to simulate the transmission patterns of ZIKV using a dynamic process-based epidemiological model written in ordinary differential equations, which incorporates the human-to-mosquito infection by bites and sewage, mosquito-to-human infection by bites, and human-to-human infection by sex. Mathematical analyses are carried out to calculate the basic reproduction number and backward bifurcation, and prove the existence and stability of the equilibria. The model is validated with infection data by applying it to the 2015–2016 ZIKV epidemic in Brazil. The results indicate that the reproduction number is estimated to be 2.13, in which the contributions by mosquito bite, sex and sewage account for 85.7%, 3.5% and 10.8%, respectively. This number and the morbidity rate are most sensitive to parameters related to mosquito ecology, rather than asymptomatic or human-to-human transmission. Multiple transmission routes and suitable temperature exacerbate ZIKV infection in Brazil, and the vast majority of human infection cases were prevented by the intervention implemented. These findings may provide new insights to improve the risk assessment of ZIKV infection.

Keywords

Zika virus
Transmission dynamics
Infection risk
Basic reproduction number
Subject terms

Infectious diseases
Risk factors
Infection
Applied mathematics
http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 12171116 Zhu Guanghu Director Fund Project of the Ministry of Education Key Laboratory of Cognitive Radio and Information ProcessingCRKL210106 Tang Tian Guangxi Key Laboratory of Cryptography and Information SecurityGCIS201707 Tang Tian issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Zika virus (ZIKV) is a Flavivirus closely related to dengue, which was first discovered in 1947 in Uganda among a certain Rhesus macaque population1. However, few human infections were reported until 2015, when ZIKV infection unexpectedly struck the Americas and spread to other countries over the next 2 years2,3. The WHO has recorded that a total of 87 countries and territories have reported evidence of autochthonous ZIKV infection, with more than 1.4 million suspected and confirmed Zika cases4. ZIKV is a serious public health problem, which has the following concerns: (1) large number of human infections with ZIKV (0.4–1.3 million cases in Brazil alone)3; (2) serious consequence of infection, in which ZIKV infection during pregnancy can cause microcephaly and other congenital anomalies in developing fetuses and newborns (nearly 6000 suspected cases of microcephaly among newborns might be linked to ZIKV infections in Brazil during 2015–20165,6); (3) about three quarters of cases of ZIKV infection are asymptomatic, who can transmate the disease but not easy to be identified7,8; (4) multiple transmission routes, where human can be infected through mosquitoes or humans, adult mosquitoes can be infected through humans, and larvae mosquitoes can be infected through the virus sewage9–15. Understanding such complex transmission patterns can provide scientific evidence to guide disease control.

Existing studies mainly use epidemiological investigation, statistical approach and mathematical models to tackle the above-mentioned issues9–19. By addressing the impacts of multiple transmission routes on ZIKV infection, it is found that (1) sexual transmission increases the risk of infection and epidemic size, and prolongs the outbreak9; (2) prevention and control efforts against ZIKV should target both the mosquito-borne and sexual transmission routes9; and (3) scenario exploration indicates that personal protection could be more effective than mosquito-reduction intervention10, and releasing male Wolbachia mosquitoes may be a good choice of disease control12. Furthermore, experimental studies have shown that urine from infected humans could be a natural ZIKV source, where Aedes mosquitoes are permissive to ZIKV infection when breeding in urine or sewage containing low concentrations of ZIKV13,14. However, there is a lack of mechanistic frameworks for modeling different routes of ZIKV transmission, and fewer evaluations of these routes on infections9,11,12,15.

To fill the knowledge gap, this study proposes a mechanistic framework for modeling ZIKV transmission patterns by addressing the following questions: How to account for the vector–virus–host interactions and the multiple routes of infection and thus to simulate the diffusion process? How to calculate the dynamics of transmission and assess the risk of infection, and thus suggest how to intervene? How to characterize the role of the dominant factors in activating or inhibiting the disease evolution? Answering these questions may provide new insights to improve ZIKV risk assessment and guide disease control. Mathematical model and data fitting method are employed to tackle these challenges. Specifically, based on the classical Ross–Macdonald theory and compartmental principle of mosquito-borne disease20, a new ZIKV transmission model is established by ordinary differential equations (ODEs), which systematically combines the dynamics of virus evolution, multiple infection routes, mosquito ecology and human behavior. Since temperature plays a vital role in ZIKV infection, its effects is included by modulating the transmission factors of virus and mosquitoes (oviposition rate, aquatic transition rate, hatching rate, mosquito bite rate and infection rate). Mathematical analysis are used to explore the transmission dynamics, including the expression of the basic reproduction number, the existence of bifurcation, and the stability of the equilibria. By using Markov Chain Monte Carlo (MCMC) methods to fit the time series of the reporting cases, the model is further validated to analyze the ZIKV outbreak in Brazil during 2015–2016. Numerical simulations and sensitivity analysis are performed to illustrate the detailed transmission patterns in Brazil.

This paper is organized as follows. The model is presented in “Modeling framework” and analyzed mathematically in “Mathematical analysis”. Main results are presented in “Model application”, including simulation ZIKV transmission in Brazil, the impact of transmission routes on the epidemic and the effectiveness of epidemic prevention and control. A brief discussion is presented in the last section.

Modeling framework

A mechanistic framework for simulating ZIKV transmission patterns is established in this section. Inspired by recently-developed mathematical models9,16–19, and based on epidemiological feature of ZIKV, a new dynamic system is proposed, which takes into account human-to-human (sexual transmission) and human-to-mosquito (bite transmission and sewage transmission) interactions by using compartmental and deterministic principle. The total numbers of human, larval and adult mosquitoes are represented by Nh, Mv and Nv. Given the stability of human demographics and the extremely low rate of ZIKV-related mortality, Nh is assumed to be a constant.

Based on the characteristics of ZIKV infection, it is further assumed that: (1) the humans are divided into five categories: susceptible Sh, latent Eh, symptomatic infected Is, asymptomatic infected Ia and recovered Rh; (2) the larval stage can be divided into two categories: uninfected Av (including eggs, larvae and pupae) and infected Jv; (3) the adult female mosquitoes are divided into three categories: susceptible Sv, latent Ev and infected Iv.

It is known that mosquito oviposition is linked to mature females and the ability of mosquitoes to develop oviposition habitats. If there are too many eggs in the oviposition habitat or too few nutrients and water resources, the females will lay fewer eggs or choose another site21,22. In addition, larvae and pupae need water or nutrients complete their development21. This biological phenomenon can be expressed by a Logistic model which explicitly incorporates the idea of limited carrying capacity resources21,22. Hence, the per capita oviposition rate is given by θ(1-(Av+Jv)/K)Nv.

When ZIKV invades an area, humans and mosquitoes there could be infected with certain probability. It is governed by the following rules. A susceptible human may be infected with ZIKV from the bite of infectious mosquitoes at rate b or through human contact with infected people at rate β. Infected humans undergo an incubation period 1/δ. After that they could be symptomatic ϕ and asymptomatic 1-ϕ, and then the former follows by an infectious period of mean duration 1/η until recovery. The larvae may be directly infected by the virus sewage at rate p and grow up to become infectious adult mosquitoes at rate α. Susceptible mosquitoes become infected at rate a by biting infectious humans, and the infected mosquito experience an extrinsic incubation period 1/γ, that is followed by an infectious state from which they do not recover. The mean larvae lifespan and adult lifespan are 1/f and 1/μ, respectively. Flow chat is shown in Fig. 1.Figure 1 Flow diagram of the ZIKV transmission among humans and mosquitoes. The black solid lines indicate the progress of infection and ecology. The yellow lines indicated the infection pathes.

Accordingly, the following ODEs are used to simulate the transmission dynamics of ZIKV:1 dShdt=dNh-bcIvNhSh-βIs+τIaNhSh-dSh,dEhdt=bcIvNhSh+βIs+τIaNhSh-δ+dEh,dIsdt=ϕδEh-η+dIs,dIadt=1-ϕδEh-η+dIa,dRhdt=ηIs+Ia-dRh,dAvdt=θ1-Av+JvKNv-pIs+qIaNhAv-α+fAv,dJvdt=pIs+qIaNhAv-α+fJv,dSvdt=αAv-acIs+kIaNhSv-μSv,dEvdt=acIs+kIaNhSv-γ+μEv,dIvdt=γEv+αJv-μIv.

Detailed explanations of the variable and parameters are shown in Table 1. All the variables and parameters are non-negative. Similar to the proofs in21,22, it is obtained that the following Ω is positively invariant and attracting under the flow described by the system (1).Ω=Sh,Eh,Is,Ia,Rh,Av,Jv,Sv,Ev,Iv∈R+10|Sh,Eh,Is,Ia,Rh,Av,Jv,Sv,Ev,Iv≥0,Sh+Eh+Is+Ia+Rh≤Nh,Av+Jv≤αK/μ,Sv+Ev+Iv≤Nv.

Temperature, as the most important factor in the modulation of mosquito-borne diseases, is considered by including in the model parameters, where the parameters are functions of temperature T. Based on observations from laboratory studies23–25, they are presented below.2 αT=70.131-0.05723T+0.01164T2-0.001341T3+0.00008723T4-0.000003017T5+5.153×10-8T6-3.42×10-10T7,

3 fT=72.13-0.3787T+0.02457T2-0.0006778T3+0.000006794T4,

4 μT=70.8692-0.159T+0.01116T2-0.0003408T3+0.000003809T4,

5 θT=7-5.4+1.8T-0.2124T2+0.01015T3-0.0001515T4,

6 aT=max0.001044TT-12.28632.461-T1/2,0,

7 γT=7/4+exp5.15-0.123T,

8 cT=70.0043T+0.0943.

Table 1 Description of parameters in the model (the time unit is week or per week).

Parameters	Definitions	Value	Source	
a	Human-to-vector transmission probability per bite	(6)	23,25	
b	Vector-to-human transmission probability per bite	[a]		
c	Number of bites per mosquito per day	(8)	23	
θ	Intrinsic oviposition rate of adult mosquitoes	(5)	23	
K	Environmental capacity of larval mosquitoes	[c]	Estimation	
k	Relative (asymptomatic) human-to-vector transmissibility	0.5	26,27	
β	Human-to-human transmission rate	[b]	Estimation	
τ	Relative (asymptomatic) human-to-human transmissibility	0.5	26,27	
p	Transmission rate from infected person fecal to larval	[b]	Estimation	
q	Relative (asymptomatic) human-to-larva transmissibility	0.5	26,27	
α	Aquatic transition rate to adult stage	(2)	23	
ϕ	Proportion of symptomatical infections	0.18	28	
1/δ	Intrinsic incubation period in human	0.5	29	
1/η	Duration of the human infection period	2	30	
1/γ	External incubation period of mosquito	(7)	23	
1/d	Human life span	3950		
1/f	Lifespan of aquatic mosquitoes	(3)	23,24	
1/μ	Lifespan of adult mosquitoes	(4)	23,24	
[a] b is determined by the effective reproduction number Rt in the case study.

[b] β and p are estimated using the MCMC approach in the case study.

[c] K is assumed to be proportional to population size, i.e.,K = φNh, where φ is estimated by MCMC method.

Mathematical analysis

The transmission dynamics of the proposed model are analyzed mathematically, in which the epidemic threshold (i.e., the basic reproduction number) is calculated by the next generation matrix method, and the evolution trends of the model solutions are investigated by stability theory.

Basic reproduction number

The basic reproduction number R0 is interpreted as the average number of secondary cases that are produced by a single primary case in a fully susceptible population, acting as the critical measure of the transmissibility31. The basic reproduction number R0 is calculated by using the theory of next generation matrix31. It is written as31: R0=ρFV-1, where F is the rate of occurring new infections, and V is the rate of transferring individuals outside the original group. Here ρ represents the spectral radius of matrix.

Let the right hand side of the system (1) equal to zero, it is obtained the disease-free equilibrium of the modelE0=Sh0,Eh0,Is0,Ia0,Rh0,Av0,Jv0,Sv0,Ev0,Iv0=Nh,0,0,0,0,Mv∗,0,Nv∗,0,0,

whereMv∗=Kαθ-μα+fαθ,Nv∗=Kαθ-μα+fμθ.

It follows from the next generation matrix method31 thatF=0ββτ00bc0000000000000pMv∗NhpqMv∗Nh0000acNv∗NhackNv∗Nh000000000,V=δ+d00000-ϕδη+d0000-1-ϕδ0η+d000000α+f000000γ+μ0000-α-γμ.

Using R0=ρFV-131, the basic reproduction number is calculated as follows:9 R0=Rhh1+Rhh12+4Rhv122,

whereRhh1=δβϕ+1-ϕτδ+dη+d,Rhv1=pbcαδMv∗ϕ+1-ϕqγ+μ+bac2γδNv∗ϕ+1-ϕkα+fNhμδ+dη+dγ+μα+f.

Some typical features can be observed from Eq. (9): (a) Rhh1 and Rhv1 represent the parts of the basic reproductive number R0 contributed by sexual transmission and mosquito-borne transmission, respectively; (b) Rhh1 is determined by human-to-human transmission; (c) Rhv1 is determine by parameters related to mosquitoes and humans, which can be further divided into two stages of larval transmission and adult transmission.

Model stability

Mosquitoes breeding in contaminated water can become infected with ZIKV, which can shorten the virus transmission cycle and promote the spread of ZIKV14. Based on this, the following mathematical analysis is divided into two cases, with or without larval infection in the virus sewage, corresponding to p=0 or p≠0.

p=0 (without larval mosquito infection in the virus sewage)

In this case there is no larval infection Jv item in the system (1). The basic reproduction number of the available model is10 R¯0=Rhh2+Rhh22+4Rhv222,

whereRhh2=δβϕ+1-ϕτδ+dη+d,Rhv2=bac2δγNv∗ϕ+1-ϕkNhμδ+dη+dγ+μ.

Theorem 3.1

When p=0, if R¯0<1, then the disease-free equilibrium of the system (1) is locally asymptotically stable.

The proof is similar to that of Theorem 3.5, so it is omitted here.

The expression of endemic equilibrium is denoted byE0∗=Sh∗,Eh∗,Is∗,Ia∗,Rh∗,Av∗,Sv∗,Ev∗,Iv∗.

Letting the right-hand side of system (1) to be zeros, direct calculation yields an equation about infectivity λh∗ asλh∗=bcIv∗Nh+βIs∗+τIa∗Nh.

Let a1=α+f,a2=γ+μ,a3=η+d,a4=δ+d.

It is further obtained that11 λh∗=bcγNv∗H1λh∗dNha2μa3a4λh∗+d+λh∗dH1+λh∗dH3a3a4λh∗+d,

where H1=acϕδ+1-ϕδk,H2=pϕδ+1-ϕδq,H3=βϕδ+1-ϕδτ. It follows from Eq. (11) that12 c0λh∗2+c1λh∗+c2=0,

wherec0=Nha2a3a4μa3a4+H1d>0,c1=dNha2a3a42μa3a4+H1d-bcγNv∗H1da3a4-dNha2H3μa3a4+H1d,c2=μNha2d2a32a421-bcγNv∗H1μa2a3a4Nh-H3a3a4.

Here λh∗ can be obtained by solving the quadratic expression (12), which also determines the expression of the endemic equilibrium. When R¯0>1, one has R¯0Rhv22+R¯0Rhh2>Rhv22+R¯0Rhh2. It is observed from Eq. (10) that Rhv22+R¯0Rhh2=R¯02.

Hence R¯0>1 yields Rhv22+Rhh2>R¯0>1. In this case,c2=μNha2d2a32a421-Rhv22-Rhh2<μNha2d2a32a421-R0_<0.

Since c0>0 and c2<0 (when R¯0>1), it is obtained a unique solutions for Eq. (12), which indicates that the system (1) has a unique endemic equilibrium when R¯0>1. Knowing that c2>0 when R0_<1, there are three scenarios under this condition: (i) If c1>0, the symmetry axis of fλh∗=c0λh∗2+c1λh∗+c2 is on the negative half axis of x, and there is no intersection point between fλh∗ and the positive half axis of x, so the system (1) has no endemic equilibrium.

(ii) If c1<0 and c12=4c0c2, there is only one intersection point between fλh∗ and the positive half axis of x, so the system (1) has a positive equilibrium point.

(iii) If c1<0 and c12>4c0c2, fλh∗ has two intersection points with the positive half axis of x, then the system (1) has two positive equilibrium points.

The above analysis concludes the following theorem.

Theorem 3.2

When p=0, if R¯0>1, then c2<0, the system (1) has a unique endemic equilibrium. If R¯0<1, the following conclusions can be drawn: (i) if c1>0, then System (1) has no endemic equilibrium;

(ii) if c1<0 and c12=4c0c2, then System (1) has a unique endemic equilibrium;

(iii) if c1<0 and c12>4c0c2, then System (1) has two endemic equilibria.

Theorem 3.3

When p=0, if R¯0=1 and dH1H3>dH1a3a4+a32a42μ, then the system (1) has a backward bifurcation; if R¯0>1, the system (1) has a forward bifurcation and the endemic equilibrium is locally asymptotically stable.

Proof

X=x1,x2,x3,x4,x5,x6,x7,x8,x9T and G=g1,g2,g3,g4,g5,g6,g7,g8,g9T to be the left-hand and right-hand sides of system (1). Thus system (1) can be written as13 dXdt=G.

The Jacobian matrix of system (1) at the disease-free equilibrium E0 isJE0=-d0-β-βτ0000-bc0-a4ββτ0000bc0ϕδ-a300000001-ϕδ0-a30000000ηη-d000000000-θNv∗K+a100000-acNv∗Nh-ackNv∗Nh0α-μ0000acNv∗NhackNv∗Nh000-a200000000γ-μ.

The probability of mosquito-to-human transmission b is selected as the branching parameter. When R¯0=1, the critical value b is solved,b=b∗=Nhμa2a3a4-H3cγNv∗H1.

Therefore, when b=b∗, the characteristic equation of the system isλE-Jb∗E0=λλ+d2λ+μλ+θNv∗K+a1λ+a3hλ,

where h(λ)=λ3+b1λ2+b2λ+b3, with b1=μ+a2+a3+a4, b2=μa2+μa3+μa4+a2a3+a2a4+a3a4-H3, and b3=μa2a3+μa2a4+μa3a4+a2a3a4-H3μ+a2. It follows from R¯0=1 thatH32a3a4+H32a3a4=H3a3a4<1.

Considering the coefficients of hλ, it is clear that b1=μ+a2+a3+a4>0, b2=μa2+μa3+μa4+a2a3+a2a4+a3a4-H3>0, b3=μa2a3+μa2a4+μa3a4+a2a3a4-H3μ+a2>μa2a3+μa2a4+μa3a4+a2a3a4-μ+a2a3a4>0, and b1b2-b3=μμ+a2+a3+a4a2+a3+a4+a3+a4a22+a2a3+a2a4+a3a4-H3>0. Hence, it follows from Rrouth-Hurwitz theorem that the real parts of the characteristic roots of the equation hλ=0 are all negative.

Furthermore, by calculating the eigenvalue of the matrix Jb∗E0, it is found that Jb∗E0 has a zero eigenvalue, and other eigenvalues have negative real parts. Therefore, the system (13) satisfies the conditions of the Central Manifold theorem. The right eigenvector and the left eigenvector corresponding to the eigenvalues λ=0 of the matrix Jb∗E0 are respectively denoted asw=w1,w2,w3,w4,w5,w6,w7,w8,w9T,v=v1,v2,v3,v4,v5,v6,v7,v8,v9,

where w1=-a4w2/d, w2>0, w3=ϕδw2/a3, w4=(1-ϕ)δw2/a3, w5=ηδw2/(da3), w6=0, w7=a2w8/μ, w8=μw9/γ, w9=w2(a4-H3/a3)/(bc), and v1=0, v2>0, v3=(βv2+acNv∗v8/Nh)/a3, v4=(βτv2+ackNv∗v8/Nh)/a3, v5=v6=v7=0, v8=γv9/a2, v9=b∗cv2/μ.

According to Castillo-Chavez and Song theorem32, it is calculatedA=∑i,j,k=19vkwiwj∂2gk0,0∂xi∂xj=∑i,j=19v2wiwj∂2g20,0∂xi∂xj+∑i,j=19v8wiwj∂2g80,0∂xi∂xj=2v2w22dH1H3-dH1a3a4-a32a42μdNha32μ,B=∑i,k=19vkwi∂2gk0,0∂xi∂b∗=v2w2γcNv∗H1μa2a3Nh>0.

The sign of B will determine the occurrence of a backward bifurcation in a given model. When b=b∗, the positive and negative of the coefficient A determines the local dynamical properties of the disease-free equilibrium32. Hence, it follows from Castillo-Chavez and Song theorem that system (1) undergoes a backward bifurcation at R¯0=1 if dH1H3>dH1a3a4+a32a42μ. When R¯0>1 and close to 1, it has A<0 and thus the system has a forward branch and the endemic equilibrium is locally asymptotically stable. □

Theorem 3.4

When p=0, if R¯0>1, the endemic equilibrium of system (1) is globally asymptotically stable.

Proof

Define the following functions14 Q1=Sh-Sh∗-Sh∗ln(Sh/Sh∗)+Eh-Eh∗-Eh∗ln(Eh/Eh∗),

15 Q2=Is-Is∗-Is∗ln(Is/Is∗),

16 Q3=Ia-Ia∗-Ia∗ln(Ia/Ia∗),

17 Q4=Sv-Sv∗-Sv∗ln(Sv/Sv∗)+Ev-Ev∗-Ev∗ln(Ev/Ev∗),

18 Q5=Iv-Iv∗-Iv∗ln(Iv/Iv∗).

Using the inequality 1-x+lnx≤0, for x>0, differentiation yields:Q1′≤βIs∗Sh∗NhIsIs∗-lnIsIs∗-EhEh∗+lnEhEh∗+βτIa∗Sh∗NhIaIa∗-lnIaIa∗-EhEh∗+lnEhEh∗+bcIv∗Sh∗NhIvIv∗-lnIvIv∗-EhEh∗+lnEhEh∗=:a1,2G1,2+a1,3G1,3+a1,5G1,5.Q2′≤ϕδEh∗EhEh∗-lnEhEh∗-IsIs∗+lnIsIs∗=:a2,1G2,1.Q3′≤(1-ϕ)δEh∗EhEh∗-lnEhEh∗-IaIa∗+lnIaIa∗=:a3,1G3,1.Q4′≤acIs∗Sv∗NhIsIs∗-lnIsIs∗-EvEv∗+lnEvEv∗+ackIa∗Sv∗NhIaIa∗-lnIaIa∗-EvEv∗+lnEvEv∗=:a4,2G4,2+a4,3G4,3.Q5′≤γEv∗EvEv∗-lnEvEv∗-IvIv∗+lnIvIv∗=:a5,4G5,4.

Figure 2 System directed graph.

With the constants aij above and A=[aij], we construct the (strongly connected) directed graph Γ(A) in Fig. 2. If and only if aij>0, there is a weighted arci,j. Along each of the cycles on the graph, one can verify that ∑Gij=0, for instance, G5,4+G1,5+G2,1+G4,2=0, and so on. Then, by Theorem 3.5 in33, there exist constants ci such that Q=∑iciQi is a Lyapunov function for system (1). To find the constants ci, we use the combinatorial identities ciaij=∑k=1pcjajk or ciaij=∑k=1pckaki. We get c2a2,1=c1a1,2+c4a4,2, c3a3,1=c1a1,3+c4a4,3 and c5a5,4=c1a1,5. Let c1=c4=1, andc2=βIs∗Sh∗+acIs∗Sv∗NhϕδEh∗,c3=βτIa∗Sh∗+ackIa∗Sv∗Nh1-ϕδEh∗,c5=bcIv∗Sh∗NhγEv∗.

Then the function Q=c1Q1+c2Q2+c3Q3+c4Q4+c5Q5 is a Lyapunov function. Its derivative along the model isQ′=Sh-Sh∗ShSh′+Eh-Eh∗EhEh′+c2Is-Is∗IsIs′+c3Ia-Ia∗IaIa′+Sv-Sv∗SvSv′+Ev-Ev∗EvEv′+c5Iv-Iv∗IvSv′.

Now we consider the set S=x∈R+9:Q′=0. When Q′=0, one can readily verify that Sh=Sh∗,Eh=Eh∗,Is=Is∗,Ia=Ia∗,Sv=Sv∗,Ev=Ev∗ and Iv=Iv∗. For the left subsystem,19 Rh′=ηIs∗+Ia∗-dRh,Av′=θ1-AvKNv∗-α+fAv.

One can show that system (19) has a unique equilibrium Rh∗,Av∗, and that this point is globally asymptotically stable for this system. Therefore, the largest and only invariant set in S is the endemic equilibrium E0∗=Sh∗,Eh∗,Is∗,Ia∗,Rh∗,Av∗,Sv∗,Ev∗,Iv∗. Using LaSalle¡¯s Invariance Principle, we conclude that the endemic equilibrium E0∗ globally asymptotically stable in Ω. □

p≠0 (mosquitoes hatched in contaminated water can be infected with ZIKV)

Theorem 3.5

When R0<1, the disease-free equilibrium of the system (1) is globally asymptotically stable.

Proof

Substituting Sh, Av and Sv by Nh-Eh-Ia-Is-Rh, Mv-Jv and Nv-Ev-Iv respectively, it is obtaineddEhdt=bcIvNh+βIs+τIaNhNh-Eh-Ia-Is-Rh-δ+dEh,dJvdt=pIs+qIaNhMv-Jv-α+fJv≤pIs+qIaNhMv-α+fJv,dJvdt=pIs+qIaNhMv-Jv-α+fJv≤pIs+qIaNhMv-α+fJv,dEvdt=acIs+kIaNhNv-Ev-Iv-γ+μEv≤acIs+kIaNhNv-γ+μEv.

Let20 dE¯hdt=bcI¯v+βI¯s+τI¯a-δ+dE¯h,dI¯sdt=ϕδE¯h-η+dI¯s,dI¯adt=1-ϕδE¯h-η+dI¯a,dJ¯vdt=pI¯s+qI¯aNhMv-α+fJ¯v,dE¯vdt=acI¯s+kI¯aNhNv-γ+μE¯v,dI¯vdt=γE¯v+αJ¯v-μI¯v.

It can be seen that the right side of the system (20) is the right side of the matrix F-V. According to R0=ρFV-1<1, it can be seen that the system (20) has only a balance point E¯h,I¯s,I¯a,J¯v,E¯v,I¯v=0,0,0,0,0,0. Therefore, every non-negative solution of (20) satisfieslimt→∞E¯ht=0,limt→∞I¯st=0,limt→∞I¯at=0,limt→∞J¯vt=0,limt→∞E¯vt=0,limt→∞I¯vt=0.

Because the system (20) is linear, the disease-free equilibrium of the system (20) is globally asymptotically stable.

According to the comparison theoremEh′t≤E¯h′t,Is′t≤I¯s′t,Ia′t≤I¯a′t,Jv′t≤J¯v′t,Ev′t≤E¯v′t,Iv′t≤I¯v′t.

Thuslimt→∞Eht=0,limt→∞Ist=0,limt→∞Iat=0,limt→∞Jvt=0,limt→∞Evt=0,limt→∞Ivt=0.

The disease-free equilibrium is globally attractive, and it is locally stable. So the disease-free equilibrium of the system (1) is globally asymptotically stable. □

Theorem 3.6

When R0>1, the system (1) exists endemic equilibrium.

Proof

It is denoted the expression of endemic equilibrium byE1∗=Sh∗,Eh∗,Is∗,Ia∗,Rh∗,Av∗,Jv∗,Sv∗,Ev∗,Iv∗.

Based on the equilibrium definition, letting the right-hand side of system (1) to be zeros and substituting E1∗, it is obtained an equation about infectivity λh∗, λv1∗ and λv2∗ asλh∗=bcIv∗Nh+βIs∗+τIa∗Nh,λv1∗=pIs∗+qIa∗Nh,λv2∗=acIs∗+kIa∗Nh.

It follows thatλh∗=bcNv∗Nhγa1λv2∗+a2μ+λv2∗λv1∗a2μ+λv2∗a1+λv1∗+λh∗dH3a3a4λh∗+d,λv1∗=λh∗dH2a3a4λh∗+d,λv2∗=λh∗dH1a3a4λh∗+d.

Substituting λv1∗ and λv2∗ into λh∗, sorted outλh∗=YX+λh∗dH3a3a4λh∗+d,

whereY=bcNv∗λh∗dγa1a3a4H1λh∗+d+a2H2μa3a4λh∗+d+H1λh∗d,X=a2Nhμa3a4λh∗+d+λh∗dH1λh∗dH2+a1a3a4λh∗+d.

Letfλh∗=Yλh∗X+dH3a3a4λh∗+d-1.

Substituting λh∗ by 0 and +∞, it follows that f+∞=-1<0, andf0=bcNv∗γa1H1+μa2H2μNha1a2a3a4+H3a3a4-1=γa1H1Nv∗bc+αa2H2Mv∗bcμNha1a2a3a4+H3a3a4-1>R0-1>0.

According to the existence theorem of zero point, when R0>1, there exists λh∗>0, such that fλh∗=0, indicating the existence of endemic equilibrium for the system (1). □

Model application

The massive outbreak of Zika in Brazil during 2015 and 2016 is used as a typical prototype to validate the proposed model. The estimate_R function in EpiEstim package of R language is used to calculate the effective reproduction number Rt based on weekly morbidity time series, intergenerational distribution and window period34. Rt is the average number of people someone infected at time t can infect over their entire infectious lifespan, which can quantify the immediate transmissibility. Here Rt is used to determine the parameter b in the model. Since clinical studies have shown that the viral load of asymptomatically infected patients with Zika is about half that of symptomatic patients, it is assumed that the transmission probability of asymptomatic people is half that of symptomatic people26,27. In addition, due to the large temperature difference between the north and south of Brazil, the average weekly temperature T=24.15∘C of the three cities of Manaus, Brasilia and Porto Alegre (divided into the north, central and south of Brazil) is employed in the model parameters expressions.

In order to verify the model of case study, MCMC method is used to quantify the uncertain parameters: ϑ, β and p. In the absence of surveillance data, it is assumed that the initial values of total mosquitoes sizes are at the positive stable level, Mv(0) and Nv(0) are given by (). The initial values of the numbers of vector in latent and infected states are also estimated by MCMC method. The MCMC is run for 200,000 iterations for each parameter and the posterior distributions are compiled from the final 80% of the iterations. The model is validated with 95% confidence interval of posterior estimations by sampling 1000 times of the posterior distributions of the estimated parameters and by incorporating them into the model. The normalized forward sensitivity and global sensitivity are used to quantify the importance of parameters related to the modeling incidence and the reproduction number R0. The normalized forward sensitivity is computed by ∂R0∂xxR0 for parameter x. The global sensitivity is realized by computing partial rank correlation coefficient (PRCC) based on Latin hypercube sampling for the model inputs and outputs35.

Study area and data collection

Brazil is located in the eastern South America, with longitudes 35∘ W to 74∘ W and latitudes 5∘ N to 35∘ S. It is the largest country in South America, whose area is about 8,514,900 square kilometers. Brazil has typical tropical climate, with the annual average temperature as 20–28 ∘C. Such climate is suitable for the growth of Aedes mosquitoes. As a result, Brazil faces public health problems related to mosquito-borne diseases.

The study use medical records of human Zika infections reported in Brazil from January 2015 to September 2016. The weekly numbers of cases is collected from the World Health Organization36 and the Brazilian Ministry of Health, which is shown in Table S2 (see Supplementary Information [SI]). Brazilian population data is obtained from the Brazilian Institute of Statistics (https://www.ibge.gov.br/). Brazil’s weekly average temperature record is extracted from the Brazilian Meteorological Agency (https://previsao.inmet.gov.br/).

Result

The reporting data shows that Brazil witnessed the first human infection in January 2015, but few case is recorded in the following 2 months. ZIKV began to expand since April, with the first epidemic peak in early July. After a continuous low incidence, human infections reached a higher peak in February 2016 and then decrease rapidly until zero in September. By fitting the epidemic curve of the cumulative number of weekly cases by the model, the estimated results of unknown parameters are shown in Table S1 and Fig. S1. As shown in Fig. 3a, it is observed that the estimated parameters enable the model to draw a good fitting capacity, except the first peak. The fitting deviation is possibly due to the temporal heterogeneity of transmission parameters and detection efficiency of human cases. Uncertainty analysis indicates that the model is robust in exploring transmission dynamics, which can draw consistent evolution of weekly accumulate incidence in case of random sampling (see Fig. S1 in SI).

Sensitivity analysis is used to quantify the response of model outputs to parameter variation. Results from Fig. 3 (b) indicate that the most sensitive parameter to the modeling infection is environmental capacity rate of mosquitoes (φ), followed by the human-to-human transmission rate (β) and the initial value of infected adult mosquitoes (Iv). Yet the model output is not sensitive to the transmission rate from infected person fecal to larval (p).Figure 3 (a) The fitting results of the ZIKV cases in Brazil from the proposed model. The light colored area is the 95% confidence intervals (CIs) for all 1000 simulations. (b) Global sensitivity analysis for weekly incidence, where the PRCCs are the mean values in each week, and ∗ indicates a significant difference from zero (with p value <0.01).

The contributions of different routes to the ZIKV infection in Brazil is estimated by splitting the values of the basic reproduction number R0. Substituting the estimated parameters (in which b is chosen as the average value of the previous 7 weeks) into Eq. (9) yields the basic reproduction number to be R0=2.13 (95% CI 1.61–2.64), in which the contribution of mosquito transmission is 2.04 (95% CI 1.53–2.55). Moreover, the transmission of the virus by mosquitoes could be divided into adult mosquitoes and larval mosquitoes in sewage, in which the latter contribution to R0 is estimated to be 0.48 (95% CI 0.26–0.71).

Results of sensitivity analysis toward R0 are presented in Fig. 4. Two different methods of sensitivity analysis show similar results, indicating the robustness of R0 to the parameters used. It is observed that the mortality of adult mosquitoes (μ), infection period of human beings (η), mosquito biting rate (c), environmental capacity rate of mosquitoes (φ), human-mosquito transmission rates (a and b) and transition rate of mosquitoes from larval to adult (α) are most sensitive parameters to determine R0. Yet R0 is less sensitive to the transmission rates from fecal to larval and from human to human (p and β), oviposition rate of adult mosquitoes (θ) proportion of symptomatical infections (ϕ), human death rate (d) and relative infectivity of asymptomatic infections (q and τ). These parameters could play minor roles in causing human infection of ZIKV.Figure 4 Sensitivity analysis of the basic reproduction number R0. (a) Normalize forward sensitivity; (b) global sensitivity with PRCC.

Figure 5 shows the effects of different transmission paths of the Zika outbreak in Brazil on the cumulative number of human infections from January 2015 to September 2016. The fitting results demonstrate that the total infections could be 304,648 (95% CI 304,096–305,199). If without human infection by sex or without larvae mosquito infection by sewage, this number could be 242,487 (95% CI 242,025–242,949) or 297,115 (95% CI 296,558–297,674). If without both of the above transmission routs, it could become 236,455 (95% CI 235,994–236,918). Of these three assumptions, the total number of human infections decreased by 20.4%, 2.47% and 22.38%, respectively. Moreover, in the absence of asymptomatic infection, the cumulative number of human infections could be 254,715 (95% CI 254,238–255,192), making human infections decrease by 16.39%. In this case, the above-mention three circumstance lead to the declines of human infections by 6.43%, 0.79% and 7.17%, respectively.Figure 5 Influence of cutting off different transmission routes on the outbreak of Zika. The cumulative numbers of humans infection in Brazil with 95% CIs are estimated in cases of different transmission routes during January 2015 and September 2016.

Figure 6 shows the time evolution of human infection in cases of different human-to-human transmission rate (β), larval mosquito transmission rate in sewage (p) and temperature. It is observed that the increase of transmission rates through sex and sewage would lead to moderately higher number of human cases and slightly faster of disease transmission, Such effect caused by high sex transmission rate is more significant. Furthermore, temperature around 28.7∘C is most favourable for ZIKV infection, and temperature away from this value could cause low morbidity and low infection risk.Figure 6 Impacts of (a) sexual transmission rate β, (b) sewage transmission rate p and (c) temperature on the number of new infections cases, which is achieved by simulating the proposed model with other parameters equal to those of the fitting results.

Figure 7 shows the effectiveness of the implementation of control measures on the spread of ZIKV infection in Brazil during January 2015 and November 2016. The intervention is measured by the effective reproduction number Rt. Here the values of Rt are fixed to be 2.17, 2.12, 1.98, 1.83, 1.70 and 1.62, which are its average values of in the previous 1–5, 1–10, 1–15, 1–20, 1–25 and 1–30 weeks, respectively. The simulation results indicate that (1) if Rt = 2.17 (few intervention measures), the cumulative number of human infections could reach 37.2 million, that is over 121 times of reported cases, with early peak of new cases as 3.7 million around the 37th week; (2) if Rt = 1.62 (limited intervention measures), the cumulative number of human cases could be 36 million, with peak number of new cases as 2.1 million around the 57th week. It is observed that the decrease of Rt causes an evident decline of humans infections and quick arriving of peak infection.Figure 7 Impacts of different effective reproduction number on the number of human infection in Brazil during January 2015 and November 2016, which is achieved by simulating the proposed model with other parameters equal to those of fitting results.

Discussion

A modeling framework for inferring ZIKV transmission patterns is attempted in this paper. Technologically, a new SEIR-AJ-SEI dynamic model is established, which couples the ZIKV circulation among/between mosquitoes and humans under potential routes, including mosquito bite, sex contact, and sewage breed. Compared with existing ZIKV model, such as using a discrete stochastic SEIR-SEI model to predict the optimal effect of bednets, infection treatment and insecticide spraying on disease transmission18, using a SEIIAR-SEI model to infer the impact of mosquito-borne and sexual transmission on ZIKV spread9, using a continuing climate-driven SEIIIR-SEI model to study threshold dynamics in a seasonal model of Zika virus disease19, using a high-dimensional ODE system to describe the joint dynamics of Zika and dengue17, and using partial differential equations model to understand how spatial heterogeneity of the vector and host populations influence ZIKV dynamics16, the proposed model is a combined update of recent works, which converges more dynamical detail.

First, the ZIKV transmission dynamics are clarified mathematically. The basic reproduction number R0 is calculated by using the next generation matrix, which is found to be determined by all the transmission routes. It is verified that the disease-free equilibrium is locally stable when the associated basic reproduction number is less than unity, and there exits endemic equilibrium when basic reproduction number is larger than unity. If without larval infection in virus sewage, central manifold theorem demonstrates that the system is capable of undergoing the phenomenon of backward bifurcation, under which the stable disease-free equilibrium would co-exist with a stable endemic equilibrium and an unstable endemic equilibrium. Such phenomenon could be caused by the combined effects of multiple transmission routes. The existence of larval infection in virus sewage would cause more complicated transmission dynamics. Dynamic details indicate that great efforts could be needed for preventing ZIKV infection.

Second, the proposed model is further validated to explore more transmission dynamics by fitting the reported cases of ZIKV infection in Brazil from January 2015 to September 2016. Two important insights based on this study may provide scientific clues for evaluating the infection risk and guiding control.

On the one hand, the impact of the transmission routes on the Zika epidemic is evaluated. It is found that mosquito-human infection by bite is still the prominent path for ZIKV occurrence, accounting for 85.7% of the basic reproduction number R0, but the contributions by sexual transmission and larval transmission in sewage are 3.5% and 10.8%, respectively, both of which are less than 1. Therefore, these two routes of infection are not enough to trigger a large-scale outbreak of ZIKV, which is consistent with previous studies9,11,15. The sensitivity analysis further confirms that ZIKV infection is dominated by the parameters related to mosquito ecology, rather than those parameters related to asymptomatic or human-human transmission. Yet the latter two transmission routes can limitedly accelerate the development of Zika epidemic and prolongs the outbreak time. Such effects would be more significant for higher concentration of ZIKV in sewage and larger probability of human-to-human infection. Therefore, the prevention and control of ZIKV should target at reducing the infection through mosquito bite, and do not ignore the infection through sex and sewage.

On the other hand, the situations of ZIKV transmission in Brazil are evaluated. The present study suggests that multiple modes of transmission and suitable temperature may be responsible for the large outbreak of ZIKV in Brazil in 2015–2016. Nevertheless, the intervention implemented in Brazil plays an important role in controlling ZIKV infections. If without intervention, the number of human infections with ZIKV in Brazil would increase rapidly37, and result in more than 37.2 million cases. After the implementation of control measures in May 2015, the number of effective reproduction Rt decreased from 1.69 to less than 1, leading to a rapid decline in morbidity. However, human infections began to rebound rapidly in November of the same year, leading to the second peak of cases in Brazil at 59th week. Meanwhile, control measures were intensified, including mobilizing Brazilian army to support community health services37, house-to-house visits and the elimination of potential Aedes breeding sites37, aerial spraying of the product to kill larvae or adult mosquito38, and reduction of breeding sites through drainage of standing water, waste management and education about mosquitoes and personal protection measures39. Under these control measures, the incidence and effective reproduction number dropped fast. The present study indicates that such comprehensive and intensified ZIKV control strategies are highly effective in curtailing ZIKV transmission.

The following limitations need to be clarified. First, since the proposed model is only used to fit the surveillance data reported in Brazil, there may be geographical differences in its application to other countries. Second, since it is impossible to obtain the true values of some model parameters, they are extracted from the literature or are estimated by MCMC method. Third, the study dose not include all the underlying factors, such as human mobility and climate (except temperature). However, the model takes into account the most influential factors and incorporates model parameterizations, providing confidence in the model output for future analysis.

Supplementary Information

Supplementary Information.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-58025-7.

Acknowledgements

This research was jointly is funded by the National Natural Science Foundation of China (12171116), Director Fund Project of the Ministry of Education Key Laboratory of Cognitive Radio and Information Processing (CRKL210106) and Guangxi Key Laboratory of Cryptography and Information Security (GCIS201707).

Author contributions

L.W.: formal analysis, methodology, visualization, funding acquisition, writing—original draft; Q.J.: data curation, methodology, software, visualization, writing—review and editing; G.Z.: resources, conceptualization, funding acquisition, writing—review and editing; G.O.: methodology, software, writing—review and editing; T.T.: project administration, visualization, supervision, funding acquisition, writing—review and editing.

Data availability

All data generated or analysed during this study are included in this article and its supplementary information files.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Simpson D Zika virus infection in man Trans. R. Soc. Trop. Med. Hyg. 1964 58 4 335 338 10.1016/0035-9203(64)90200-7 14175744
Simpson, D. et al. Zika virus infection in man. Trans. R. Soc. Trop. Med. Hyg. 58(4), 335–338. 10.1016/0035-9203(64)90200-7 (1964).14175744 10.1016/0035-9203(64)90200-7
2. Musso D Nilles E Cao-Lormeau VM Rapid spread of emerging Zika virus in the Pacific area Clin. Microbiol. Infect. 2014 20 10 O595 O596 10.1111/1469-0691.12707 24909208
Musso, D., Nilles, E. & Cao-Lormeau, V. M. Rapid spread of emerging Zika virus in the Pacific area. Clin. Microbiol. Infect. 20(10), O595–O596. 10.1111/1469-0691.12707 (2014).24909208 10.1111/1469-0691.12707
3. Bogoch II Brady OJ Kraemer MU German M Creatore MI Kulkarni MA Brownstein JS Mekaru SR Hay SI Groot E Anticipating the international spread of Zika virus from Brazil Lancet 2016 387 10016 335 336 10.1016/S0140-6736(16)00080-5 26777915
Bogoch, I. I. et al. Anticipating the international spread of Zika virus from Brazil. Lancet 387(10016), 335–336. 10.1016/S0140-6736(16)00080-5 (2016).26777915 10.1016/S0140-6736(16)00080-5
4. Pan American Health Organization (PAHO). Zika virus infection. https://www.paho.org/en/documents/25-august-2017-zika-epidemiological-update-0.
5. Mlakar J Korva M Tul N Popović M Poljšak-Prijatelj M Mraz J Kolenc M Resman Rus K Vesnaver Vipotnik T Fabjan Vodušek V Zika virus associated with microcephaly N. Engl. J. Med. 2016 374 10 951 958 10.1056/NEJMoa1600651 26862926
Mlakar, J. et al. Zika virus associated with microcephaly. N. Engl. J. Med. 374(10), 951–958. 10.1056/NEJMoa1600651 (2016).26862926 10.1056/NEJMoa1600651
6. Cauchemez S Besnard M Bompard P Dub T Guillemette-Artur P Eyrolle-Guignot D Salje H Van Kerkhove MD Abadie V Garel C Association between Zika virus and microcephaly in French Polynesia, 2013–15: A retrospective study Lancet 2016 387 10033 2125 2132 10.1016/S0140-6736(16)00651-6 26993883
Cauchemez, S. et al. Association between Zika virus and microcephaly in French Polynesia, 2013–15: A retrospective study. Lancet 387(10033), 2125–2132. 10.1016/S0140-6736(16)00651-6 (2016).26993883 10.1016/S0140-6736(16)00651-6
7. Duffy MR Chen TH Hancock WT Powers AM Kool JL Lanciotti RS Pretrick M Marfel M Holzbauer S Dubray C Zika virus outbreak on yap island, federated states of micronesia N. Engl. J. Med. 2009 360 24 2536 2543 10.1056/NEJMoa0805715 19516034
Duffy, M. R. et al. Zika virus outbreak on yap island, federated states of micronesia. N. Engl. J. Med. 360(24), 2536–2543. 10.1056/NEJMoa0805715 (2009).19516034 10.1056/NEJMoa0805715
8. Musso D Nhan T Robin E Roche C Bierlaire D Zisou K Yan AS Cao-Lormeau V Broult J Potential for Zika virus transmission through blood transfusion demonstrated during an outbreak in French Polynesia, november 2013 to february 2014 Eurosurveillance 2014 19 14 20761 10.2807/1560-7917.ES2014.19.14.20761 24739982
Musso, D. et al. Potential for Zika virus transmission through blood transfusion demonstrated during an outbreak in French Polynesia, november 2013 to february 2014. Eurosurveillance 19(14), 20761. 10.2807/1560-7917.ES2014.19.14.20761 (2014).24739982 10.2807/1560-7917.ES2014.19.14.20761
9. Gao D Lou Y He D Porco TC Kuang Y Chowell G Ruan S Prevention and control of zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis Sci. Rep. 2016 6 1 1 10 10.1038/srep28070 28442746
Gao, D. et al. Prevention and control of zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis. Sci. Rep. 6(1), 1–10. 10.1038/srep28070 (2016).28442746 10.1038/srep28070
10. Agusto FB Bewick S Fagan W Mathematical model of Zika virus with vertical transmission Infect. Dis. Model. 2017 2 2 244 267 10.1016/j.idm.2017.05.003 29928740
Agusto, F. B., Bewick, S. & Fagan, W. Mathematical model of Zika virus with vertical transmission. Infect. Dis. Model. 2(2), 244–267. 10.1016/j.idm.2017.05.003 (2017).29928740 10.1016/j.idm.2017.05.003
11. Wang L Zhao H Modeling and dynamics analysis of Zika transmission with contaminated aquatic environments Nonlinear Dyn. 2021 104 845 862 10.1007/s11071-021-06289-3
Wang, L. & Zhao, H. Modeling and dynamics analysis of Zika transmission with contaminated aquatic environments. Nonlinear Dyn. 104, 845–862. 10.1007/s11071-021-06289-3 (2021).10.1007/s11071-021-06289-3
12. Wang L Zhao H Oliva SM Zhu H Modeling the transmission and control of Zika in Brazil Sci. Rep. 2017 7 1 7721 10.1038/s41598-017-07264-y 28798323
Wang, L., Zhao, H., Oliva, S. M. & Zhu, H. Modeling the transmission and control of Zika in Brazil. Sci. Rep. 7(1), 7721. 10.1038/s41598-017-07264-y (2017).28798323 10.1038/s41598-017-07264-y
13. Bonaldo MC Ribeiro IP Lima NS Dos Santos AA Menezes LS da Cruz SO de Mello IS Furtado ND de Moura EE Damasceno L Isolation of infective Zika virus from urine and saliva of patients in Brazil PLoS Negl. Trop. Dis. 2016 10 6 e0004816 10.1371/journal.pntd.0004816 27341420
Bonaldo, M. C. et al. Isolation of infective Zika virus from urine and saliva of patients in Brazil. PLoS Negl. Trop. Dis. 10(6), e0004816. 10.1371/journal.pntd.0004816 (2016).27341420 10.1371/journal.pntd.0004816
14. Du S Liu Y Liu J Zhao J Champagne C Tong L Zhang R Zhang F Qin CF Ma P Aedes mosquitoes acquire and transmit Zika virus by breeding in contaminated aquatic environments Nat. Commun. 2019 10 1 1 11 10.1038/s41467-019-09256-0 30602773
Du, S. et al. Aedes mosquitoes acquire and transmit Zika virus by breeding in contaminated aquatic environments. Nat. Commun. 10(1), 1–11. 10.1038/s41467-019-09256-0 (2019).30602773 10.1038/s41467-019-09256-0
15. Towers S Brauer F Castillo-Chavez C Falconar AK Mubayi A Romero-Vivas CM Estimate of the reproduction number of the Zika virus outbreak in Barranquilla, Colombia, and estimation of the relative role of sexual transmission Epidemics 2015 17 2016 50 55 10.1016/j.epidem.2016.10.003
Towers, S. et al. Estimate of the reproduction number of the Zika virus outbreak in Barranquilla, Colombia, and estimation of the relative role of sexual transmission. Epidemics 17(2016), 50–55. 10.1016/j.epidem.2016.10.003 (2015).10.1016/j.epidem.2016.10.003
16. Cai Y Ding Z Yang B Peng Z Wang W Transmission dynamics of Zika virus with spatial structure—a case study in Rio de Janeiro, Brazil Phys. A 2019 514 729 740 10.1016/j.physa.2018.09.100
Cai, Y., Ding, Z., Yang, B., Peng, Z. & Wang, W. Transmission dynamics of Zika virus with spatial structure—a case study in Rio de Janeiro, Brazil. Phys. A 514, 729–740. 10.1016/j.physa.2018.09.100 (2019).10.1016/j.physa.2018.09.100
17. Wang L Zhao H Dynamics analysis of a Zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement Phys. A 2019 522 248 273 10.1016/j.physa.2019.01.099
Wang, L. & Zhao, H. Dynamics analysis of a Zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement. Phys. A 522, 248–273. 10.1016/j.physa.2019.01.099 (2019).10.1016/j.physa.2019.01.099
18. Bonyah E Khan MA Okosun K Islam S A theoretical model for Zika virus transmission PLoS One 2017 12 10 e0185540 10.1371/journal.pone.0185540 28977007
Bonyah, E., Khan, M. A., Okosun, K. & Islam, S. A theoretical model for Zika virus transmission. PLoS One 12(10), e0185540. 10.1371/journal.pone.0185540 (2017).28977007 10.1371/journal.pone.0185540
19. Ibrahim MA Dénes A Threshold dynamics in a model for Zika virus disease with seasonality Bull. Math. Biol. 2021 83 4 1 28 10.1007/s11538-020-00844-6
Ibrahim, M. A. & Dénes, A. Threshold dynamics in a model for Zika virus disease with seasonality. Bull. Math. Biol. 83(4), 1–28. 10.1007/s11538-020-00844-6 (2021).10.1007/s11538-020-00844-6
20. Reiner RC Jr Perkins TA Barker CM A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010 J. R. Soc. Interface 2013 10 81 20120921 10.1098/rsif.2012.0921 23407571
Reiner, R. C. Jr. et al. A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010. J. R. Soc. Interface 10(81), 20120921. 10.1098/rsif.2012.0921 (2013).23407571 10.1098/rsif.2012.0921
21. Moulay D Alaoui MAA Cadivel M The Chikungunya disease: Modelling, vector and transmission global dynamics Math. Biosci. 2011 229 50 63 10.1016/j.mbs.2010.10.008 21070789
Moulay, D., Alaoui, M. A. A. & Cadivel, M. The Chikungunya disease: Modelling, vector and transmission global dynamics. Math. Biosci. 229, 50–63. 10.1016/j.mbs.2010.10.008 (2011).21070789 10.1016/j.mbs.2010.10.008
22. Traore B Koutou O Sangare B A global mathematical model of malaria transmission dynamics with structured mosquito population and temperature variations Nonlinear Anal. RWA 2020 53 103081 10.1016/j.nonrwa.2019.10308
Traore, B., Koutou, O. & Sangare, B. A global mathematical model of malaria transmission dynamics with structured mosquito population and temperature variations. Nonlinear Anal. RWA 53, 103081. 10.1016/j.nonrwa.2019.10308 (2020).10.1016/j.nonrwa.2019.10308
23. Wesolowski A Qureshi T Boni MF Impact of human mobility on the emergence of dengue epidemics in Pakistan Proc. Natl. Acad. Sci. 2015 112 38 11887 11892 10.1073/pnas.1504964112 26351662
Wesolowski, A. et al. Impact of human mobility on the emergence of dengue epidemics in Pakistan. Proc. Natl. Acad. Sci. 112(38), 11887–11892. 10.1073/pnas.1504964112 (2015).26351662 10.1073/pnas.1504964112
24. Yang H Macoris M Galvani K Andrighetti M Wanderley D Assessing the effects of temperature on the population of Aedes Aegypti, the vector of dengue Epidemiol. Infect. 2009 137 8 1188 1202 10.1017/S0950268809002040 19192322
Yang, H., Macoris, M., Galvani, K., Andrighetti, M. & Wanderley, D. Assessing the effects of temperature on the population of Aedes Aegypti, the vector of dengue. Epidemiol. Infect. 137(8), 1188–1202. 10.1017/S0950268809002040 (2009).19192322 10.1017/S0950268809002040
25. Lambrechts L Paaijmans KP Fansiri T Impact of daily temperature fluctuations on dengue virus transmission by Aedes Aegypti Proc. Natl. Acad. Sci. 2011 108 18 7460 7465 10.1073/prias.1101377108 21502510
Lambrechts, L. et al. Impact of daily temperature fluctuations on dengue virus transmission by Aedes Aegypti. Proc. Natl. Acad. Sci. 108(18), 7460–7465. 10.1073/prias.1101377108 (2011).21502510 10.1073/prias.1101377108
26. Aubry M Richard V Green J Broult J Musso D Inactivation of Zika virus in plasma with amotosalen and ultraviolet a illumination Transfusion 2016 56 1 33 40 10.1111/trf.13271 26283013
Aubry, M., Richard, V., Green, J., Broult, J. & Musso, D. Inactivation of Zika virus in plasma with amotosalen and ultraviolet a illumination. Transfusion 56(1), 33–40. 10.1111/trf.13271 (2016).26283013 10.1111/trf.13271
27. Musso D Gubler DJ Zika virus Clin. Microbiol. Rev. 2016 29 3 487 524 10.1056/NEJMra1602113 27029595
Musso, D. & Gubler, D. J. Zika virus. Clin. Microbiol. Rev. 29(3), 487–524. 10.1056/NEJMra1602113 (2016).27029595 10.1056/NEJMra1602113
28. Haby MM Pinart M Elias V Reveiz L Prevalence of asymptomatic Zika virus infection: A systematic review Bull. World Health Organ. 2018 96 6 402 10.2471/BLT.17.201541 29904223
Haby, M. M., Pinart, M., Elias, V. & Reveiz, L. Prevalence of asymptomatic Zika virus infection: A systematic review. Bull. World Health Organ. 96(6), 402. 10.2471/BLT.17.201541 (2018).29904223 10.2471/BLT.17.201541
29. Charrel RN Leparc-Goffart I Pas S Lamballerie XD Reusken C State of knowledge on Zika virus for an adequate laboratory response Bull. World Health Organ. 2017 94 8 10.2471/BLT.16.171207
Charrel, R. N., Leparc-Goffart, I., Pas, S., Lamballerie, X. D. & Reusken, C. State of knowledge on Zika virus for an adequate laboratory response. Bull. World Health Organ. 94, 8. 10.2471/BLT.16.171207 (2017).10.2471/BLT.16.171207
30. Paz-Bailey G Rosenberg ES Doyle K Munoz-Jordan J Santiago GA Klein L Perez-Padilla J Medina FA Waterman SH Adams LE Persistence of Zika virus in body fluids N. Engl. J. Med. 2018 379 13 1234 1243 10.1056/NEJMoa1613108 28195756
Paz-Bailey, G. et al. Persistence of Zika virus in body fluids. N. Engl. J. Med. 379(13), 1234–1243. 10.1056/NEJMoa1613108 (2018).28195756 10.1056/NEJMoa1613108
31. Van den Driessche P Watmough J Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Math. Biosci. 2002 180 1–2 29 48 10.1016/S0025-5564(02)00108-6 12387915
Van den Driessche, P. & Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48. 10.1016/S0025-5564(02)00108-6 (2002).12387915 10.1016/S0025-5564(02)00108-6
32. Castillo-Chavez C Song B Dynamical models of tuberculosis and their applications Math. Biosci. Eng. 2004 1 2 361 10.3934/mbe.2004.1.361 20369977
Castillo-Chavez, C. & Song, B. Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361. 10.3934/mbe.2004.1.361 (2004).20369977 10.3934/mbe.2004.1.361
33. Shuai Z van den Driessche P Global stability of infectious disease models using lyapunov functions SIAM J. Appl. Math. 2013 73 4 1513 1532 10.1137/120876642
Shuai, Z. & van den Driessche, P. Global stability of infectious disease models using lyapunov functions. SIAM J. Appl. Math. 73(4), 1513–1532. 10.1137/120876642 (2013).10.1137/120876642
34. Thompson R Stockwin J van Gaalen RD Polonsky J Kamvar Z Demarsh P Dahlqwist E Li S Miguel E Jombart T Improved inference of time-varying reproduction numbers during infectious disease outbreaks Epidemics 2019 29 100356 10.1016/j.epidem.2019.100356 31624039
Thompson, R. et al. Improved inference of time-varying reproduction numbers during infectious disease outbreaks. Epidemics 29, 100356. 10.1016/j.epidem.2019.100356 (2019).31624039 10.1016/j.epidem.2019.100356
35. Marino S Hogue IB Ray CJ Kirschner DE A methodology for performing global uncertainty and sensitivity analysis in systems biology J. Theor. Biol. 2008 254 1 178 196 10.1016/j.jtbi.2008.04.011 18572196
Marino, S., Hogue, I. B., Ray, C. J. & Kirschner, D. E. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254(1), 178–196. 10.1016/j.jtbi.2008.04.011 (2008).18572196 10.1016/j.jtbi.2008.04.011
36. World Health Organization (WHO). Zika cases from the World Health Organization. http://www.who.int/emergencies/zika-virus/situation-report/en/.
37. Heukelbach J Alencar CH de Kelvin AA de Oliveira WK Góes Cavalcanti LP Zika virus outbreak in Brazil J. Infect. Dev. Countr. 2016 10 2 116 120 10.3855/jidc.8217
Heukelbach, J., Alencar, C. H., de Kelvin, A. A., de Oliveira, W. K. & Góes Cavalcanti, L. P. Zika virus outbreak in Brazil. J. Infect. Dev. Countr. 10(2), 116–120. 10.3855/jidc.8217 (2016).10.3855/jidc.8217
38. Schoch-Spana M Watson C Ravi S Meyer D Pechta LE Rose DA Lubell KM Podgornik MN Sell TK Vector control in Zika-affected communities: Local views on community engagement and public health ethics during outbreaks Prev. Med. Rep. 2020 18 101059 10.1016/j.pmedr.2020.101059 32154093
Schoch-Spana, M. et al. Vector control in Zika-affected communities: Local views on community engagement and public health ethics during outbreaks. Prev. Med. Rep. 18, 101059. 10.1016/j.pmedr.2020.101059 (2020).32154093 10.1016/j.pmedr.2020.101059
39. Stetson A Zika Virus: Control Measures for an Emerging Pathogen in Brazil 2017 Boston University
Stetson, A. Zika Virus: Control Measures for an Emerging Pathogen in Brazil (Boston University, 2017).
