Viral infections spread by mosquitoes are a growing threat to human health and welfare. Zika virus (ZIKV) is one of them and has become a global worry, particularly for women who are pregnant. To study ZIKV dynamics in the presence of demographic stochasticity, we consider an established ZIKV transmission model that takes into consideration the disease transmission from human to mosquito, mosquito to human, and human to human. In this study, we look at the local stability of the disease-free and endemic equilibriums. By conducting the sensitivity analysis both locally and globally, we assess the effect of the model parameters on the model outcomes. In this work, we use the continuous-time Markov chain (CTMC) process to develop and analyze a stochastic model. The main distinction between deterministic and stochastic models is that, in the absence of any preventive measures such as avoiding travel to infected areas, being careful from mosquito bites, taking precautions to reduce the risk of sexual transmission, and seeking medical care for any acute illness with a rash or fever, the stochastic model shows the possibility of disease extinction in a finite amount of time, unlike the deterministic model shows disease persistence. We found that the numerically estimated disease extinction probability agrees well with the analytical probability obtained from the Galton–Watson branching process approximation. We have discovered that the disease extinction probability is high if the disease emerges from infected mosquitoes rather than infected humans. In the context of the stochastic model, we derive the implicit equation of the mean first passage time, which computes the average amount of time needed for a system to undergo its first state transition.

1
D.
Gao
,
Y.
Lou
,
D.
He
,
T. C.
Porco
,
Y.
Kuang
,
G.
Chowell
, and
S.
Ruan
, “
Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis
,”
Sci. Rep.
6
(
1
),
28070
(
2016
).
2
A.
Ali
,
B.
Wahid
,
S.
Rafique
, and
M.
Idrees
, “
Advances in research on Zika virus
,”
Asian Pac. J. Trop. Med.
10
(
4
),
321
331
(
2017
).
3
B.
Wahid
,
A.
Ali
,
S.
Rafique
, and
M.
Idrees
, “
Zika: As an emergent epidemic
,”
Asian Pac. J. Trop. Med.
9
(
8
),
723
729
(
2016
).
5
M.
Khalid
and
F. S.
Khan
, “
Stability analysis of deterministic mathematical model for Zika virus
,”
Br. J. Math. Comput. Sci.
19
(
4
),
1
10
(
2016
).
6
J.
Cohen
, “Zika’s long, strange trip into the limelight,” Science, 2016.
7
S.
Lequime
,
J.-S.
Dehecq
,
S.
Matheus
,
F.
de Laval
,
L.
Almeras
,
S.
Briolant
, and
A.
Fontaine
, “
Modeling intra-mosquito dynamics of Zika virus and its dose-dependence confirms the low epidemic potential of Aedes albopictus
,”
PLoS Pathog.
16
(
12
),
e1009068
(
2020
).
8
G.
Grard
,
M.
Caron
,
I. M.
Mombo
,
D.
Nkoghe
,
S.
Mboui Ondo
,
D.
Jiolle
,
D.
Fontenille
,
C.
Paupy
, and
E. M.
Leroy
, “
Zika virus in gabon (Central Africa)–2007: A new threat from Aedes albopictus?
,”
PLoS Neglected Trop. Dis.
8
(
2
),
e2681
(
2014
).
9
M. R.
Duffy
,
T.-H.
Chen
,
W. T.
Hancock
,
A. M.
Powers
,
J. L.
Kool
,
R. S.
Lanciotti
,
M.
Pretrick
,
M.
Marfel
,
S.
Holzbauer
,
C.
Dubray
, and
L.
Guillaumot
, “
Zika virus outbreak on Yap Island, federated states of Micronesia
,”
N. Engl. J. Med.
360
(
24
),
2536
2543
(
2009
).
10
N.
Dahiya
,
M.
Yadav
,
A.
Yadav
, and
N.
Sehrawat
, “
Zika virus vertical transmission in mosquitoes: A less understood mechanism
,”
J. Vector Borne Dis.
59
(
1
),
37
44
(
2022
).
11
A. J.
Kucharski
,
S.
Funk
,
R. M.
Eggo
,
H.-P.
Mallet
,
W. J.
Edmunds
, and
E. J.
Nilles
, “
Transmission dynamics of Zika virus in island populations: A modelling analysis of the 2013–14 French Polynesia outbreak
,”
PLoS Neglected Trop. Dis.
10
(
5
),
e0004726
(
2016
).
12
E.
Bonyah
,
M. A.
Khan
,
K.
Okosun
, and
S.
Islam
, “
A theoretical model for Zika virus transmission
,”
PLoS One
12
(
10
),
e0185540
(
2017
).
13
E. O.
Alzahrani
,
W.
Ahmad
,
M. A.
Khan
, and
S. J.
Malebary
, “
Optimal control strategies of Zika virus model with mutant
,”
Commun. Nonlinear Sci. Numer. Simul.
93
,
105532
(
2021
).
14
N.
Sharma
,
R.
Singh
,
J.
Singh
, and
O.
Castillo
, “
Modeling assumptions, optimal control strategies and mitigation through vaccination to Zika virus
,”
Chaos, Solitons Fractals
150
,
111137
(
2021
).
15
S.
Rezapour
,
H.
Mohammadi
, and
A.
Jajarmi
, “
A new mathematical model for Zika virus transmission
,”
Adv. Differ. Equ.
2020
(
1
),
589
.
16
F. B.
Agusto
,
S.
Bewick
, and
W.
Fagan
, “
Mathematical model of Zika virus with vertical transmission
,”
Infect. Dis. Model.
2
(
2
),
244
267
(
2017
).
17
B.
Hasan
,
M.
Singh
,
D.
Richards
, and
A.
Blicblau
, “
Mathematical modelling of Zika virus as a mosquito-borne and sexually transmitted disease with diffusion effects
,”
Math. Comput. Simul.
166
,
56
75
(
2019
).
18
O.
Maxian
,
A.
Neufeld
,
E. J.
Talis
,
L. M.
Childs
, and
J. C.
Blackwood
, “
Zika virus dynamics: When does sexual transmission matter?
,”
Epidemics
21
,
48
55
(
2017
).
19
C. J.
Kuhlman
,
Y.
Ren
,
B.
Lewis
, and
J.
Schlitt
, “Hybrid agent-based modeling of Zika in the United States,” in 2017 Winter Simulation Conference (WSC) (IEEE, 2017), pp. 1085–1096.
20
S.
Maity
and
P. S.
Mandal
, “
A comparison of deterministic and stochastic plant-vector-virus models based on probability of disease extinction and outbreak
,”
Bull. Math. Biol.
84
(
3
),
1
29
(
2022
).
21
M.
Maliyoni
, “
Probability of disease extinction or outbreak in a stochastic epidemic model for west nile virus dynamics in birds
,”
Acta Biotheor.
69
(
2
),
91
116
(
2021
).
22
L. J.
Allen
, “
A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis
,”
Infect. Dis. Model.
2
(
2
),
128
142
(
2017
).
23
M. Z.
Ndii
and
A. K.
Supriatna
, “
Stochastic mathematical models in epidemiology
,”
Information
20
,
6185
6196
(
2017
).
24
P. S.
Mandal
,
L. J.
Allen
, and
M.
Banerjee
, “
Stochastic modeling of phytoplankton allelopathy
,”
Appl. Math. Model.
38
(
5-6
),
1583
1596
(
2014
).
25
M.
Fahimi
,
K.
Nouri
, and
L.
Torkzadeh
, “A stochastic model for Zika virus transmission,” in 52nd Annual Iranian Mathematics Conference (Shahid Bahonar University of Kerman, 2021).
26
M.
Zevika
and
E.
Soewono
, “
Deterministic and stochastic CTMC models from Zika disease transmission
,”
AIP Conf. Proc.
1937
(
1
),
020023
(
2018
).
27
E.
Soewono
and
G.
Lahodny
, “
On the effect of postponing pregnancy in a Zika transmission model
,”
Adv. Differ. Equ.
2021
(
1
),
1
14
(
2021
).
28
L.
Esteva
and
C.
Vargas
, “
Analysis of a dengue disease transmission model
,”
Math. Biosci.
150
(
2
),
131
151
(
1998
).
29
V.
Margiotta
,
L.
Oglesby
,
T.
Portone
, and
B.
Stephenson
, “Analysis of the spread of malaria disease,” Science, 2010.
30
H. M.
Ali
and
I. G.
Ameen
, “
Optimal control strategies of a fractional order model for Zika virus infection involving various transmissions
,”
Chaos, Solitons Fractals
146
,
110864
(
2021
).
31
G.
Adamu
,
M.
Bawa
,
M.
Jiya
, and
U.
Chado
, “
A mathematical model for the dynamics of Zika virus via homotopy perturbation method
,”
J. Appl. Sci. Environ. Manag.
21
(
4
),
615
623
(
2017
).
32
N.
Goswami
and
B.
Shanmukha
, “
A mathematical analysis of Zika virus transmission with optimal control strategies
,”
Comput. Methods Differ. Equ.
9
(
1
),
117
145
(
2021
).
33
Z.
Yue
and
F. M.
Yusof
, “
A mathematical model for biodiversity diluting transmission of Zika virus through competition mechanics
,”
Discrete Contin. Dyn. Syst.-B
27
(8),
4429
(
2021
).
34
H. R.
Thieme
,
Mathematics in Population Biology
(
Princeton University Press
,
2018
), Vol. 12.
35
P.
Van den Driessche
and
J.
Watmough
, “Further notes on the basic reproduction number,” in Mathematical Epidemiology (Springer, 2008), pp. 159–178.
36
A.
Perasso
, “
An introduction to the basic reproduction number in mathematical epidemiology
,”
ESAIM: Proc. Surv.
62
,
123
138
(
2018
).
37
P.
Van den Driessche
and
J.
Watmough
, “
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission
,”
Math. Biosci.
180
(
1–2
),
29
48
(
2002
).
38
O.
Diekmann
,
J. A. P.
Heesterbeek
, and
J. A.
Metz
, “
On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations
,”
J. Math. Biol.
28
(
4
),
365
382
(
1990
).
39
O.
Diekmann
,
J.
Heesterbeek
, and
M. G.
Roberts
, “
The construction of next-generation matrices for compartmental epidemic models
,”
J. R. Soc. Interface
7
(
47
),
873
885
(
2010
).
40
U.
Sanusi
,
S.
John
,
J.
Mueller
, and
A.
Tellier
, “
Quiescence generates moving average in a stochastic epidemiological model with one host and two parasites
,”
Mathematics
10
(
13
),
2289
(
2022
).
41
K. P.
Hadeler
,
M. C.
Mackey
, and
A.
Stevens
,
Topics in Mathematical Biology
(
Springer
,
2017
).
42
N. G.
Hairston Jr
and
B. T.
De Stasio Jr
, “
Rate of evolution slowed by a dormant propagule pool
,”
Nature
336
(
6196
),
239
242
(
1988
).
43
G. A.
Korn
and
T. M.
Korn
,
Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review
(
Courier Corporation
,
2000
).
44
S.
Marino
,
I. B.
Hogue
,
C. J.
Ray
, and
D. E.
Kirschner
, “
A methodology for performing global uncertainty and sensitivity analysis in systems biology
,”
J. Theor. Biol.
254
(
1
),
178
196
(
2008
).
45
S. M.
Blower
and
H.
Dowlatabadi
, “
Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example
,”
Int. Stat. Rev./Revue Internationale de Statistique
62
,
229
243
(
1994
).
46
L. J. S.
Allen
,
An Introduction to Stochastic Processes with Applications to Biology
(
CRC Press
,
2010
).
47
L. J. S.
Allen
, “An introduction to stochastic epidemic models,” in Mathematical Epidemiology (Springer, 2008), pp. 81–130.
48
L. J.
Allen
and
G. E.
Lahodny Jr
, “
Extinction thresholds in deterministic and stochastic epidemic models
,”
J. Biol. Dyn.
6
(
2
),
590
611
(
2012
).
49
L. J.
Allen
and
P.
van den Driessche
, “
Relations between deterministic and stochastic thresholds for disease extinction in continuous-and discrete-time infectious disease models
,”
Math. Biosci.
243
(
1
),
99
108
(
2013
).
50
L. J.
Allen
, “Branching processes,” in Encyclopedia of Theoretical Ecology (University of California Press, 2012), pp. 112–119.
51
S.
Ditlevsen
and
A.
Samson
, “Introduction to stochastic models in biology,” in Stochastic Biomathematical Models (Springer, 2013), pp. 3–35.
52
P.
Sarathi Mandal
and
S.
Maity
, “
Impact of demographic variability on the disease dynamics for honeybee model
,”
Chaos
32
(
8
),
083120
(
2022
).
53
M.
Maliyoni
,
F.
Chirove
,
H. D.
Gaff
, and
K. S.
Govinder
, “
A stochastic epidemic model for the dynamics of two pathogens in a single tick population
,”
Theor. Popul. Biol.
127
,
75
90
(
2019
).
54
M.
Maliyoni
,
F.
Chirove
,
H. D.
Gaff
, and
K. S.
Govinder
, “
A stochastic tick-borne disease model: Exploring the probability of pathogen persistence
,”
Bull. Math. Biol.
79
(
9
),
1999
2021
(
2017
).
55
G. E.
Lahodny
and
L. J. S.
Allen
, “
Probability of a disease outbreak in stochastic multipatch epidemic models
,”
Bull. Math. Biol.
75
(
7
),
1157
1180
(
2013
).
56
A. L.
Lloyd
,
J.
Zhang
, and
A. M.
Root
, “
Stochasticity and heterogeneity in host–vector models
,”
J. R. Soc. Interface
4
(
16
),
851
863
(
2007
).
57
H.
Farooq
,
M. S.
Parwez
, and
A.
Imran
, “Continuous time Markov chain based reliability analysis for future cellular networks,” in 2015 IEEE Global Communications Conference (GLOBECOM) (IEEE, 2015), pp. 1–6.
58
V. G.
Kulkarni
,
Introduction to Modeling and Analysis of Stochastic Systems
(
Springer
,
2011
), Vol. 1, no. 3.3.
59
D. T.
Gillespie
, “
A general method for numerically simulating the stochastic time evolution of coupled chemical reactions
,”
J. Comput. Phys.
22
(
4
),
403
434
(
1976
).
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