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An SIR model of Zika virus (ZIKV) spread is formulated that includes ZIKV infections to newborns. Analytically, the model has one disease free and one endemic equilibrium point. The free one is stable for some conditions when
R
_{0} and unstable when
R
_{0}>1. In Brazil, when
R
_{0}≈2>1 ZIKV infections expand and for
R
_{0} = 0.875<1 diminishes. There were 67 micro-cephaly babies confirmed to have linked with ZIKV infections as of February 2016 in Brazil. Simulation predicts that the number rose to not less than 2100 in January 2017 and expected to increase around 4350 by January 2018 if not controlled. Simulation of the endemic equilibrium point indicates that there is more number of individuals in the recovered class than in the infectious class contrasting the initial state of the epidemic. This recovered class serves as a silent natural resistance against spread of the epidemic. Measures to control ZIKV infections have been suggested by analyzing parameters linked to transmission. The base is controlling basic reproduction number (
R
_{0}) of the model. There are parameters for human-mosquito transmission and some for sexual-transmission factor. It appears that controlling spread of ZIKV infections by human-mosquito transmission may greatly reduce the value of
R
_{0}.

On February 1^{st}, 2016, the World Health Organization declared ZIKV epidemic in the Americas as a Public Health Emergency of International Concern. The foremost reason is due to an emerging linkage with congenital birth anomalies such as microcephaly and Guillain-Barre syndrome [

Despite the fact that attack of ZIKV shows moderate disease symptoms, like that of the clinical presentation and transmission cycles in the epidemiology of Chikunganya and Dengue, symptoms of ZIKV disease encompasses conjunctivitis, skin rash, fever, muscle pains in various joints, non comforting nuisance and headache, which can last normally between 2 to 7 days time period. Not limited to those, it has been found that about 80% of individuals infected by ZIKV are asymptomatic [

The causes are numerous, including chromosomal abnormalities, intrauterine infections, exposure to toxic substances during pregnancy, and severe malnutrition [

No specific antiviral medication is available for ZIKV disease. Treatment is generally supportive and can include rest, fluids, and use of analgesics and antipyretics. Aspirin including other nonsteroidal anti-inflammatory drugs should be avoided until dengue can be ruled out to reduce the risk of hemorrhage. Febrile pregnant women should be treated with acetaminophen. Persons infected with ZIKV, dengue, or Chikungunya virus should be protected from further mosquito exposure during the first few days of illness to reduce the risk for local transmission [

The very first incidence of ZIKV was identified in 1947 from species of monkey in the African forest called Zika from Uganda. And the first isolated case identified from humans in 1952 also from Uganda and then later from a neighbor country, the United Republic of Tanzania. However, comprehensive genomic comparison showed different sub-classes reflecting the existence of two main lineages, one African and the other from Asia lineage [

ZIKV has been steadily increasing its geographical domain, ever since its initial appearance. The pace of ZIKV outbreaks as epidemic in the Americas is on rise. Similarly, the spread of this monster is reported in more regions of the world that were previously completely unaffected, that includes Europe [

Brazil’s first confirmed Zika infection was in March 2015. Over five years preceding 2015, the country saw between 130 and 170 cases of microcephaly each year [

Regarding general ZIKV infections, data announced on December 2015, by the ECDC estimate that between 440,000 and 1.3 million cases by autochthonous transmission of ZIKV have occurred in Brazil (

Basing on the nature of epidemic, though with a relatively long history since its knowledge in 1947, few researches have been conducted. Probably, the main reason is that its effects where not vividly clear to human being as of now. However, this does not rule out that in the past people had not been affected by the problem. About three years ago, a number of researches have been working to explore the disease down so as to come up with some means to control it. According to Rojas DP et al. [

more researches in medicine and other discipline. This paper is more concerned with mathematical models exploring the epidemic. One of the model was done by Adam J. Kucharski et al. in early 2016 [

The model consists of two categories, the humans and mosquitoes with total population

Susceptible individual moves into the infectious class after effective interaction with infected mosquitoes, or sexual contact with and individual in the infectious class. Newborns with ZIKV infection by MTCT enters infected class. The mosquitoes moves from the susceptible to their infectious-class after effective infection through biting an infectious human.

In this context, in all cases the subscript

Parameter | Definition |
---|---|

constant rate of incoming susceptible | |

natural death rates | |

birth rate of newborn who are infected | |

rate of transfer I individuals to R | |

rate of infection by mosquitoes | |

rate of infections through sexual transmission | |

rate of mosquitoes infected by humans | |

effective bites between infected mosquitoes and susceptible humans | |

effective contacts from humans to humans | |

effective bites between infected humans to susceptible mosquitoes |

Assign

Consider to evaluate the basic reproduction number,

Equations i and iv are solved simultaneously, so that

From the system of Equation (3),

Summary of infection rates, gains or losses in classes

The loss-terms matrix,

Calculations for

The eigenvalues,

For endemic equilibrium,

Solving this simultaneously, the equilibrium point is: (5)is obtained from the equation: (6)The equilibrium point reduces to a disease free when.4.1.1. Existence of the Endemic EquilibriumConsider Equation (6):. Assign. Endemic equilibrium exists whenever there is a positive solution (i.e. value of). When,.When,. Therefore at some point in the interval there exists, which proves the existence of endemic equilibrium. As an example, consider a simulation: Choose any; say

By definition, the lower boundaries of the population are zero. Yet, the upper bounds can be deduced from the given dynamical equations. As it is for

That is

Therefore,

A similar approach is used to show that

In order to understand the local stability of the model, Jacobian matrix,

Recall the disease free equilibrium (4),

The characteristic equation,

That is

Proof: Recall that:

For

Choose

Which suggests that

On the other hand when

The characteristic equation is Hurwitz provided that

In the system of Equation (8),

is to estimate the number of newborns who have been infected with ZIKV since beginning of the epidemic in a given community. The assumption is that the rate of infections (

order to estimate the number of infected newborns (

There are a approximations and assumptions for estimating parameters that have been used to simulate the model. Due to inaccessibility of some parameters, assumptions have been made for some in order to fit the model with current trend of epidemic. For the purpose of simulation in the Americas, Brazil is considered as illustrative example. Model is suggesting possible outcomes but also is providing us with the virtual of ZIKV disease features.

Parameter | Units (% per year) | Description | Source |
---|---|---|---|

0.83 | susceptible recruitment rate | geoba.se | |

0.180 | mosquitoes recruitment rate | assumptions | |

1.666 | natural birth rate | geoba.se | |

0.644 | natural death rate | geoba.se | |

0.600 | recovering rate | assumptions | |

36 | mosquito natural death rate | assumptions | |

0.756 | rate of ZVD newborns | assumptions | |

0.375 | Effective contact rate H to | [ | |

1.125 | Effective contact rate H to H | [ | |

1.125 | Effective contact rate | [ |

Initials (2016) | Amount × 10^{6} | Initial value for | Source |
---|---|---|---|

210 | N | worldmeters.com | |

209 | S | worldmeters.com | |

0.672 | I | Estimations [ | |

0.000067 | M | [ | |

0.00 | R | Assumptions | |

10.0 | Estimations | ||

7.0 | Estimations |

The graphs in

For M estimations, the differential equation

simulated for 25 months (

total number of children who has been inflicted by ZIKV. It is clear that this accounts for all children including the initial amount as of January 2016. Up to February 27, 2016 according to Brazilian Ministry of Health, out of all reported microcephaly cases only 67 fulfill the laboratory criteria for ZIKV infection [

Then, the February 2016 cases of microcephaly is assumed to be initial value for January 2016 in this simulation. Estimation for all newborns regardless of those who die later after birth, so long as had been diagnosed with microcephaly linking to ZIKV then are part of this integral. After 25 months since January 2016, the number of ZIKV infected newborns in the same month 2017 is approximated above 2100 (

At the latest, if the situation is not attended or interrupted, naturally may lead to

equilibrium as in

Among other reasons that can lead to microcephaly situation side track the societies attention to ZIKV disease outbreak especially when the infections are low and unnoticeable is not keeping records, especially in most villages of the developing countries. Therefore, appreciation should be given to the Americas, like Brazil for keeping records of microcephaly born children which is one of the reasons lead to notice the break of epidemic [

According to the model (2), two main factors to be controlled in order to diminish or eliminate the epidemic basing on the model include reducing both unsafe sexual-interaction and mosquitoes bites against humans. Controlling the two factors to the extent that

The model (2) estimates that in 15 years for

As for women who want to get pregnant but are not sure of ZIKV infections (no symptoms) but they have been exposed to ZIKV infected regions or had sex with ZIKV diagnosed partners, should wait at least eight weeks after exposure. For those with diagnosed ZIKV infections, wait after eight weeks since the start of symptoms (

The fundamental part in the analysis of model (2) is the basic reproduction number (

Model simulations have been conducted, considering Brazil as an example. However, values of the parameters have been a challenge, some range between values, some no information and others just varies. Parameters for simulation in

this case have been assumed to be constant, therefore some assumptions or estimations have been done in some way to fit the current trend of the epidemic in Brazil. A population of more than 208 million as of January 2016 is found with 672,000 infected individual. Up to February 27, 2016; 67 newborns with microcephaly were confirmed to have been due to ZIKV infections. On simulations, there are at least 2100 in total of babies who have been ZIKV infected by January 2017, this number is expected to rise up about 4350 by January 2018 (

The value of

The study has a lot open for further studies. Analytically, the disease free equilibrium has not fully been analyzed, the obtained disease free equilibrium holds for some conditions leaving one side open for investigations. The endemic equilibrium exists but its stability has been discussed numerically, hence calling for further insights in analytically. As for model simulation, assumptions and estimations for current parameters have been used to assign values. Shall it be possible to finely obtain better parameter estimations, model simulation would produce more reliable results. Nevertheless the model can be modified by reducing assumptions so as to make it more realistic. For example, some rates are not constant, therefore may be made non-constant accordingly to fit the reality hence improving the model. Therefore, for some reasons this study calls for more research into the epidemic of ZIKV infections.

Sincere thanks to the members of SCIRP for their professional performance, and special thanks to managing editor Karen ZHANG for a rare attitude of high quality. Finally, but not the least it is grateful to the National Natural Science Foundation of China (No. 11371161, 11471133) and the Foundation Research Funds for the Central Universities, Central China Normal University (No. CCNU15A02055) for partially supporting this research financially.

Kibona, I.E. and Yang, C.H. (2017) SIR Model of Spread of Zika Virus Infections: ZIKV Linked to Microcephaly Simulations. Health, 9, 1190-1210. https://doi.org/10.4236/health.2017.98086

CDC Centers for Disease Control and Prevention in USA

ECDC European Centre for Disease Prevention and Control

MTCT Mother-to-child-transmission

PAHO Pan American Health Organization

SIR Susceptible Infectives and Recovery

WHO World Health Organization

ZIKV Zika virus

ZVD Zika virus disease

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