Analysis of model for the transmission dynamics of Zika with sterile insect technique

One of the major reason for the persistence of Zika and other vector borne diseases has been lack of effective mosquito control techniques. Sterile insect technique (SIT) is a non polluting biological method of mosquito control, where sterile mosquitoes are predominantly non reproductive. We present a new deterministic model for the transmission dynamics of Zika, by incorporating both human and mosquito population, with fraction of mosquitoes being sterilized. We consider both aquatic and non-aquatic stages of mosquitoes, so as to evaluate the effect of mosquito control in the transmission of the disease. We computed the basic reproduction number ($R_{0}$), and theoretically analysed the stability properties of the disease-free equilibrium (DFE) and the endemic-equilibrium (EE). In addition, effect of human-human transmission, and other important parameters were assessed. Numerical simulations to support the results will be presented.


Introduction
Zika fever is a mosquito-borne Flavivirus caused by Zika virus (Zv).The virus was first identified in Uganda in 1947, through a monitoring network of sylvatic yellow fever in rhesus monkeys.Five years later, human infection was identified in Uganda and Tanzania.Since then, Zika outbreaks have been recorded in Africa, Americas, Asia and the Pacific [9,16,20].Zv is primarily transmitted by Aedes mosquito, mainly Aedes aegypti (which also transmits Yellow fever, Dengue, Chikungunya, West Nile, and Japanese encephalitis viruses) [20,23].However, sexual transmissions between humans have also been reported [9,19].
Controlling mosquito population is one of the most important tool in the control and/or prevention of vector borne diseases.Sterile insect technique is a promising and non polluting method of vector control, it has been thought of in the control of mosquito borne diseases [29].In this work, we attempt to investigate the impact of sterilization on vector control and disease transmission, by considering the interaction of both mosquito (where fraction of males are sterilized) and human population, this allows us to assess the potential impact of sterilization.

Model formulation
Although complex, life cycle of mosquito is generally divided into two stages; aquatic and non-aquatic.The aquatic stage is denoted by a single compartment A, (this is similar to the assumption in other models in the literature, such as those in [1,11,13,14,17] and some of the references therein).
The non-aquatic stage is divided into seven compartments, consisting of young females (Y ), fertilized non-sterilized females (those who could lay eggs and hatch due to mating with non-sterile male mosquitoes) (F N ), fertilized sterilized females (those who could lay eggs but do not hatch due to mating with sterile male mosquitoes) (F S ), non-sterilized fertilized infected females (F NI ), sterilized fertilized infected females (F SI ), and non-sterilized male mosquitoes (M N ).Sterilized male mosquitoes (M S ) are introduced into the population at a constant rate.Thus, the total mosquito population is given by Human population is divided into susceptible (S H ), infected (I H ) and recovered humans (R H ). The total human population is given by

Incidence function
The incidence in human and mosquito populations are respectively, given by and where, β 1 (N H , N V ) = ρ MH ξ 1 is the adequate contact rate of infectious mosquito with susceptible human.Probability that contact is made with an infectious mosquito is given by 1 N V (F NI + η 1 F SI ), with η 1 ∈ (0, 1) as a modification parameter.β 2 = ρ HH ξ 2 is the adequate contact (sexual) rate of infectious human with susceptible human, 1  N H (I H + η 2 R H ) is the probability that contact is made with an infectious human (while η 2 ∈ (0, 1) is a modification parameter).β 3 = ρ HM ξ 3 is the adequate contact rate of an infectious human with a susceptible female mosquito, I H N H is the probability that contact was made with an infectious human.
For number of bites to be conserved, the total number of bites made by mosquitoes must be equal to the total number of bites received by humans [2,3,21].Thus, the following equation holds Therefore,

Dynamics of mosquito population
The aquatic stage (A) involves egg, larva and pupa.The population is increased from oviposition by reproductive mosquitoes at the rate φ .It reduces due to natural death at the rate µ V , it is assumed that natural death occurs in all mosquito compartments (except sterilized male mosquitoes) at the rate µ V , density dependence death at the rate µ, they mature and move out of aquatic stage at the rate b V .So that, Mosquitoes at the non-aquatic stage are considered adult.Young female mosquito evolve from aquatic stage at the rate rb V A, they mate with non-sterilized male mosquitoes and progress to the compartment of fertilized, non-sterilized susceptible female mosquitoes (F N ) at the rate α N M N M , or with a sterilized male mosquito and move to the compartment of fertilized, sterilized susceptible female mosquito (F S ), at the rate α S M S M .Therefore, this gives Fertilized, non-sterilized susceptible female mosquitoes are generated from compartment Y by mating with a non-sterilized male mosquito (M N ).In order to nourish their eggs before oviposition, they need blood, they will probably bite an infectious human and move to the compartment of fertilized, non-sterilized infectious female mosquitoes (F NI ) at the rate λ V , or they bite a susceptible human and stay in the compartment, in which case, they can lay eggs.They die naturally at the rate µ V .So that, Similarly, the compartment of fertilized, sterilized susceptible female mosquitoes are generated through the fertilization of young female mosquitoes by sterilized male mosquitoes.This population is reduced by infection they acquired following contact with infectious humans and progress to the F SI compartment at a rate λ V .This gives Fertilized, infectious non-sterilized female mosquitoes are generated by the infection of mosquitoes in F N class, and are decreased by natural death.Hence, A fertilized, infectious sterilized female mosquito population is generated from F S after biting an infectious human.Thus, Non-sterilized male mosquitoes evolve directly from aquatic stage at the rate (1 − r)b V , and decreased due to natural death, so that, Sterilized male mosquitoes (M S ) are released into the population at rate ω(t) at time t.However, due to some environmental and geographical factors that may affect the mixing of sterilized and wild mosquitoes, such as location of mosquito breeding site, it is convenient to assume that, only a fraction p of the released mosquitoes will join wild mosquito population, likewise, because of differences in the physiology of wild and sterilized mosquitoes, a parameter g is used to capture the mean mating competitiveness of sterilized mosquito, so that the actual number of sterilized male mosquitoes competing with wild mosquitoes is M S pg , and therefore, the growth rate of sterilized male mosquitoes is pgω(t).This formulation is similar to that in [1].
The population is reduced due to natural death at the rate µ M .Hence,

Dynamics of human population
The population of susceptible humans are generated by birth or immigration at a constant rate b H .This population is decreased by acquiring infection after receiving substantive amount of bites capable of disease transmission from an infectious mosquito (F NI or F SI ) at a rate λ H1 .However, due to recent developments on the potentiality of human-to-human transmission [9,18], we assumed a direct (horizontal) transmissions from an infected human to a susceptible human at a rate λ H2 , and they are reduced by natural death at a rate µ H .It is worth mentioning that, Zika is the first Flavivirus known to be transmitted sexually from an infectious human to a susceptible human [19].They are represented by The population of infectious human is generated by infection of susceptible humans at the rate λ H , and decreases due to recovery (at a rate γ), natural death at the rate µ H and disease induced death at a rate δ , so that, Finally, the population of recovered individuals is generated by the recovery of infected individuals (at a rate γ) and reduces due to natural death at a rate µ H .This gives It is worth mentioning that, high viral load was found in the semen and saliva of recovered patients weeks after recovery, hence, there is high chance of human-tohuman vaginal or oral sex transmission by a recovered human [9,16,18].

Model equation
Similar to the formulation in [1], we assumed that, the mating competitiveness of both sterilized and non-sterilized mosquitoes are equal, that is α S = α N = α.Furthermore, since there are only two mating possibilities, either with a sterilized or non-sterilized mosquitoes, we let for the model is illustrated in Figure 1.

Fig. 1: Flow diagram of mosquito-human interaction
Therefore, Zika transmission model is given by the following system of non-linear differential equations with the description of the parameters and variables given in Tables 1 and 2): The last equation of ( 5) is independent of other compartments and has the following solution M S (t) = e −µ M t M S (0) + t 0 e µ M j pgω( j)d j . (6)

Basic properties
Lemma 1.The solution of model (5) uniquely exists.In addition it is positive and bounded.
Proof.The right hand side of model ( 5) is C 1 in R 10 + , hence solution exists locally and it is unique.
In general, for all x i = 0, f i = x i > 0 whenever x j > 0 (monotone increasing with respect to x j ), so that, by Proposition B.7. of [30], solution to the system is positive.Also, The equation of aquatic mosquitoes population can be considered as having logistic growth, with carrying capacity K .So that, A(t) ≤ K .Let the total mosquito population at the non-aquatic stage be given by Then, Applying Gronwall lemma on ( 7) and (8) gives and which are bounded and therefore solutions exists for all t ≥ 0.
Theorem 1.The following biologically-feasible region of the model (5) is positivelyinvariant where K is the carrying capacity of the aquatic compartment.
Proof.It is clear from ( 9) and ( 10) that, Consequently, every solution initiated in Ω remain in Ω for all time t ≥ 0.

Existence and stability of equilibria
be defined as the basic offspring number of the mosquito population.It is the average number of offspring produced by a single female mosquito that mated with a nonsterilized male mosquito in its entire lifespan.

Disease-free equilibrium
Let N 0 > 1, that is, on average, a reproductive female mosquito reproduces more than one offspring in her entire life span, then there exist a non trivial, positive disease free equilibrium denoted by E 0 , such that, where

Local stability of E 0
The local stability of the DFE (E 0 ) can be established using the next generation operator.The F matrix is given by Therefore using the method described in [10], the next generation matrix (NGM) with large domain K L is given by , so that the next generation matrix is a 3 × 3 matrix given by Following [10,15], the associated reproduction number of the system model ( 5) denoted by R 0 is given by where are the reproduction numbers associated with Zika transmission from human to human, human to vector and vector to human, respectively.

Interpretation of R 0
In the absence of direct transmission, the threshold quantity (R 1 ) defined by is the expected number of secondary cases generated by an infected case introduced into a population, consisting of completely susceptible.It is interpreted as follows.
Susceptible humans get infection from an infectious mosquito (F NI or F SI ).The number of infections caused by non-sterilized infectious mosquitoes (near DFE) is given by the product of its infection rate , with the average duration in the infectious (non-sterilized) class 1 µ V , and the probability that a female mosquito survives the fertilized non-sterilized class (F N ), and move to fertilized non-sterilized and infectious compartment α(1−θ ) Similarly, the number of human infections generated by sterilized mosquitoes (near DFE) is given by the product of the infection rate of sterilized infectious mosquitoes , with the average duration in the infectious (sterilized) class 1 µ V , and the probability that a female mosquito survives the fertilized and sterilized class (F S ), and move to fertilized, sterilized and infectious compartment αθ µ V , so that Therefore, the sum of ( 16) and (17) gives the average number of new human infections generated by infectious mosquitoes (sterilized or non-sterilized).
Mosquito infection is caused by infectious humans only, the number of mosquitoes infection caused by infectious human (near the DFE), is the product of its infection rate , with the average duration in infectious class 1 K 1 , hence (with The geometric mean of ( 18) and (19) gives the associated reproduction number (since . The number of new human-human infections (sexual), generated by an infectious human (I H ) (near DFE) is given by the product of infection rate of infectious human , and the average duration in the infectious class 1 K 1 , this gives Similarly, the number of new human infections (sexual) generated by a recovered human, is given by the product of its infectious rate , the probability that human survives the infectious stage γ Hence, the sum of ( 20) and (21) gives the threshold quantity associated with the human-human Zika transmission Thus, the associated reproduction number for the Zika model with direct transmission is given by It is worth mentioning that, R 0 is consistent with those obtained by Brauer et al [2] and Chitnis et al [6], Gao et al [20] for epidemic models of vector borne diseases with direct or vertical transmissions.

Endemic equilibrium
represents an arbitrary positive endemic equilibrium point (EE) of the model (5).Furthermore, let

Sensitivity analysis
Sensitivity analysis is a tool used in studying the variation of an output of a model due to change in the input parameters.Using elasticity index, we perform local sensitivity analysis (where all other parameters are held at a certain baseline) for the basic reproduction number (R 0 ).The method is used to measure the percentage change of a parameter say α, with respect to a percentage change of a quantity say R 0 (α).The basic reproduction number (R 0 ) can be used to measure the potential impact of a disease.The normalized sensitivity index (elasticity indices) of R 0 (α) Using data in Table 1, we give in Table 2 the sensitivity index of the parameters for low and high baseline values.For both low and high transmission regions, R 0 is most negatively correlated to µ V , with ϒ R 0 µ V = −0.78228 in low region and ϒ R 0 µ V = −0.62677.Similarly, R 0 is most positively correlated to β 2 in both cases.

Numerical simulations
The Zika model ( 5) is simulated using parameter values in Table 1 together with the following initial conditions S H (0) = 600, I H (0) = 20, R H (0) = 0, Low baseline values give R 0 = 0.5606 < 1, while high baseline values give R 0 = 13.0160> 1. Fig. 2 illustrates the solution profile of the model ( 5), showing cumulative number of new human cases with different values of θ (probability of a female mosquito mating with a sterilized male).For α = 0.7 and θ = 0.2, the rate of sterilization (αθ ) = 0.14, that is (20%) of young female mosquitoes mate with sterilized male mosquitoes, the percentage increase (which depends on θ ) is negatively correlated to the cumulative number of new human cases.4 show population of infected humans with different initial conditions.Fig. 3 shows the convergence to a non zero equilibrium when R 0 = 13.0160> 1, while Fig. 4 shows the convergence to DFE of infected humans when R 0 = 0.5606 < 1. Disease prevalence (ratio of infected humans by the total human population), for both high and low baseline values are respectively shown in Fig. 5 and Fig. 6.Fig. 5 shows convergence of solution to endemic equilibrium, while Fig. 6 shows convergence to DFE with different initial conditions.5) showing total number of reproductive mosquitoes, with different values of θ , such that, for θ = 0.2, the population spikes four times and faded, while the number of spike reduces as the value of θ increases, that is to say, increasing the chance of mating with sterilized mosquito reduces the spate in population of reproductive mosquitoes, consequently reducing the number of mosquito population in an environment.Fig. 8 depict total adult mosquitoes with different oviposition rate.

Conclusion
A Zika model where a fraction of male mosquitoes are being sterilized is studied.In addition to human-mosquito and mosquito-human transmission, the model which adopts a standard incidence formulation also allows direct (human-human) transmission.Both aquatic and non-aquatic stages of mosquito development were considered.The basic offspring number of the mosquito population was computed, as well as the basic reproduction number for the case when the basic offspring num-ber is greater than one.Using elasticity index (local sensitivity analysis), it is shown that, the most effective parameter for the control of the basic reproduction number in both areas of high and low transmission is mosquito death rate, while the adequate contact rate between infectious and susceptible humans is the most positively correlated parameter.

K 1 and
move to recovered class, and the average duration in the recovered class1  µ H , this gives H + I * * H + R * * H be the associated forces of infections.Solving equations of model (5) at the steady state gives

Fig. 2 :
Fig. 2: Simulation of (5) showing cumulative number of new cases in humans with different values of θ

Fig. 7
Fig.7depicts the solution profile of the model(5) showing total number of reproductive mosquitoes, with different values of θ , such that, for θ = 0.2, the population spikes four times and faded, while the number of spike reduces as the value of θ increases, that is to say, increasing the chance of mating with sterilized mosquito reduces the spate in population of reproductive mosquitoes, consequently reducing the number of mosquito population in an environment.Fig.8depict total adult mosquitoes with different oviposition rate.

Table 1 :
Two sets of parameter values used in numerical simulations, with low baseline values that give R 0 = 0.1489 < 1, while R 0 = 4.7033 > 1 for the high baseline values

Table 2 :
Sensitivity index of R 0 with respect to parameters of the model (5) for R 0 = 0.5606 < 1 and R 0 = 13.0160> 1 using the values of Table1