The results are significant extension of the results of paper [9]

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Tomul LXVI, 2020, f. 2

Positively invariant semi-implicit discrete model for malaria propagation

Istv´an Farag´o · Rahele Mosleh

Abstract In this paper, we consider the malaria propagation process for infected populations for humans and mosquitoes with an extension of the classical Ross model. The numerical model is constructed by using the semi-implicitθ-method. We show that the numerical solution and the total populations of the extended Ross model are positively invariant in specific intervals for some step sizes. We demonstrate the validity of the theoretical results of the semi-implicit method by numerical simulations for some examples.

The results are significant extension of the results of paper [9].

Keywords Ross model· Extended Ross model· Positively invariant · Positivity preservation·Semi-implicit method·Malaria propagation

Mathematics Subject Classification (2010) 65L05·34C60·92D30

1 Introduction

Malaria is an infectious and lethal host-vector disease that is transmitted by the bites of infectious female Anopheles mosquitoes as vectors to humans as hosts. It is an ancient disease and individuals have to deal with this disease since there is no effective vaccine. This phenomenon depends heavily on climatic factors, including temperature, altitude, precipitation, and raised humidity. This implies that the distribution and transmission of malaria is increased by high temperatures, rainy seasons and increased humidity, the reason it is spreaded out in tropical and subtropical regions such as Africa, Asia, Latin America and also some parts of Europe like Hungary, Austria, Italy [1], [13].

Some mathematical models such as Ross, Ross-Macdonald, delayed Ross- Macdonald, Anderson and May models and others have been built to gain a better insight into the spread of malaria and decrease its effects in the world. As an SIS-Type model, the Ross model is discussed for humans and mosquitoes in constant population sizes and the solutions are densities in

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the interval [ 0,1] (c.f. [7]). In this analysis, we consider an expansion of the Ross model to take more effective malaria factors into account.

As the solutions of the extended Ross mode are the number of the individuals, they should be positive. Analytically, it is shown that the extended Ross model is well-posed and the solutions are positively invariant in certain intervals (c.f. [6]). Moreover, there are two equilibrium points, disease-free and endemic, which under certain constraints, are globally stable (c.f. [2], [5], [8]). The numerical model obtained by using the explicit Euler method is investigated in [9] and the condition of the positivity preservation property is given. This approach allows us to use only relatively small step sizes.

Therefore in this paper we suggest to use implicite discretization method which enables us to select bigger step-size. The fully implicit method for the extended Ross model is rigid to prove the positivity preservation for the solutions, therefore a semi-implicit method is applied. We show that this kind of discretization allows us to select bigger step-size under which the postovoty is preserved. Moreover, in this paper we also investigate the upper bound preservation property of the solution.

As a brief outline, section 2 interprets the extended Ross model biologically and qualitatively. In section 3 we consider a semi-implicit method to prove the positively invariant solutions for humans and mosquitoes. Section 4 simulates the theoretical results of the semi-implicit method numerically with examples. Section 5 gives a summary of the results.

2 Mathematical models for malaria transmission

A diversity of mathematical models of malaria transmission have been de- veloped to explain the dynamics of malaria spread as epidemiological models to some extent. Ronald Ross [12] introduced the first model of malaria transmission. In this model, the human and mosquito populations are split into two subclasses, infected and susceptible individuals, and the birth and mortality rates are the same.

2.1 Ross model

According to the Ross model, the dynamics of malaria transfer is described through a system of ordinary differential equations as below:









S˙_h(t) =r−αI_m(t)(1−I_h(t))−rS_h(t) I˙h(t) =αIm(t)(1−Ih(t))−rIh(t) S˙m(t) =µ−βI_h(t)(1−Im(t))−µSm(t) I˙_m(t) =βI_h(t)(1−I_m(t))−µI_m(t)

(2.1)

with given nonnegative initial values satisfying the following conditions:

Sh(0) +Ih(0) = 1

Sm(0) +Im(0) = 1 (2.2)

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Here S_h(t) denotes the density of host susceptible humans at time t, the density of infected human at time t is defined by I_h(t), Sm(t) and Im(t), respectively, signify the densities of susceptible and infected mosquitoes at time t. Biologically, the other parametersα, β, r andµ are defined as the proportion of bites that produce infection in humans, the proportion of bites by which one susceptible mosquito becomes infectious, the rate of the death for humans and the rate of the death for mosquitoes, respectively.

Since the solutions of the Ross models are densities, they should take values in the interval [0,1]. This property is proven in [7] analytically and numerically. The Ross model (2.1) fails to introduce an adequate model for malaria transmission by the following reasons:

– There are no vital dynamics for the total populations. In other words, the birth and mortality rates are assumed to be equal, outcoming the total population sizes are constant for humans and mosquitoes.

– As malaria parasites have some latency period in humans and mosquitoes bodies, there is an intermediate state between the two susceptible and infected compartments known exposed. It means that the transition from the susceptible state to the infected state is not direct.

– Since the Ross model (2.1) is aSIS-type model, the recovery state,R(t), is not considered for humans.

2.2 Extended Ross model

In this section, we discuss an extended Ross model which is able to dispel the aforementioned deficits of the Ross model (2.1) to some extent (c.f. [10]). In this model we divide the humans population into four groups: susceptible humans, exposed humans, infectious humans and recovered humans.The population of mosquitoes is split into three subclasses, unlike the humans population. Due to the short life time, mosquitoes do not have enough time to meloriate and they die after infection. By some biological interpretations [10] the malaria propagation is sketched as below:











S˙_h(t) =Λ_h−^bβ₁₊^h^S_ν^h_h⁽^t_I⁾_m^I^m₍_t⁽₎^t⁾−µ_hS_h(t) +ωR_h(t) E˙_h(t) = ^bβ₁₊^h^S_ν^h⁽^t⁾^I^m⁽^t⁾

hIm(t) −(α_h+µ_h)E_h(t) I˙_h(t) =α_hE_h(t)−(r+µ_h+δ_h)I_h(t) R˙_h(t) =rI_h(t)−(µ_h+ω)R_h(t) S˙_m(t) =Λ_m−^bβ₁₊^m^S_ν^m_m⁽_I^t⁾_h^I₍^h_t₎⁽^t⁾−µ_mS_m(t) E˙m(t) = ^bβ₁₊^m^S_ν^m⁽^t⁾^I^h⁽^t⁾

mIh(t) −(αm+µm)Em(t) I˙m(t) =αmEm(t)−(µm+δm)Im(t).

(2.3)

The following initial conditions are applied:

Sh(0) =S₀h, Ih(0) =I₀h, Eh(0) =E₀h, Rh(0) =R₀h,

Sm(0) =S0m, Em(0) =E0m, Im(0) =I0m. (2.4)

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Here functionS_h(t) denotes the number of the host susceptible humans at time t, the number of the host exposed humans to malaria infection at time t is denoted byEh(t), the number of the infected and the recovered humans at time t are characterized byIh(t) andRh(t) respectively.

In order to interpret a more accurate model, the model (2.3) takes into account population dynamics by considering various birth and death rates.

We assume that all the newly born children are healthy, so the recruitment terms, (Λ_h, Λ_m), are the birth rates applied as inputs for the susceptible class (S_h, Sm).

The biting rate of the mosquito is denoted by b and the parasites injected by the mosquito into the human blood system with some probabilityβhspend some a latent period in human body then the susceptible human moves to the exposed class Eh(t). After the latency period the exposed human is proceeded to the infected class with some α_h rate. The infected human transfers to the recovery class with some rate r. The recovered stateR_h(t) contains the recovered humans. The recovered human has some immunity to the disease however, after a while the individual loses the immunity and becomes susceptible again by ω pace. Each class of humans is decreased by the natural death rate, µh, according to the model (2.3), and only the infectious class is decreased by the death rate caused by disease,δh. In a similar way, βm is the probability of the parasites transitioning into the susceptible mosquitoS_m(t) through the bites of an infected human and the susceptible mosquito moves to the exposed class E_m(t). After a given time it becomes infectiousI_m. The mosquitoes population is decreased by natural death rateµmor disease induced death rateδm.

The ratio 1

1 +ν_hI_m(t)denotes a saturating feature that prevents the force of infection from infected mosquitoes to susceptible humans in whichν_h∈[0,1]

is the proportion of antibodies produced in humans bodies in response to the conflict of antigens produced by infected mosquitoes. By similar inter- pretation for mosquitoes,νm∈[0,1] is the rate of antibodies generated by infectious humans against the antigens contacted.

Since the malaria transmission occurs in an inharmonious population, the epidemiological model (2.3) must partition the population into groups, in which the members have similar characteristics such as mode of transmission, contact patterns, latent period, infectious period, genetic susceptibility or resistance. Accordingly, the total humans population size at time t,V_h(t), is defined as

Vh(t) =Sh(t) +Eh(t) +Ih(t) +Rh(t). (2.5) The same definition holds for the total size of the mosquitoes population, V_m(t), at time t

V_m(t) =S_m(t) +E_m(t) +I_m(t). (2.6) Analytically, it is proven [6,10] that solutions of the extended Ross model (2.3) are invariant in the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^m

m] at time t for humans and mosquitoes respectively. According to the aforementioned intervals, when

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the rate of natural mortality for either humans or mosquitoes vanishes, the size of the corresponding total population is unbounded. Biologically, it means when there is no natural death, then the number of the total population blows up.

We note that 2.3-2.4 yields a Cauchy-problem for the system of nonlinear ordinary differential equations. Since the analytical solution cannot be defined, we have to use some numerical method, which results in the discrete model got the malaria propagation.

3 Semi-implicit method applied for the extended Ross model At the first step we apply the standard implicitθ- method for the extended Ross model to approximate the numerical solutions of the system (2.3) as below:











S_hⁱ⁺¹−Shⁱ

∆t = (1−θ)(Λ_h−^bβ₁₊^h_ν^S_h^hⁱ_I^I_mⁱ^mⁱ −µ_hS_hⁱ +ωRⁱ_h) +θ(Λ_h−^bβ₁₊^h^S_ν^hⁱ⁺¹_h_I_mⁱ⁺¹^I^mⁱ⁺¹ −µ_hS_hⁱ⁺¹+ωRⁱ_h⁺¹)

Eⁱ⁺¹_h −Ehⁱ

∆t = (1−θ)(^bβ₁₊^h_ν^S^hⁱ^I^mⁱ

hI_mⁱ −(α_h+µ_h)E_hⁱ) +θ(^bβ^h^S^hⁱ⁺¹^Iⁱ⁺¹^m

1+νhImⁱ⁺¹ −(α_h+µ_h)E_hⁱ⁺¹)

I_hⁱ⁺¹−Ihⁱ

∆t = (1−θ)(α_hE_hⁱ −(r+µ_h+δ_h)I_hⁱ) +θ(α_hE_hⁱ⁺¹−(r+µ_h+δ_h)I_hⁱ⁺¹)

Rⁱ⁺¹_h −Rⁱ_h

∆t = (1−θ)(rI_hⁱ −(µh+ω)Rⁱ_h) +θ(rI_hⁱ⁺¹−(µh+ω)Rⁱ_h⁺¹)

S_mⁱ⁺¹−S_mⁱ

∆t = (1−θ)(Λm−^bβ₁₊^m_ν^S_mⁱ^m_I^Iⁱ^hⁱ

h −µmS_mⁱ ) +θ(Λ_m−^bβ₁₊^m^S_νⁱ⁺¹^m_m_Iⁱ⁺¹^I^hⁱ⁺¹

h −µ_mS_mⁱ⁺¹)

Eⁱ⁺¹_m −E_mⁱ

∆t = (1−θ)(^bβ₁₊^m_ν^S^mⁱ^I^hⁱ

mI_hⁱ −(α_m+µ_m)E_mⁱ ) +θ(^bβ^m^S^mⁱ⁺¹^I^hⁱ⁺¹

1+νmIⁱ⁺¹_h −(αm+µm)E_mⁱ⁺¹)

I_mⁱ⁺¹−Imⁱ

∆t = (1−θ)(αmE_mⁱ −(µm+δm)I_mⁱ ) +θ(α_mE_mⁱ⁺¹−(µ_m+δ_m)I_mⁱ⁺¹).

(3.1)

However, the system (3.1) is a system of nonlinear algebraic equations.

Therefore, there are some difficulties to prove the positivity preservation of the solutions by algebraic tools. Since the nonlinearity is a specific structure appearing in the force of infection terms only, we apply the following semi- implicit method a nonlocal discretization of the implicit method (3.1) :

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









S_hⁱ⁺¹−Shⁱ

∆t = (1−θ)(Λ_h−^bβ₁₊^h^S_ν_hⁱ^h_I^I_mⁱ^mⁱ −µ_hS_hⁱ +ωRⁱ_h) +θ(Λh−^bβ₁₊^h^S_ν^hⁱ⁺¹_h_I_mⁱ^I^mⁱ −µhS_hⁱ⁺¹+ωRⁱ_h)

Eⁱ⁺¹_h −E_hⁱ

∆t = (1−θ)(^bβ₁₊^h^S_νⁱ^h^I^mⁱ

hI_mⁱ −(αh+µh)E_hⁱ) +θ(^γ₁₊^h^Sⁱ⁺¹^h_ν ^I^mⁱ

hI_mⁱ −(α_h+µ_h)E_hⁱ⁺¹)

I_hⁱ⁺¹−I_hⁱ

∆t = (1−θ)(α_hE_hⁱ −(r+µ_h+δ_h)I_hⁱ) +θ(α_hE_hⁱ⁺¹−(r+µ_h+δ_h)I_hⁱ⁺¹)

Rⁱ⁺¹_h −Rⁱ_h

∆t = (1−θ)(rI_hⁱ−(µ_h+ω)Rⁱ_h) +θ(rI_hⁱ⁺¹−µhRⁱ_h⁺¹−ωR_hⁱ)

S_mⁱ⁺¹−S_mⁱ

∆t = (1−θ)(Λ_m− ^bβ₁₊^m_ν^S_m^mⁱ_I^I_hⁱ^hⁱ −µ_mS_mⁱ ) +θ(Λ_m− ^bβ₁₊^m^S_ν^mⁱ⁺¹_m_Iⁱ⁺¹^I^hⁱ⁺¹

h −µ_mSⁱ_m⁺¹)

Eⁱ⁺¹_m −Emⁱ

∆t = (1−θ)(^bβ₁₊^m_ν^S^mⁱ^I^hⁱ

mI_hⁱ −(α_m+µ_m)E_mⁱ ) +θ(^bβ^m^S^mⁱ⁺¹^I^hⁱ⁺¹

1+νmI_hⁱ⁺¹ −(αm+µm)E_mⁱ⁺¹)

I_mⁱ⁺¹−I_mⁱ

∆t = (1−θ)(αmE_mⁱ −(µm+δm)I_mⁱ ) +θ(α_mEⁱ_m⁺¹−(µ_m+δ_m)I_mⁱ⁺¹).

(3.2)

The system (3.2) is a linear system for which we are able to define the solutions explicitly as below:

S_hⁱ⁺¹= S_hⁱ(1−∆t(1−θ)(₁₊^bβ_ν^h^I^mⁱ

hI_mⁱ +µ_h)) +∆t(Λ_h+ωRⁱ_h) 1 +∆tθ(₁₊^bβ_ν^h^I^mⁱ

hI_mⁱ +µ_h) , (3.3) E_hⁱ⁺¹= E_hⁱ(1−∆t(1−θ)(α_h+µh)) +∆t(1−θ)^bβ₁₊^h^S_ν^hⁱ^I^mⁱ

hI_mⁱ +∆tθ^bβ₁₊^h^S_ν^hⁱ⁺¹^I^mⁱ

hI_mⁱ

1 +∆tθ(α_h+µ_h) ,

(3.4) I_hⁱ⁺¹=I_hⁱ(1−∆t(1−θ)(r+µ_h+δ_h)) +∆t(1−θ)α_hE_hⁱ +∆tθα_hE_hⁱ⁺¹

1 +∆tθ(r+µ_h+δ_h) , (3.5) Rⁱ_h⁺¹= Rⁱ_h(1−∆t((1−θ)µ_h+ω)) +∆t(1−θ)rI_hⁱ +∆tθrI_hⁱ⁺¹

1 +∆tθµh

, (3.6) S_mⁱ⁺¹= S_mⁱ (1−∆t(1−θ)(₁₊^bβ_ν^m^I^hⁱ

mIⁱ_h+µ_m)) +∆tΛ_m 1 +∆tθ( ^bβ^m^I^hⁱ⁺¹

1+νmI_hⁱ⁺¹ +µ_m)

, (3.7)

E_mⁱ⁺¹=

E_mⁱ (1−∆t(1−θ)(αm+µm))+∆t(1−θ)^bβ₁₊^m_ν^S^mⁱ^Iⁱ^h

mI_hⁱ +∆tθ^bβ^m^Sⁱ⁺¹^m ^I^hⁱ⁺¹

1+νmI_hⁱ⁺¹

1+∆tθ(αm+µm) ,

(3.8)

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I_mⁱ⁺¹= I_mⁱ (1−∆t(1−θ)(µ_m+δm)) +∆t(1−θ)α_mE_mⁱ +∆tθαmE_mⁱ⁺¹

1 +∆tθ(µ_m+δm) .

(3.9) Since analytically we proved the solutions and total populations of the extended Ross model (2.3) are invariant on the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^m

m] for humans and mosquitoes, respectively (c.f. [6], [8]), we shall consider this property for the discrete model (3.2), too. For this aim, we need to guar- antee that for any S_hⁱ, E_hⁱ, I_hⁱ, Rⁱ_h, and V_hⁱ on the interval (0,^Λ_µ^h

h] and for any S_mⁱ , E_mⁱ , I_mⁱ , andV_mⁱ on the interval (0,^Λ_µ^m

m] the discrete model (3.2) results the solutionsS_hⁱ⁺¹,E_hⁱ⁺¹, I_hⁱ⁺¹, Rⁱ_h⁺¹, and V_hⁱ⁺¹ on (0,^Λ_µ^h

h] and S_mⁱ⁺¹, E_mⁱ⁺¹,I_mⁱ⁺¹, andV_mⁱ⁺¹on (0,^Λ_µ^m

m]. At the first step we consider the positivity preservation of the solutions.

Lemma 1.LetΛ_h, Λ_m>0 and the initial data of the system (3.2) are on the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^m

m], respectively. If

∆t

(≤ ₍₁_−θ₎₍_bβ_h^µ_Λ^m_m⁺₊^ν_µ^h_h^Λ₍_µ^m_m₊_ν_h_Λ_m₎₎ θ∈[0,1)

is any θ= 1, (3.10)

thenS_hⁱ⁺¹ of the system (3.2) is positive.

Proof. From the first equation of the system (3.2) we have S_hⁱ⁺¹(1+∆tθ( bβ_hI_mⁱ

1 +ν_hI_mⁱ +µ_h)) =S_hⁱ+∆t(Λ_h−(1−θ)(bβhS_hⁱI_mⁱ

1 +ν_hI_mⁱ +µ_hS_hⁱ)+ωRⁱ_h).

(3.11) At the first step, we assume thatθ does not equal one, i.e., 1−θ6= 0 If the right side of (3.11) is positive, then the required inequalityS_hⁱ⁺¹>0 holds. This means the condition

S_hⁱ > ∆t((1−θ)(bβhS_hⁱI_mⁱ

1 +νhI_mⁱ +µ_hS_hⁱ)−(Λ_h+ωRⁱ_h)). (3.12) When

(1−θ)(bβhS_hⁱI_mⁱ

1 +ν_hI_mⁱ +µ_hS_hⁱ)−(Λ_h+ωRⁱ_h)≤0, (3.13) then the inequality (3.12) is valid for any step size∆t. Otherwise, we have the bound

∆t < S_hⁱ

(1−θ)(^bβ₁₊^h_ν^S^hⁱ^I^mⁱ

hI_mⁱ +µhS_hⁱ)−(Λh+ωRⁱ_h). (3.14)

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Clearly, S_hⁱ (1−θ)(^bβ₁₊^h^S_νⁱ^h^I^mⁱ

hI_mⁱ +µhS_hⁱ) < S_hⁱ (1−θ)(^bβ₁₊^h^S_ν^hⁱ^I^mⁱ

hI_mⁱ +µhS_hⁱ)−(Λh+ωRⁱ_h). (3.15) Therefore, the condition

∆t≤ 1 +ν_hI_mⁱ

(1−θ)(bβhI_mⁱ +µh(1 +νhI_mⁱ )) (3.16) is a sufficient condition of the positivity for S_hⁱ⁺¹. Since the right side of (3.16) is a monotonically decreasing function ofI_mⁱ andI_mⁱ ∈(0,^Λ_µ^m

m], under the condition

∆t≤ µm+νhΛm

(1−θ)(bβhΛm+µh(µm+νhΛm)), (3.17) the required positivity property for S_hⁱ⁺¹ holds. For θ = 1, we have the following equation

S_hⁱ⁺¹= S_hⁱ +∆t(Λ_h+ωRⁱ_h) 1 +∆t(₁₊^bβ_ν^h^I^mⁱ

hI_mⁱ +µ_h). (3.18) Obviously, for this case,S_hⁱ⁺¹ is positive unconditionally for any step size

∆t.

u t In the following, we analyze the positivity of the other classes for humans.

The second equation of the system (3.2) yields E_hⁱ⁺¹(1+∆tθ(α_h+µ_h)) =E_hⁱ+∆t((1−θ)(bβhS_hⁱI_mⁱ

1+ν_hI_mⁱ −(α_h+µ_h)Eⁱ_h)+θbβ_hS_hⁱ⁺¹I_mⁱ 1+ν_hI_mⁱ ).

(3.19) The value of E_hⁱ⁺¹ is positive provided that the right side of the equation (3.19) is positive. Hence, when

(1−θ)(bβhS_hⁱI_mⁱ

1 +νhI_mⁱ −(α_h+µ_h)E_hⁱ) +θbβ_hSⁱ_h⁺¹I_mⁱ

1 +νhI_mⁱ ≥0, (3.20) thenE_hⁱ⁺¹is positive for any step size∆t. Otherwise, for 1−θ6= 0 we have the bound

∆t < E_hⁱ

(1−θ)((αh+µh)E_hⁱ −^bβ₁₊^h^S_ν_hⁱ^h_I^I_mⁱ^mⁱ )−θ^bβ₁₊^h^S_ν^hⁱ⁺¹^I^mⁱ

hI_mⁱ

. (3.21)

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Provided that (3.10) is satisfied, the following estimation is valid E_hⁱ

(1−θ)(αh+µh)E_hⁱ ≤ E_hⁱ

(1−θ)((α_h+µ_h)E_hⁱ −^bβ₁₊^h_ν^S_h^hⁱ_I^I_mⁱ^mⁱ )−θ^bβ₁₊^h^S_νⁱ⁺¹^h ^I^mⁱ

hI_mⁱ

. (3.22) Therefore,

∆t≤ 1

(1−θ)(α_h+µ_h) (3.23)

is a sufficient condition to attain the positiveE_hⁱ⁺¹>0. Forθ= 1 we get

Eⁱ_h⁺¹= E_hⁱ +∆t^bβ₁₊^h^S_νⁱ⁺¹^h ^I^mⁱ

hI_mⁱ

1 +∆t(α_h+µ_h) . (3.24) Since in this caseS_hⁱ⁺¹ (3.18) is positive unconditionally for any step size, E_hⁱ⁺¹>0 for all step sizes. Similarly, we get I_hⁱ⁺¹>0 provided that (3.23) is satisfied and

∆t≤ 1

(1−θ)(r+µ_h+δ_h), (3.25) for 1−θ6= 0 and for all step sizes whenθ= 1. Moreover, we obtainRⁱ_h⁺¹>0 as long as (3.25) holds and

∆t≤ 1

(1−θ)µ_h+ω (3.26)

for 1−θ6= 0 and

∆t≤ 1

ω (3.27)

forθ= 1. Hence, we can summarize the results as follows.

Lemma 2.Assume thatΛ_h, Λm>0 and the initial data of the system (3.2) are on the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^m

∆t ≤







min{₍₁₋_θ₎₍_bβ_h_Λ^µ^m_m⁺₊^ν_µ^h_h^Λ₍_µ^m_m₊_ν_h_Λ_m₎₎,₍₁₋_θ₎₍¹_α

h+µh),

1

(1−θ)(r+µh+δh),₍₁₋_θ₎¹_µ

h+ω} θ∈[0,1)

1

ω θ= 1,

(3.28)

thenE_hⁱ⁺¹, I_hⁱ⁺¹, and Rⁱ_h⁺¹of the system (3.2) are positive.

Similar results can be formulated for mosquitoes, as well.

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Lemma 3.SupposeΛ_h, Λ_m>0 and the initial data of the system (3.2) are on the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^m

∆t

(≤min{₍₁₋_θ₎₍_r₊¹_µ_h₊_δ_h₎,₍₁₋_θ₎₍_bβ ^µ^h⁺^ν^m^Λ^h

mΛh+µm(µh+νmΛh))} θ∈[0,1)

is any θ= 1, (3.29)

thenS_mⁱ⁺¹ of the system (3.2) is positive.

Proof. From the fifth equation of the system (3.2) we have S_mⁱ⁺¹(1+∆tθ( bβ_mI_hⁱ⁺¹

1 +νmI_hⁱ⁺¹+µ_m)) =Sⁱ_m+∆t(Λ_m−(1−θ)(bβmS_mⁱ I_hⁱ

1 +νmI_hⁱ +µ_hS_mⁱ )).

(3.30) When 1−θ 6= 0 and (3.25) is satisfied, S_mⁱ⁺¹ is positive provided that the right side of the equation (3.30) is positive, i.e.,

S_mⁱ > ∆t((1−θ)(bβ_mS_mⁱ I_hⁱ

1 +νmI_hⁱ +µmS_mⁱ )−Λm). (3.31) When

(1−θ)(bβmS_mⁱ I_hⁱ

1 +ν_mI_hⁱ +µ_mS_mⁱ )−Λ_m≤0, (3.32) then the inequality (3.31) is valid for any step size∆t satisfied in the condition (3.25). Otherwise, we have the bound

∆t≤ S_mⁱ

(1−θ)(^bβ₁₊^m_ν^S^mⁱ^I^hⁱ

mI_hⁱ +µmS_mⁱ )−Λm

. (3.33)

Visibly,

S_mⁱ (1−θ)(^bβ₁₊^m_ν^S^mⁱ^I^hⁱ

mIⁱ_h +µmS_mⁱ ) < S_mⁱ (1−θ)(^bβ₁₊^m_ν^Sⁱ^m^I^hⁱ

mI_hⁱ +µmS_mⁱ )−Λm

. (3.34) Accordingly, the condition

∆t≤ 1 +νmI_hⁱ

(1−θ)((1 +νmI_hⁱ)µm+bβmI_hⁱ) (3.35) is a sufficient condition of the positivity for S_mⁱ⁺¹. Since the right side of (3.35) is a decreasing function ofI_hⁱ andI_hⁱ ∈(0,^Λ_µ^h

h], we attain

∆t≤ µh+νmΛh

(1−θ)(bβ_mΛh+µm(µ_h+νmΛh)). (3.36) Supposeθ= 1, then we have the following expression

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S_mⁱ⁺¹= Sⁱ_m+∆tΛm

1 +∆t( ^bβ^m^I^hⁱ⁺¹

1+νmI_hⁱ⁺¹ +µ_m)

. (3.37)

As in this case I_hⁱ⁺¹ is positive unconditionally for any step size, S_mⁱ⁺¹ is positive for any step size∆t.

u t The sixth equation of the system (3.2), yields

E_mⁱ⁺¹(1 +∆tθ(αm+µm)) =E_mⁱ +∆t((1−θ)(bβ_mS_mⁱ I_hⁱ

1 +νmI_hⁱ −(αm+µm)E_mⁱ )+

θbβmS_mⁱ⁺¹I_hⁱ⁺¹

1 +νmI_hⁱ⁺¹ ). (3.38) The value E_mⁱ⁺¹ is positive as long as the right side of (3.38) is positive.

When

(1−θ)(bβ_mS_mⁱ I_hⁱ

1 +ν_mI_hⁱ −(α_m+µ_m)E_mⁱ ) +θbβ_mS_mⁱ⁺¹I_hⁱ⁺¹

1 +νmI_hⁱ⁺¹ ≥0, (3.39) E_mⁱ⁺¹ is positive for any step size∆t. Otherwise, for 1−θ6= 0 we have the bound

∆t≤ E_mⁱ

(1−θ)((αm+µm)E_mⁱ −^bβ₁₊^m_ν^S_mⁱ^m_I^I_hⁱ^hⁱ)−θ^bβ^m^S^mⁱ⁺¹^Iⁱ⁺¹^h

1+νmI_hⁱ⁺¹ )

. (3.40)

If (3.25) and (3.29) are satisfied, the following estimation is valid E_mⁱ

(1−θ)(αm+µm)E_mⁱ ≤ E_mⁱ

(1−θ)((αm+µm)Eⁱ_m−^bβ₁₊^m_ν^S_mⁱ^m_I^Iⁱ^hⁱ

h )−θ^bβ^m^S^mⁱ⁺¹^Iⁱ⁺¹^h

1+νmI_hⁱ⁺¹ ) . (3.41) Hence,

∆t≤ 1

(1−θ)(αm+µm) (3.42)

is a sufficient condition to getE_mⁱ⁺¹>0. Ifθ= 1, we acquire

E_mⁱ⁺¹=

Eⁱ_m+∆t^bβ^m^S^mⁱ⁺¹^I^hⁱ⁺¹

1+νmI_hⁱ⁺¹

1 +∆t(αm+µm) . (3.43)

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Since I_hⁱ⁺¹ and S_mⁱ⁺¹ are positive for all step sizes, E_mⁱ⁺¹ is positive unconditionally for any step size. Similarly,I_mⁱ⁺¹ is positive provided that (3.42) holds and

∆t≤ 1

(1−θ)(δm+µm) (3.44)

for 1−θ6= 0 and without any restriction for any step size forθ= 1. There- fore, we can summarize the results as follows.

Lemma 4.SupposeΛ_h, Λ_m>0 and the initial data of the system (3.2) are on the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^h

h], respectively. If

∆t

(≤min₁_−θ¹ {₍_bβ_m_Λ_h^µ₊^h_µ⁺_m^ν₍^m_µ^Λ_h₊^h_ν_m_Λ_h₎₎,_r₊_µ¹

h+δh,_α ¹

m+µm,_µ ¹

m+δm} θ∈[0,1)

is any θ= 1,

(3.45) then the solutionsE_mⁱ⁺¹ andI_mⁱ⁺¹of the system (3.2) are positive.

Lemmas 1-4 give sufficient conditions for the positivity preservation property of the model (10). In the following we show a sharper property of the model.

Theorem 3.1 Assume that Λ_h, Λ_m >0 and the initial data of the system (3.2) are on the intervals(0,^Λ_µ^h

h] and (0,^Λ_µ^m

h^∗=min{h₁, h₂}, (3.46) where

h1 =min{ µ_m+ν_hΛ_m

(1−θ)(bβ_hΛm+µ_h(µm+ν_hΛm)), 1

(1−θ)(α_h+µ_h), 1

(1−θ)(r+µ_h+δ_h), 1

(1−θ)µ_h+ω} (3.47) and

h₂=min 1

1−θ{ µ_h+νmΛ_h

(bβmΛ_h+µm(µ_h+νmΛ_h)), 1

(αm+µm), 1 (µm+δm)},

(3.48) for the semi-implicit model (3.2), then the solutions and total populations are invariant on(0,^Λ_µ^h

h] and(0,^Λ_µ^m

m]for humans and mosquitoes respectively for the discretization step size∆t ∈ (0, h^∗] for 1−θ 6= 0 and for step size

∆t∈(0,_ω¹]when θ= 1.

Proof. In this part, we shall prove invariance of the positivity preservation of the total populations on the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^m

m] for humans and mosquitoes, respectively. For this aim, we should demonstrate if the number

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of the total populations at time t_i are on the intervals V_hⁱ ∈ (0,^Λ_µ^h

h] and V_mⁱ ∈ (0,^Λ_µ^m

m] for humans and mosquitoes respectively, the discrete model (3.2) guarantees that V_hⁱ⁺¹∈ (0,^Λ_µ^h

h] andV_mⁱ⁺¹∈(0,^Λ_µ^m

m], too. The number of the total population in the discrete model (3.2) for humans at timeti+1

is defined as

V_hⁱ⁺¹=S_hⁱ⁺¹+E_hⁱ⁺¹+I_hⁱ⁺¹+Rⁱ_h⁺¹. (3.49) According to Lemma 1 and Lemma 2, if the initial data of the system (3.2) are on the intervals (0,^Λ_µ^h

h] and (0,^Λ_µ^m

m], respectively and (3.28) satisfy, the values ofS_hⁱ⁺¹,E_hⁱ⁺¹, I_hⁱ⁺¹, andRⁱ_h⁺¹ are positive, thereforeV_hⁱ⁺¹>0.

By adding the terms of the system (3.2) and the number of the total population definition for humans (3.49) we attain

V_hⁱ⁺¹(1+∆tθµ_h) =V_hⁱ+∆t(Λ_h−(1−θ)µ_hV_hⁱ−(1−θ)δ_hI_hⁱ−θδ_hI_hⁱ⁺¹)). (3.50) If 1−θ6= 0 and (3.28) is satisfied, we have the bound

V_hⁱ⁺¹(1 +∆tθµ_h)≤V_hⁱ(1−∆t(1−θ)µ_h) +∆tΛ_h. (3.51) LetV_hⁱ≤ ^Λµh^h, from the right side of the inequality (3.51) we obtain

V_hⁱ(1−∆t(1−θ)µh) +∆tΛh< Λ_h

µ_h(1−∆t(1−θ)µh+∆tµh). (3.52) According to (3.51)-(3.52) we get

V_hⁱ⁺¹(1 +∆tθµ_h)≤ Λ_h

µ_h(1 +∆tθµ_h). (3.53) Consequently,

V_hⁱ⁺¹< Λh

µh

. (3.54)

Whenθ= 1, the value ofI_hⁱ⁺¹is positive for any step size. Therefore from (3.50) we have the following bound

V_hⁱ⁺¹(1 +∆tµ_h)≤V_hⁱ+∆tΛ_h (3.55) IfV_hⁱ≤ ^Λ_µ_h^h, we attain

V_hⁱ⁺¹< Λh

µh

. (3.56)

The relationV_hⁱ≤ ^Λ_µ_h^h is valid for any i. Hence, for the initial total populations for humans we haveV_h⁰≤ ^Λ_µ^h_h implying the invariancy of the number of the total populations for humans in the interval (0,^Λ_µ^h

h]. As a result, the solutions of the system (3.2) for humans are bounded and positively invariant in (0,^Λ_µ^h

h]. Similarly, the solutions for mosquitoes are bounded and positively invariant in (0,^Λ_µ^m

m].

u t