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Vertical transmission and reproduction rate: modeling a common strategy for two related diseases
 Abba Mahamane Oumarou^{1} and
 Yannick Tchaptchie Kouakep^{2, 3}Email authorView ORCID ID profile
 Received: 20 November 2015
 Accepted: 4 April 2016
 Published: 14 April 2016
Abstract
Motivated by (Goyal and Murray in PLoS One 9(10):e110143, 2014) we consider a partially agestructured model simulating the dynamic of two infectious diseases vertically transmitted almost independently with horizontal coinfection and a common agestructured vaccination strategy. We study influence of parameters on existence and uniqueness of solutions and epidemiological equilibria. Impact of vertical transmission on basic reproduction rate is also presented.
Keywords
 Vaccination
 Basic reproduction rate
 Differential infectivity
Mathematics Subject Classification
 35Q92
 92D30
 92D25
Introduction
This paper investigates influence of parameters on existence and uniqueness of solutions and equilibria in an age structured model. This model mimics the dynamic of two diseases vertically transmitted almost independently with horizontal coinfection and a common vaccination strategy. For example Goyal and Murray (2014) notes a decrease in Hepatitis B virus (HBv) prevalence as vaccination coverage increases and it is possible to eradicate both HBv and HDv (hepatitis D virus) using high vaccination coverage. Age structure a added to the continuous time t brings some improvements in the comprehension of the disease dynamics. For the case of HBv and HDv, age plays also a great role in the vaccination strategy (Goyal and Murray 2014). We follow methods of Djidjou et al. (2014), Yang et al. (2014), Brauer et al. (2013) or Inaba (1990) for quantitative (wellposedness with semigroup theory) studies. One can see also CastilloChavez and Feng (1998), Greenhalgh (2010) and references Hadeler and Muller (1996), Kouakep and Houpa (2014), Muller (1998, 2000), and Pasquini and Cvjetanović (1988) therein for a good review. Most of time in Africa (WHO 2014), vaccination campaigns concern more than one disease. This study starts with the case of two diseases and forthcoming works will deal with more than two diseases.
We study impact on basic reproduction rate (with a common vaccination strategy) of vertical transmission. Our goal is to bring our contribution with quantitative results concerning special cases in the context of nonlinear dynamics of infectious diseases (in the context of the “Appendix 1”) and non local boundary conditions. We included vaccination ignored in Djidjou et al. (2014), vertical transmission neglected in CastilloChavez and Feng (1998) and show explicitly the basic reproduction rate theoretical shown with an equivalent number in term of asymptotic properties (threshold parameter) in Djidjou et al. (2014).
The paper is organised as follow: second, third and fourth sections are respectively devoted to problem formulations, primary material on the model, some asymptotic results with impacts on basic reproduction rate induced by vertical transmission. We conclude with a discussion and conclusions.
Problem formulations
\(\Lambda _I\) and \(\Lambda _E\) traduce respectively the proportional influx of new infectives of W and Ytypes coming from vertical transmission (see “Appendix 1”). We will track their influences on basic reproduction rate.
To perform our analysis we shall assume that the contact between individuals is homogeneous so that \(\beta _i(a,a')\equiv \beta _I>0{\text { and }}\beta _e(a,a')\equiv \beta _E\ge 0\) and vaccination strategies are timeindependent \(\Psi (t,a)\equiv \Psi (a) \ge 0\).
Like Republic of Niger’s or Cameroonian Governments (see Ministry 2014) we choose the situation with no newborn baby vaccination: \(v(t,0)=0\). Technically in most part of this work, the maximum lifetime \(\omega\) will be taken as \(+\infty\) (through coefficients’ supports) for sake of simplicity.
Preliminary materials
We point out that the case \(i=2\) with \(\sigma =0\) has been partially investigated in Kouakep and Houpa (2014) using integrated semigroup theory. We focus later on case \(i=1\). In the sequel \(\omega \in (0;+\infty ]\): it represents biologically the human maximum lifetime.
Abstract formulation
 1.the operator A satisfy Hille–Yosida property: \((\mu ,\infty )\subset \rho (A)\) withMoreover we have for each \(\lambda >\mu\): (1) \((\lambda A)^{1}X^{+}\subset X_{0+}\);$$\begin{aligned} \Vert (\lambda A)^{1}\Vert _{{\mathcal {L}}(X)}\le \frac{1}{\lambda +\mu },\;\;\forall \lambda >\mu . \end{aligned}$$(7)
 2.
A is generator of a \(C_0\)semigroup of linear bounded operators;
 3.
the domain D(A) of operator A is dense in \(X_0:=\overline{D(A)}\) and A is a closed operator.
Existence and uniqueness of solutions
For arbitrary \(\phi _0\in X_{0+}\), we solve (8) as mild solution of the integrated equation (see Djidjou et al. 2014): \(\phi (t)= \phi _0+A\int ^t_0 \phi (s)ds +\int ^t_0 F\left( \phi (s)\right) ds\;\;,\forall t\ge 0\). We obtain the following lemma.
Lemma 1
 (a) :

The operator A is generator of a \(C_0\) semigroup of linear bounded operators and the domain D(A) of operator A is dense in \(X_0\) and A is a closed operator.
 (b) :

Moreover, the nonlinear operator F from X to X is continuous and locally Lipschitz.
 (c) :

(8) generates a strongly continuous positive semiflow \(\{U(t)\}_{t\ge 0}\) on \(X_{0+}\). This means that for each \(x=\phi (0)\in X_{0+}\), the continuous map \(t\rightarrow U(t)x\) defined from \([0,\infty )\) into \(X_{0+}\) is a weak solution of (8), that is (for the integrated problem)$$\begin{aligned} \int _0^t \phi (s)ds\in D(A),\;\;\forall t\ge 0,\;\; \phi (t)=x+A\int _0^t \phi (s)ds+\int _0^t F\left( \phi (s)\right) ds,\;\;\forall t\ge 0. \end{aligned}$$
 (d) :

It satisfies the following boundeddissipative estimates for each \(x=\phi (0)\in X_{0+}\) (with \(\omega =+\infty\) for sake of simplicity) and each \(t\ge 0\):or$$\begin{aligned} \Vert x\Vert _X e^{\overline{\Lambda } t}\le \Vert U(t)x\Vert _{X}\le \frac{\Vert x\Vert _X}{\underline{\Lambda }}\left( 1e^{\underline{\Lambda } t}\right) +\Vert x\Vert _X e^{\underline{\Lambda } t}, \end{aligned}$$(9)with \(\underline{\Lambda }=\mu +min\left\{ \left( \epsilon _1\Lambda _I\right) ,\left( \epsilon _2\Lambda _E\right) \right\} ,\) and \(\overline{\Lambda }=\mu +max\left\{ \left( \epsilon _1\Lambda _I\right) ,\left( \epsilon _2\Lambda _E\right) \right\}\).$$\begin{aligned} \Vert x\Vert _X e^{\overline{\Lambda } t}\le \Vert U(t)x\Vert _{X}\le \frac{\Vert x\Vert _X}{\mu }\left( 1e^{\mu t}\right) +\Vert x\Vert _X e^{\mu t}, \end{aligned}$$(10)
 (e) :

Theorem 1.4 in Pazy (1983) proves that for \(\phi _0\in X_{0+}=\left[ L^1_+\left( 0,\omega ;[0,+\infty )\right) \right] ^2\times [0,+\infty )^3\) there exists a unique bounded continuous solution \(\phi\) to the integrated problem defined on \([0,+\infty )\) with values in \(X_{0+}\).
Proof
The proof of \([abce]\) is rather standard. Indeed it is easy to check that operator A satisfies the Hille–Yosida property. Then standard methodologies apply to provide the existence and uniqueness of mild solution for System (1)–(2) (see for instance Djidjou et al. 2014, Pazy 1983 and the references therein).
We define the total population P(t) at time t by \(P(t)=\int ^{+\infty }_0 \left( \phi _1(t,a)+\phi _2(t,a)\right) da+\phi _3(t)+\phi _4(t)+\phi _5(t)\) and use the fact that \(P(0)=\Vert x\Vert _X\). The proof of [d] is immediate from the integration of the Eqs. (1)–(2) using formal differentiation of P(t) in respect to t and assumptions made: \(\varepsilon \in L_{+}^{\infty }\left( 0,\omega ;{\mathbb {R}}\right)\), \(\Lambda _I\le \epsilon _1\) and \(\Lambda _E\le \epsilon _2\).\(\square\)
Remark 1
Under assumptions of Lemma 1, one could show that the semiflow \(\{U(t)\}_{t\ge 0}\) is asymptotically smooth on X by using results derived by Sell and You (2002).
From Lemma 1 and above Remark 1, one deduces using the results of Hale (1989), Smith and Thieme (2011), and Magal and Zhao (2005) the following results:
Lemma 2
Assume that \(\varepsilon \in L^{\infty }_+\left( 0,\omega ;{\mathbb {R}}\right)\), \(\Lambda _I\le \epsilon _1\) and \(\Lambda _E\le \epsilon _2\). The semiflow \(\{U(t)\}_{t\ge 0}\) provided by Lemma 1 has a nonempty compact global attractor \(\mathcal A\subset X_{0+}\). It means that \(\mathcal A\) is compact, invariant and attracts all bounded set \(B\subset X_{0+}\), such that for each \(B\subset X_{0+}\) bounded subset, one has \(d\left( U(t)B,\mathcal A\right) \rightarrow 0\) as \(t\rightarrow \infty\) where d(B, A) denotes the semi distance from B to A defined by \(d(B,A)=\sup _{y\in B}\inf _{x\in A} \Vert yx\Vert _X.\)
Asymptotic properties: impact on basic reproduction rate of vaccination efficiency and vertical transmission
In all this section we assume that \(\varepsilon \in L^{\infty }\left( 0,\omega ;{\mathbb {R}}\right)\), \(\Lambda _I<\epsilon _1\) and \(\Lambda _E<\epsilon _2\). We will see that the basic reproduction rate \(R_0\) is a decrease function of \(\Psi\) as noticed by Goyal and Murray (2014).
Steady states (DFE and EE)
Here we provide some information on steady states for (1)–(2).
Lemma 3
 (i) :

If \(R^{\Psi }_0\le 1\), then System (1)–(2)(\(i=1\)) has a unique stationary statewhere$$\begin{aligned} x_F=\left( s_F(a),v_F(a),0,0,0\right) ^T\in X_+ \end{aligned}$$and \(v_F(a)=\int _0^a \Psi (s)s_F(s)exp\left( \mu \left( as\right) \right) ds\).$$\begin{aligned} \begin{aligned} s_F(a)&=f_i\left( \int _0^\infty \left( 1\varepsilon (u)\right) \left[ s_F(u)+l_1 v_F(u)\right] da\right) \\&\quad \times \,exp\left( \mu a\int ^{a}_0 \Psi (s)ds\right) ,\quad a\ge 0, \end{aligned} \end{aligned}$$(12)
 (ii) :

If \(R^{\Psi }_0>1\), then system (1)–(2)(\(i=1\)) has two stationary states: Disease free equilibrium (DFE) \(x_F\in X_+\) and Endemic Equilibrium (EE) \(x_E=\left( s_E(.), v_E(.),I_E, E_E, R_E\right) ^T\) withwhere \(\lambda _E>0\) is the unique solution of the equation$$\begin{aligned} \begin{aligned} s_E(a)&=f_i\left( \int _0^\infty \overline{\varepsilon }(s)\left[ s_E(s)+l_1 v_E(s)\right] ds\right. \\&\left. +\,l_2 R_E+l_3 I_E+l_4 E_E\right) \times exp\left( \left( \lambda _E+\mu \right) a\int ^{a}_0 \Psi (s)ds\right) ,\\ v_E(a)&=\int _0^a \Psi (s)s_E(s)exp\left( \left( \delta \lambda _E+\mu \right) \left( as\right) \right) ds,\\ I_E&=\frac{\lambda _E}{\nu _I +\sigma \Lambda _I}\int _0^\infty p(a)\left( s_E(a)+\delta v_E(a)\right) da,\\ E_E&=\frac{\lambda _E}{\nu _E\Lambda _E}\int _0^\infty q(a)\left( s_E(a)+\delta v_E(a)\right) da +\frac{\sigma }{\nu _E \Lambda _E}I_E\\ R_E&=\frac{(\mu _I\epsilon _1)}{\mu }I_E+\frac{(\mu _E\epsilon _2)}{\mu }E_E. \end{aligned} \end{aligned}$$(13)$$\begin{aligned} \begin{aligned} 1&=\int _0^\infty \left( \frac{\beta _I}{\nu _I+\sigma \Lambda _I}p(a)+\frac{\beta _E}{\nu _E\Lambda _E}q(a)\right. \\&\quad +\left. \frac{\beta _E\sigma p(a)}{(\nu _E\Lambda _E)(\nu _I+\sigma \Lambda _I)} \right) \left( s_E(a)+\delta v_E(a)\right) da \end{aligned} \end{aligned}$$(14)
Rewriting Eq. (13) provides (see also Inaba 2001) a coupled integral equations system. The existence and uniqueness of continuous solutions \((s_E,v_E,I_E,E_E,R_E)\) for this type of Volterra like system is given by Gurtin and MacCamy (1974) (see a special case in “DFE special case (s*, v*, I* = 0, E* = 0, R* = 0): integral equation” and “EE special case (s*, v*, I* ≠ E* ≠ 0, R* ≠ 0): integral equation” sections in “Appendix 2”.
Threshold number explained as basic reproduction rate
Remark 2
It is obvious that the vertical transmission increases the basic reproduction rate if \(\forall u\in \left\{ I,E\right\} ,\;\Lambda _u < \epsilon _{d(u)}\le \mu _u\le \nu _u\) [more deaths than births, \(d(I)=1\) and \(d(E)=2\)]. We focus on the case (\(i=1)\).
Theorem 1
Discussion
The works of CastilloChavez and Feng (1998) and Djidjou et al. (2014) are more general by considering agedependent death rates and birth fertility. But, our work connects these two important works in some of their complementary lacks and strength in order to study the impact on basic reproduction rate (with influence vertical transmission) of a common vaccination strategy inducing the stability of steady states of two related diseases. We saw that the basic reproduction rate \(R_0\) is a decrease function of \(\Psi\) confirming the decrease in Hepatitis B virus (HBv) prevalence as vaccination coverage increases (Goyal and Murray 2014): it is then possible to eradicate both HBv and HDv (hepatitis D virus) using high vaccination coverage. In further work we will include migrations in the infected individuals’ classes. One could biologically suspect the cases (\(\nu _u \le \Lambda _u, \forall u\in \left\{ E,I\right\}\)) in Lemma 1 to be critical since we would like to avoid blowup of solutions in order to obtain global in time solutions. We said nothing in the cases: \(\nu _E> \Lambda _E> \epsilon _2 {{{\text { or }}}} \nu _I+ \sigma> \Lambda _I > \epsilon _1\) What arises in Theorem 1 if \(h=0\)?
Conclusions
The main objective of this work is study the impact of vertical transmission on basic reproduction rates in the case of coinfection like HBV(hepatitis B)/HDV(hepatitis D) coinfection. We found that vertical transmission increases the basic reproduction rate. Beside this, we studied the influence of the influx by migration on the wellposedness of the mathematical problem: there is a tradeoff between entries balanced by mortalities and wellposedness for long term dynamic of our agestructured model. Some asymptotic relations between the mean of the fertility rate and other biological parameters are derived in endemic or free epidemic situations (“Appendices 1 and 2”). A perspective could be to introduce diffusion in our model and evaluate a minimal speed for travelling wave solutions.
Declarations
Authors' contributions
AMO contributes for the modeling of the diseases dynamics, computation of steady states and the discussion around impacts of vertical transmission on basic reproduction rate. While YTK contributes in the model construction, basic reproduction rate design and the proof of the wellposedness of the model with developments in the “Appendices 1 and 2”. Both the authors read and approved the final manuscript.
Acknowledgements
We would like to thank the two anonymous reviewers and our advisors (Prof B. Mampassi, Prof B. Saley, Prof Békollè D., Prof Houpa D. D. E.), Dr A. Tall (AIMSSn), Dr Ntyam A., LYCLAMO/MINESEC with GDMMIAP student group for their valuable comments or advices. YTK was supported with an AMMSI scholarship through an IMUCDC Grant in 2014. Research supported also for YTK by Canada’s International Development Research Centre (IDRC), and within the framework of the AIMS Research for Africa Project, while AMO was supported by a Post AIMSSenegal Grant in 2014.
Competing interests
Both authors declare no competing interest.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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