1 Introduction

Tick-borne diseases in livestock have a significant economic impact in particular in the sub-Saharan region where Rift Valley fever (RVF) prevalence is endemic (Rich and Wanyoike 2010; Chevalier et al. 2004). Ticks are vectors of a number of both human and animal diseases including Lyme disease, colorado tick fever, Rocky Mountain spotter fever, African tick bite fever, bovine anaplasmosis and tick paralysis, just to mention a few. Virus isolation and laboratory investigations have implicated ticks in the transmission of RVF (Linthicum et al. 1989; Fontenille et al. 1998; Chevalier et al. 2004; Pepin et al. 2010). Ticks attach to the skin of humans and animals and feed on blood causing direct loss through sucking blood (Nchu and Rand 2013), limiting livestock production and improvement (Mwambi 2002).

In this paper, we propose a model that investigates the possible implications of ticks in the transmission of RVF (Linthicum et al. 1989; Nchu and Rand 2013) and the resulting epidemiological consequences in efforts for controlling RVF epidemics. We adapt previous RVF transmission models (Gaff et al. 2007; Pedro et al. 2014) to include ticks compartments. Hyalomma truncatum is both a two-host and three-host tick depending on the hosts species (Magano et al. 2000). This means that it must feed on two different hosts as larva and adult or three different hosts as larva, nymph and adult respectively (Mwambi 2002). Therefore, we include in our model attached and detached compartments and combine both immature stages and adults in one compartment to keep the model tractable.

The paper is set out as follows. In Sects. 1.1 and 1.2 we describe the epidemiology of the disease and its important features as well as the biology of ticks and their possible involvement in RVF transmission. In addition, we discuss previous work and contribution in the sphere of understanding disease transmission and persistence. Section 2 follows with model formulation, including the definition of a domain where the model is mathematically and epidemiologically well posed. In Sect. 3, we derive an explicit formula for the basic reproductive number, \(R_0\), an important critical condition for quantifying initial disease invasion, then we prove the existence and global stability of the disease-free equilibrium. In Sect. 4 we simulate the system to understand various underlying RVF dynamics and possible contribution of ticks to the disease spread and persistence. Finally in Sect. 5, we explore global sensitivity analysis followed by local sensitivity analysis of the model output with respect to input parameters.

1.1 RVF Epidemiology Mechanism

Rift Valley fever virus (RVFV), a member of the phlebovirus genus, and family Bunyaviridae, is an enveloped virus with a segmented, RNA genome. RVF is a viral disease that primarily affects both domestic and wild animals but is also capable of infecting humans (Xu et al. 2007; Jupp et al. 2002). Major host disease amplifiers are sheep, cattle and goats but the disease also affects camels, buffaloes and other mammalian species (Métras et al. 2011), causing high mortality, abortion and significant morbidity in domestic livestock (Anyamba et al. 2010). The virus has been isolated from at least 30 mosquito species in the field (Davies and Martin 2006), biting midges, blackflies and ticks (Linthicum et al. 1989; Fontenille et al. 1998; Pepin et al. 2010). However, major vectors are certain species of mosquitoes, most commonly of the genera Aedes, Culex, Eretmopodites and Mansonia among others (Jupp and Cornel 1988; Fontenille et al. 1998).

The RVF epizootics and epidemics in East Africa in particular have been largely correlated with heavy rainfall and flooding that provide suitable habitats for the development and proliferation of immature Aedes and Culex mosquitoes (Linthicum et al. 1985; Logan et al. 1991). Aedes Mcintosh are thought to be the reservoir of the virus as they have the ability to transmit the pathogen transovarially to their offspring (Linthicum et al. 1985; Pepin et al. 2010), leading to virus persistence during dry season/inter-epidemic periods in the endemic cycle (Linthicum et al. 1985). In periods of rainfall activities, transovarially infected adult mosquitoes may emerge and transmit RVFV to nearby domestic animal populations (Anyamba et al. 2010; Munyua et al. 2010). High viremias in these animals may then lead to infection of secondary arthropod vector species including various Culex species (Sang et al. 2010), and probably ticks species which are also capable of carrying RVFV (Linthicum et al. 1989; Fontenille et al. 1998) which further disperse the virus causing an outbreak.

1.2 Ticks and Their Possible Role on the Transmission of RVF

Hlyalomma truncatum is a tick species of the family Ixodidae, widely distributed within the African tropical geographical region (Apanaskevich and Horak 2008), which is found throughout the entire geographic range of RVFV (Linthicum et al. 1989). Ticks require blood meals to survive at each of their four life stages: egg, larvae, nymph and adult. Their hosts include humans, mammals, birds, reptiles and amphibians. However, most ticks have a variant of animal hosts at each stage of their life and disease transmission occurs through the process of feeding. As pointed out in the introduction, Hyalomma truncatum is both a two-host and three-host tick depending on the hosts species (Magano et al. 2000). Thus, a susceptible larvae, nymph or adult may acquire the disease when feeding on an infected host, drop off and switch to another host while in the same stage, and infect that host. Alternatively, susceptible larvae or nymph may acquire the disease by feeding from an infected host then transmit the disease in a later stage to the new susceptible host (Mwambi 2002). The present research study is motivated by a review undertaken by Nchu and Rand (Nchu and Rand 2013) on possible implications of Hyalomma truncatum on the dynamics of RVF outbreaks and it aims to hypothetically evaluate this phenomenon by means of mathematical modelling. This research builds on important features that underline the biology and the ecology of ticks in particular in Sub-Saharan Africa where RVF is endemic. The features are as follows:

  1. 1.

    Wild animals including rodents and livestock (sheep, cattle and goats) are the same hosts for both ticks and mosquitoes that transmit RVF (Pepin et al. 2010; Gora et al. 2000). In addition, immature stages of ticks prefer feeding on hares and rodents, extending the range of feeding hosts (Nchu and Rand 2013).

  2. 2.

    Ticks can be transported over long distances on their vertebrate hosts, hence serving as possible hosts for RVFV (Linthicum et al. 1989; Fontenille et al. 1998).

  3. 3.

    Ticks are widely distributed in the entire geographic range of RVFV.

  4. 4.

    Ticks can often survive for long time between blood meals (Sonenshine 1993) and the virus can persist in ticks for their whole lifespan (Davies et al. 1986). Thus, long life for ticks means a long time of persistence for the virus increasing the chances of contact between ticks and hosts (Nchu and Rand 2013).

  5. 5.

    When feeding on blood, a tick excretes substances in its saliva which have several effects. One is to modulate the hosts immune system. Viruses can benefit from this mechanism, helping it to infect co-feeding ticks on the same hosts vertebrate (Labuda and Nuttall 2004).

  6. 6.

    Ticks cause direct loss through sucking blood (Nchu and Rand 2013), reducing host production and increasing host vulnerability to diseases.

  7. 7.

    Mosquito population peaks generally coincide with availability of pastures for domestic animals and abundance of adult H. truncatum ticks (Nchu and Rand 2013).

Putting together the above ticks, mosquito and host’s epidemiological and ecological features we hypothesize that ticks may be contributing to RVF spreading and endemicity. In spite of all these inherent complexities, mathematical models can provide some very useful informative indicators regarding the potential contribution of ticks in the transmission of RVF. Disease outbreaks in livestock in a particular site are very brief (Munyua et al. 2010) and the peak is likely to pass undetected or under-reported. This may be due to interruption in the rainy events or the duration of the rain in a particular site as well as to animal incubation period which is very short (Pepin et al. 2010), and the resulting acquired immunity. In livestock the peak is likely to occur after the second or third week after the onset of the epidemic while in humans it is likely to occur between the fourth and sixth week after the first human case (Munyua et al. 2010; Anyamba et al. 2010; CDC 2007; Nguku et al. 2010). RVF modelling studies have also suggested that arthropods other than Aedes and Culex species may be contributing to RVF transmission by accelerating the cause of the outbreak (Pedro et al. 2014). Thus, this study aims to assess factors leading to this accelerated exponential phase of RVF outbreaks. Nchu and Rand (2013) highlighted that in addition to mosquitoes, optimum climatic conditions, international trade of livestock and animal products, ticks could be implicated in the spread of RVF. This would affect the dynamics of the disease, including the number of infected host, host extrinsic incubation period and size of the epidemic. In recent years RVF disease models have been developed for addressing a variety of questions related to disease transmission, maintenance and propagation across geographical regions (Gaff et al. 2007, 2011a; Mpeshe et al. 2011; Niu et al. 2012; Xue et al. 1012; Chitnis et al. 2013; Mpeshe and Luboobi 2014; Pedro et al. 2014). However, none of these models has included ticks compartments in order to access the possible implications of these blood feeding arthropods in the spread and endemicity of RVF. Characterization of tick host preference at different life stages by means of attached and detached compartments makes this modelling framework unique for investigating how the above ecological tick features could affect the transmission and persistence of the disease. To explain this, we extend previous deterministic epidemic models with two modes of disease transmission: horizontal (host-vector) transmission and vertical transmission from a female Aedes to its eggs to include compartments of ticks according to their questing and feeding behaviour. Then, we thoroughly investigate the system analytical and numerically and show that certain model parameters are relevant to the start of an outbreak, exponential phase of an outbreak, the prevalence of RVFV and the epidemic size of an outbreak. Some conclusions may also apply to other vector-borne diseases in which ticks are thought to participate in transmission of the pathogen as additional or secondary vectors. Our analysis provides general qualitative insights on the importance of the time ticks spend attached to a particular host and their host life cycle preference. These results suggest that it is possible to diminish the impact of ticks in the transmission of RVF by either inhibiting ticks to attach to a host or by enhancing the immunity of the host to avoid passage of the infection when a tick feeds on the host.

2 RVF Model Development

Three vector species, Aedes, Culex mosquitoes and ticks and one host livestock population (not necessarily cattle, sheep or goat) are considered in the model to investigate the role of ticks on RVF disease dynamics. Female infectious Aedes mosquitoes not only transmit RVFV to susceptible animals, but also to their own eggs (Gaff et al. 2007; Linthicum et al. 1985). Culex mosquitoes and ticks acquire RVFV during blood meals on an infected animal and then amplify the transmission while feeding on other susceptible animals. The animal host, Aedes and Culex compartments follow the same structure as in (Pedro et al. 2014) while the ticks sub-model follows the structure proposed in (Mwambi et al. 2000) and successfully applied by (Mwambi 2002; Rosà and Pugliese 2007), which is according to the questing and feeding behaviour of ticks.

In livestock, the majority of animals do not manifest clinical signs even in regions with severe RVF outbreaks (Munyua et al. 2010). However, for tractability, in our model we do not include an asymptomatic class. Thus, the density of animal population is divided into three classes according to the following epidemiological status: susceptible, \(S_2\), infectious, \(I_2\), and recovered (immune), \(R_2\). Animals enter the susceptible class through a constant recruitment rate, then become infectious, \(I_2\) when fed upon by infectious vectors (Aedes, Culex and ticks). After some time infected animals move to the recovered class, \(R_2\) and obtain life immunity or die due to the disease.

Aedes and Culex mosquito populations follow the same epidemiological structure ( susceptible, \(S_a\), exposed, \(E_a\), and infectious, \(I_a\)), although only an Aedes female is capable of transmitting the virus transovarially. The subscripts \(a = 1\) and \(a = 3\) represent Aedes and Culex mosquitoes, respectively. Mosquitoes enter a susceptible class through birth and are horizontally infected when biting an infectious animal. Once infected, mosquitoes remain infected for life. To account for transovarial transmission in the Aedes species, compartments for uninfected \(P_1\) and infected \(U_1\) eggs are included. As the Culex species cannot transmit RVF vertically, only uninfected eggs \(P_3\) are included (Gaff et al. 2007). Both mosquitoes and animals exit the population through a per capita natural death rate, however, infected animals may leave the population via a disease induced death rate. The size of each adult mosquito population is \(N_1 = S_1 + E_1 + I_1\) for adult Aedes and \(N_3 = S_3 + E_3 + I_3\) for adult Culex, whilst for animal hosts it is \(N_2=S_2+I_2+R_2\). Ticks have unique life histories that create epidemics which differ from other vector-borne diseases. Depending on the host species Hyalomma truncatum has both two-host and three-host life cycles (Magano et al. 2000). Meaning that they may need to attach to two or three hosts during their developmental stages for a blood meal at least once at the stages of larvae, nymph and or adult (Mwambi 2002). We assume no transovarial transmission, meaning that no disease transmission from mother to eggs. Thus, ticks become infected when feeding on infectious animals. As a result only the nymph and adult are able to transmit the infection when feeding on a susceptible animal.

The structure of the tick sub-model follows the model framework of Mwambi (2002). In this framework, we assume that the population of ticks interacts with a population of hosts where both tick and hosts may be infected or uninfected. For mathematical tractability, we combine the larvae, nymphs and adults into one compartment. However, we make distinction between attached and detached ticks such that adult female ticks lay their eggs after detaching. The oviposition rate is proportional to the number of newly attached ticks, \(A(t)=S_a(t)+I_a(t)\) as we do not keep track of the time a tick stays attached or matures (Mwambi 2002). After birth they join the susceptible detached compartment, \(S_d(t)\) and ticks die only while detached from the host at rate \(d_t\). Both uninfected and infected ticks reproduce or lay eggs after having detached from the host upon successful feeding or by rejection mechanisms by the host or by human intervention. Ticks prefer feeding in some special areas of the host, making it difficult for an extra tick to attach where one is already attached (Mwambi 2002). Thus, we postulate that ticks attach with a constant rate depending on host density and ticks already attached. This rate is a decreasing function of the total number of attached ticks A(t) and an increasing function of the host population \(N_2\), and it is defined to be \(\alpha N_2/(1+A(t))\). The constant is set equal to unity such that if the number of attached ticks is zero then the overall attachment rate of ticks will be \(\alpha N_2\). On the other hand detaching ticks will increase with increasing number of tick-susceptible livestock grazing in a given area. Thus, instead of a constant detachment rate we set it as a linear function of \(N_2\), namely \(\delta N_2\). The size of the animal host population may be reduced due to the disease-induced mortality reducing the interaction between hosts and vectors. For the tick-submodel we assume that interaction between questing ticks and host is governed by mass-action, since we do not keep track of which tick lives on which host category (Mwambi et al. 2000; Rosà and Pugliese 2007). This way, we model the encounter between ticks and host population as separate but interacting populations similar to a prey-predator systems (Mwambi 2002). As a result we are assuming that both host to ticks and ticks to host infections occur exactly at the beginning of the blood feeding. Hence, there will be a constant rate of infection from tick to host and vice versa such that the infection rate of hosts is proportional to the number of infected attached ticks and the infection rate of ticks is proportional to the number of infected hosts. In other words, we are assuming that the infection rate \(\beta\) ( implicitly the product of contact rate and transmission probability) scales up with the animal host population size \(N_2\), implying that the transmission is density-dependent. This means that as the population size of mammalian host increases, so does the contact rate. The rationale is the same as that of the questing rate assumption, which says that the rate at which ticks attach is an increasing function of the host population. One would argue that it would be convenient to set the infection rate from ticks to host to be proportional to the rate at which infectious ticks attach to susceptible host. However, we chose to take a different approach in order to keep the model tractable and within its practical purposes by incorporating such effects in the dynamics of ticks population. This is accomplished by allowing ticks to attach according to a rate depending on host density and ticks already attached. Although in this setting, information regarding the distribution of ticks on the host population may be lost, our approach is critical for determining important threshold conditions regarding persistence of ticks on the host and for the persistence of the disease on the tick-host system. The approach has been successfully applied in other studies of tick-host disease transmission (Mwambi et al. 2000; Mwambi 2002; Rosà and Pugliese 2007), and the density-dependent transmission mode is generally well accepted for animal diseases (McCallum and Dobson 2002; Keeling and Rohani 2007). A complete summary of the above mosquito-host and tick-host RVF transmission processes is given by a schematic representation of the flow of individuals between epidemiological classes (see Fig. 1).

Fig. 1
figure 1

Flow diagram of RVFV transmission with each species, namely, Aedes mosquitoes, Culex mosquitoes, ticks and livestock (the solid lines represent the transition between compartments and the dashed lines represent the transmission between different interacting species)

Full size image

2.1 Mathematical Model of RVF Transmission with Three Vectors

Here we develop the mathematical representation of the RVF transmission processes by making use of ordinary differential equations. Consider livestock population settled in regions close to mosquito habitats, particularly species of genus Aedes. When it rains mosquito eggs hatch and transovarially infected eggs emerge and transmit RVFV to nearby domestic animals (Anyamba et al. 2010). High circulation of the virus in these animals may then lead to infection of secondary arthropod vectors including various Culex species (Sang et al. 2010; Sindato et al. 2014), and probably some ticks species, enough to trigger an outbreak. The mosquito-host and vice-versa disease transmission depends on both the vector and host population sizes (Chitnis et al. 2013). The flow diagram in Fig. 1 with state variables described in Table 1 and parameters interpreted in Table 2 satisfy the following system of ordinary differential equations:

2.2 Aedes

$$\begin{aligned} \dot{P_1}(t)&= \mu _1(A_0-q_1I_1)-\theta _1 P_1, \\ \dot{U_1}(t)&= \mu _1q_1I_1-\theta _1 U_1, \\ \dot{S_1}(t)&=\theta _1 P_1-\displaystyle \frac{\sigma _1\sigma _2\beta _{12}}{\sigma _1 N_1+\sigma _2 N_2}I_2S_1-\mu _1S_1,\\ \dot{E_1}(t)&=\displaystyle \frac{\sigma _1\sigma _2\beta _{12}}{\sigma _1 N_1+\sigma _2 N_2}I_2S_1-\gamma _1 E_1-\mu _1E_1 ,\\ \dot{I_1}(t)&=\gamma _1 E_1 +\theta _1 U_1- \mu _1I_1 , \end{aligned}$$
(1)

2.3 Livestock

$$\begin{aligned} \dot{S_2}(t)&= \mu _2L_0-\displaystyle \frac{\sigma _1\sigma _2\beta _{21}}{\sigma _1 N_1+\sigma _2 N_2}I_1S_2-\displaystyle \frac{\sigma _3\sigma _2\beta _{23}}{\sigma _3 N_3+\sigma _2 N_2}I_3 S_2-\beta _{2t}I_a S_2-\mu _2S_2 ,\\ \dot{I_2}(t)&= \displaystyle \frac{\sigma _1\sigma _2\beta _{21}}{\sigma _1 N_1+\sigma _2 N_2}I_1S_2+\displaystyle \frac{\sigma _3\sigma _2\beta _{23}}{\sigma _3 N_3+\sigma _2 N_2}I_3 S_2+\beta _{2t}I_a S_2-\varepsilon _2 I_2-\mu _2I_2 -m_2 I_2,\\ \dot{R_2}(t)&= \varepsilon _2 I_2- \mu _2R_2 , \end{aligned}$$
(2)

2.4 Culex

$$\begin{aligned} \dot{P_3}(t)&= \mu _3C_0-\theta _3 P_3, \\ \dot{S_3}(t)&=\theta _3 P_3-\displaystyle \frac{\sigma _3\sigma _2\beta _{32}}{\sigma _3 N_3+\sigma _2 N_2}I_2S_3-\mu _3S_3 ,\\ \dot{E_3}(t)&=\displaystyle \frac{\sigma _3\sigma _2\beta _{32}}{\sigma _3 N_3+\sigma _2 N_2}I_2S_3-\gamma _3E_3-\mu _3E_3,\\ \dot{I_3}(t)&=\gamma _3 E_3 - \mu _3I_3, \end{aligned}$$
(3)

2.5 Ticks

$$\begin{aligned} \dot{S_a}(t)&= \displaystyle \frac{\alpha N_2 S_d}{1+S_a+I_a}-\beta _{t2}I_2 S_a -\delta N_2 S_a, \\ \dot{S_d}(t)&= b_t(S_a+I_a)-\displaystyle \frac{\alpha N_2 S_d}{1+S_a+I_a} +\delta N_2 S_a-d_t S_d,\\ \dot{I_a}(t)&=\beta _{t2}I_2 S_a+\displaystyle \frac{\alpha N_2 I_d}{1+S_a+I_a}-\delta N_2 I_a,\\ \dot{I_d}(t)&=\delta N_2 I_a-\displaystyle \frac{\alpha N_2 I_d}{1+S_a+I_a}-d_t I_d.\end{aligned}$$
(4)
Table 1 State variables of the RVF model
Full size table

For Aedes-Livestock-Culex RVF horizontal transmission we assume that the total number of bites varies with both the livestock and mosquito population sizes [for more details see (Chitnis et al. 2013)]. While for the tick-livestock RVF horizontal transmission we assume that the livestock and ticks separate populations that interact in a similar manner as in a prey-predator system (Mwambi 2002), such that the number of newly infected ticks is proportional to infected livestock and the number of newly infected livestock is proportional to attached infected ticks (Rosà and Pugliese 2007). Therefore, the forces of infections are given by:

$$\begin{aligned} \lambda _{1}= \lambda _{12}=\displaystyle \frac{\sigma _1\sigma _2 N_2}{\sigma _1 N_1+\sigma _2 N_2}\beta _{12}\frac{I_2}{N_2}= \displaystyle \frac{\sigma _1\sigma _2\beta _{12}I_2}{\sigma _1 N_1+\sigma _2 N_2},\\ \lambda _2=\lambda _{21}+\lambda _{23}+\lambda _{2t}=\displaystyle \frac{\sigma _1\sigma _2\beta _{21}I_1}{\sigma _1 N_1+\sigma _2 N_2}+\displaystyle \frac{\sigma _3\sigma _2\beta _{23}I_3}{\sigma _3 N_3+\sigma _2 N_2}+\beta _{2t}I_a,\\ \lambda _{3}=\lambda _{32}=\displaystyle \frac{\sigma _3\sigma _2 N_2}{\sigma _3 N_3+\sigma _2 N_2}\beta _{32}\frac{I_2}{N_2}= \displaystyle \frac{\sigma _3\sigma _2\beta _{32}I_2}{\sigma _3 N_3+\sigma _2 N_2},\\ \lambda _4 = \lambda _{t2}=\beta _{t2}I_2. \end{aligned}$$
(5)
Table 2 Parameters of the RVF model
Full size table

To establish the positivity and feasibility of solutions of the model system (1–4) we discuss two invariant sub-systems as follows:

  1. 1.

    A sub-system of uninfected livestock, attached and detached ticks such that:

    $$\begin{aligned} \dot{S_2}(t)&= \mu _2 L_0-\mu _2 S_2,\\ \dot{S_a}(t)&= \displaystyle \frac{\alpha N_2 S_d}{1+S_a} -\delta N_2 S_a, \\ \dot{S_d}(t)&= b_tS_a-\displaystyle \frac{\alpha N_2 S_d}{1+S_a} +\delta N_2 S_a-d_t S_d. \end{aligned}$$
    (6)

    In the absence of the disease \(S_2 = L_0\), thus the livestock population is at equilibrium. Now we can investigate the remaining two equations of sub-system (6) describing the dynamics of ticks as in (Mwambi 2002). Clearly there is one free-tick equilibrium \((S_a,S_d)=(0,0)\), however, we are interested on the non-trivial one. Adding the last two equations of (6) and equating the sum to zero we obtain:

    $$S_a = \frac{d_t}{b_t}S_d.$$
    (7)

    Then after substituting Eq. (7) into the last equation of (6) gives

    $$S_d = \frac{\alpha b_t^2 - \delta b_t d_t}{\delta d_t^2},$$
    (8)

    such that the quantities \(S_a\) and \(S_d\) are only positive if and only if

    $$\frac{\alpha b_t}{\delta d_t} > 1.$$
    (9)

    Clearly the equilibrium \((S_a,S_d)=(0,0)\) is unstable if Eq. (9) holds. The trace of the Jacobian matrix for the \((S_a,S_d)\) system is negative. Thus, by the negative criteria of Bendixon, periodic orbits do not exist which implies that every trajectory goes to a stationary point (Li and Muldowney 1993; Mwambi 2002).

  2. 2.

    Now we consider a system made of livestock, Aedes and Culex mosquitoes only. However, when no confusion arises we establish the positivity of solutions of this sub-system together with the tick sub-system to avoid repetition.

Hence, we reorganize the system (1–4) and write it in matrix form as

$$\frac{dX}{dt}=M(x)X+F$$
(10)

where \(X=(P_1,U_1,S_1,E_1,I_1,S_2,I_2,R_2,P_3,S_3,E_3,I_3,S_a,S_d,I_a,I_d)\). M(x) is a 16 by 16 matrix and F is a column matrix. Substituting \(I_1=N_1-S_1-E_1\) we have

\(\dot{P_1}(t)=\mu _1A_0(1-q_1)+\mu _1q_1A_0S_1+\mu _1q_1A_0E_1-\theta _1P_1\). Thus

$$\begin{aligned} M(x)= \begin{pmatrix} M_1(x) &{} 0 &{} 0 &{} 0\\ 0&{}M_2(x)&{}0&{}0\\ 0&{}0&{}M_3(x)&{}0\\ 0&{}0&{}0&{} M_4(x) \end{pmatrix}, \end{aligned}$$
(11)

where

$$\begin{aligned} M_1= & {} \begin{pmatrix} -\theta _1 &{}0 &{} \mu _1q_1A_0 &{} \mu _1q_1A_0 &{} 0\\ 0 &{} -\theta _1 &{} 0 &{} 0 &{} \mu _1q_1\frac{A_0}{N_1} \\ \theta _1 &{} 0 &{}-g_1I_2-\mu _1 &{} 0 &{} 0\\ 0 &{} 0 &{} g_1 I_2 &{} -(\gamma _1+\mu _1) &{} 0\\ 0 &{} \theta _1 &{} 0 &{} \gamma _1 &{} -\mu _1 \end{pmatrix}, M_2= \begin{pmatrix} -g_3I_1-g_4I_3-\beta _{2t}I_A-\mu _2 &{} 0 &{} 0\\ g_3I_1+g_4I_3+\beta _{2t}I_A &{}-(\varepsilon _2+m_2+\mu _2) &{} 0\\ 0 &{} \varepsilon _2 &{} -\mu _2 \end{pmatrix},\end{aligned}$$
(12)
$$\begin{aligned} M_3= & {} \begin{pmatrix} -\theta _3 &{}0 &{} 0 &{} 0\\ \theta _3 &{}-g_5I_2-\mu _3 &{} 0 &{} 0\\ 0 &{} g_5 I_2 &{} -(\gamma _3+\mu _3) &{} 0\\ 0 &{} 0 &{} \gamma _3 &{} -\mu _3 \end{pmatrix}, M_4= \begin{pmatrix} -b_{t2}I_2-\delta N_2 &{} \frac{\alpha N_2}{1+S_a+I_a} &{} 0 &{} 0\\ b_t+\delta N_2 &{} - \frac{\alpha N_2}{1+S_a+I_a} &{} b_t &{} 0\\ \beta _{t2}I_2 &{} 0 &{} -\delta N_2-d_t &{} \frac{\alpha N_2}{1+S_a+I_a}\\ 0 &{} 0 &{} \delta N_2 &{} -\frac{\alpha N_2}{1+S_a+I_a}-d_t \end{pmatrix} \end{aligned}$$
(13)

and \(F=(\mu _1A_0(1-q_1),0,0,0,0,\mu _2L_0,0,0,\mu _3C_0,0,0,0,0,0,0,0)^T\).

Combining all matrices together, M(x) is a Metzler matrix, i.e. a matrix such that off diagonal terms are non-negative, for all \({\mathbb {R}}_+^{16}\). F is non-negative given the fact that \(1-q_1\ge 0\) and F is Lipschitz continuous. Thus, system (10) is positively invariant in \({\mathbb {R}}_+^{16}\). Then, the feasible region for the model system is the set

$$\begin{aligned} \varPhi =\left\{ (P_1,U_1,S_1,E_1,I_1,S_2,I_2,R_2,P_3,S_3,E_3,I_3,S_a,S_d,I_a,I_d)\ge 0\in {\mathbb {R}}_+^{16}\right\} . \end{aligned}$$
(14)

The solution remains in the feasible region \(\varPhi\) if it starts in this region. Hence, the system is epidemiologically and mathematically well posed and it is sufficient to study the dynamics of the model in \(\varPhi\).

3 Model Analysis and Results

3.1 Existence and Stability of Model Steady States

We analyse model system (1–4) to obtain equilibrium points of the system and their stability. Let \(X(P_1^*,U_1^*,S_1^*,E_1^*,I_1^*,S_2^*,I_2^*,R_2^*,P_3^*,S_3^*,E_3^*,I_3^*,S_a^*,S_d^*,I_a^*,I_d^*)\) be an arbitrary equilibrium point of system (1-4). At the equilibrium point, we have

$$\begin{aligned} P'_1=U'_1=S'_1=E'_1=I'_1=S'_2=I'_2=R'_2=P'_3=S'_3=E'_3=I'_3=S'_a=S'_d=I'_a=I'_d=0. \end{aligned}$$
(15)

3.1.1 Disease-Free Equilibrium (DFE), \(X^0\)

In the absence of the disease, that is, \(U_1^0=E_1^0=I_1^0=I_2^0=E_3^0=I_3^0=I_a^0=I_d^0=0\), model system (1–4) has an equilibrium point called the disease-free equilibrium, \(X^0\). When solving for the equilibria Eq. (15) for the tick sub-model we obtain

$$\begin{aligned} \{S_d^0,S_a^0\}=\{0,0\}\,\text {or}\,\left\{ \frac{(b_t\alpha -\delta d_t)b_t}{d_t^2\delta },\frac{b_t\alpha -\delta d_t}{\delta d_t}\right\} . \end{aligned}$$
(16)

The later is biologically significant whenever \(b_t\alpha > d_t\delta\). At equilibrium the birth is equal to the death rate, hence the inequality \(b_t\alpha > d_t\delta\) can be written as \(\alpha> \delta \Leftrightarrow \frac{\alpha }{\delta }>1\). \(1/\delta\) refers to the time ticks spend attached to the host and \(\alpha /\delta\) is the rate that gives rise to the number of newly attached ticks. This shows that we have two possible disease-free equilibria: one when we do not have a tick population at all, that is a system without ticks; another one with the presence of ticks. Since we are interested in studying the role of ticks in the spread of RVF among livestock, we consider \(S_d^0>0\) and \(S_a^0>0\). Therefore the disease-free equilibrium of the system is given by

$$\begin{aligned} X^0&=\left( P_1^0,0,S_1^0,0,0,S_2^0,0,0,P_3^0,S_3^0,0,0,S_a^0,S_d^0,0,0 \right) \\ &=\left( \frac{\mu _1A_0}{\theta _1},0,A_0,0,0,L_0,0,0,\frac{\mu _3 C_0}{\theta _3},C_0,0,0,\frac{b_t\alpha -\delta d_t}{\delta d_t},\frac{(b_t\alpha -\delta d_t)b_t}{d_t^2\delta },0,0\right) . \end{aligned}$$
(17)

In order to establish the linear stability of the model equilibria states, we employ the next generation matrix approach of Driessche and Watmough (2002). A reproduction number obtained in this way determines the local stability of the disease-free equilibrium for \(R_0 < 1\) and instability for \(R_0 > 1\). \(R_0\) represents the average number of individuals infected by a single infected individual during his or her entire infectious period, in a population which is entirely susceptible (Heesterbeek 2002; Heffernan et al. 2005). When \(R_0 < 1\), if a disease is introduced there is high probability that the disease will not invade the population. For the opposite case \(R_0 > 1\), the disease may become endemic, implying low probability of stochastic fade out of the disease. Following the notation in Driessche and Watmough (2002), if \({\mathscr {F}}_{i}\) is the rate of appearance of new infections in compartment i, and \({\mathscr {V}}_{i}\) is the rate of transfer of infections from one compartment to another, then the next generation operator is given by \({FV}^{-1}\) where F is nonnegative and V is a nonsingular matrix. For more details on the derivation of the matrices F and V see Appendix 1. The basic reproduction number, \(R_0\) is the largest eigenvalue of the spectral radius of \({FV}^{-1}\) and is given by

$$\begin{aligned} R_0=\frac{1}{2}R_{0,V}+\frac{1}{2}\sqrt{R_{0,V}^{2}+4R_{0,H}^{2}}, \end{aligned}$$
(18)

where

$$\begin{aligned} R_{0,V}=\frac{\mu _1q_1}{\mu _1} \end{aligned}$$
(19)

and

$$\begin{aligned} \begin{array}{lll} R_{0,H}= \displaystyle \sqrt{\frac{\gamma _1}{\gamma _1+\mu _1}\frac{g_3S_2^0}{\mu _1}\times \frac{g_1S_1^0}{\varepsilon _2+m_2+\mu _2}+\frac{\gamma _3}{\gamma _3+\mu _3}\frac{g_4S_2^0}{\mu _3}\times \frac{g_5S_3^0}{\varepsilon _2+m_2+\mu _2}+\left( \frac{\beta _{2t}S_2^0\alpha }{d_t(1+S_a^0)\delta }+\frac{\beta _{2t}S_2^0}{\delta N_2}\right) \frac{\beta _{t2}S_a^0}{\varepsilon _2+m_2+\mu _2}}, \end{array} \end{aligned}$$
(20)

where \(g_1=\displaystyle \frac{\sigma _1\sigma _2\beta _{12}}{\sigma _1A_0+\sigma _2L_0},\,g_3=\displaystyle \frac{\sigma _1\sigma _2\beta _{21}}{\sigma _1A_0+\sigma _2L_0},\,g_4=\displaystyle \frac{\sigma _3\sigma _2\beta _{23}}{\sigma _3C_0+\sigma _2L_0}\) and \(g_5=\displaystyle \frac{\sigma _3\sigma _2\beta _{32}}{\sigma _3C_0+\sigma _2L_0}\).

In the absence of vertical transmission, \(q_1=0\), \(R_0\) is the geometric mean of the number of new infections in livestock from infected Aedes, Culex mosquitoes and ticks, and the number of new infections in both mosquitoes and ticks from an infected animal in the limiting case that both livestock and vector populations are fully susceptible.

3.1.2 Biological Interpretation of \(R_0\)

From the expression for \(R_{0,H}\) we obtain the following sub-reproduction numbers:

$$\begin{aligned} \bar{R}_0^1=\displaystyle \frac{\gamma _1}{\gamma _1+\mu _1}\frac{g_3S_2^0}{\mu _1}\times \frac{g_1S_1^0}{\varepsilon _2+m_2+\mu _2} \end{aligned}$$

is the basic reproduction number for the model without vertical transmission, Culex mosquitoes and ticks;

$$\begin{aligned} R_0^3=\frac{\gamma _3}{\gamma _3+\mu _3}\frac{g_4S_2^0}{\mu _3}\times \frac{g_5S_3^0}{\varepsilon _2+m_2+\mu _2} \end{aligned}$$

is the basic reproduction number for the model without Aedes mosquitoes and ticks, and

$$\begin{aligned} R_0^t=\left( \frac{\beta _{2t}S_2^0\alpha }{d_t(1+S_a^0)\delta }+\frac{\beta _{2t}S_2^0}{\delta L_0}\right) \frac{\beta _{t2}S_a^0}{\varepsilon _2+m_2+\mu _2} \end{aligned}$$

is the basic reproduction number for the model without Aedes and Culex mosquitoes. \(\bar{R}_0^1\) is the product of \(R_{21}\times R_{12}\), where \(R_{21}\) is the number of new infections in livestock from one infected Aedes mosquito and is given by

$$\begin{aligned} R_{21}=\displaystyle \frac{\gamma _1}{\gamma _1+\mu _1}\times \frac{g_3S_{2}^{0}}{\mu _1}=\displaystyle \frac{\gamma _1}{\gamma _1+\mu _1}\times \frac{\sigma _1\sigma _2\beta _{21}S_2^0}{\sigma _1A_0+\sigma _2L_0}\times \frac{1}{\mu _1}, \end{aligned}$$

representing the product of the probability that an Aedes mosquito survives the exposed stage \(\frac{\gamma _1}{\gamma _1+\mu _1}\), the number of bites on livestock per mosquito \(\frac{\sigma _1\sigma _2}{\sigma _1A_0+\sigma _2L_0}S_{2}^{0}\), the probability of transmission per bite \(\beta _{21}\), and the infectious lifespan of an Aedes mosquito \(1/\mu _1\). \(R_{12}\) is the number of new infections in Aedes mosquitoes from one infected animal, and is given by

$$\begin{aligned} R_{12}=\displaystyle \frac{g_1S_1^0}{\varepsilon _2+m_2+\mu _2}=\frac{\sigma _1\sigma _2\beta _{12}S_1^0}{\sigma _1A_0+\sigma _2L_0}\times \frac{1}{\varepsilon _2+m_2+\mu _2}, \end{aligned}$$

which describe the product of the number of bites an animal receives \(\frac{\sigma _1\sigma _2}{\sigma _1A_0+\sigma _2L_0}S_{1}^{0}\), the probability of transmission per bite \(\beta _{12}\) from an infected animal and the duration of the infective period \(\frac{1}{\varepsilon _2+m_2+\mu _2}\) for an animal.

\(R_0^3\) is the product of \(R_{23}\times R_{32}\), where \(R_{23}\) is the number of new infections in livestock from one infected Culex mosquito and is given by

$$\begin{aligned} R_{23}=\displaystyle \frac{\gamma _3}{\gamma _3+\mu _3}\times \frac{g_4S_{2}^{0}}{\mu _3}=\displaystyle \frac{\gamma _3}{\gamma _3+\mu _3}\times \frac{\sigma _3\sigma _2\beta _{23}S_2^0}{\sigma _3C_0+\sigma _2L_0}\times \frac{1}{\mu _3}, \end{aligned}$$

which is the product of the probability that a Culex mosquito survives the exposed stage \(\frac{\gamma _3}{\gamma _3+\mu _3}\), the number of bites on livestock per mosquito \(\frac{\sigma _3\sigma _2}{\sigma _3C_0+\sigma _2L_0}S_{2}^{0}\), the probability of transmission per bite \(\beta _{23}\), and the infectious lifespan of a Culex mosquito \(1/\mu _3\). \(R_{32}\) is the number of new infections in Culex mosquitoes from an infected animal and is given by

$$\begin{aligned} R_{32}=\displaystyle \frac{g_5S_3^0}{\varepsilon _2+m_2+\mu _2}=\displaystyle \frac{\sigma _3\sigma _2\beta _{32}S_3^0}{\sigma _3L_0+\sigma _2L_0}\times \frac{1}{\varepsilon _2+m_2+\mu _2}. \end{aligned}$$

This is the product of number of bites one animal receives \(\frac{\sigma _3\sigma _2}{\sigma _3C_0+\sigma _2L_0}S_{3}^{0}\), the probability of transmission per bite \(\beta _{32}\) an infected animal and the duration of the infective period \(\frac{1}{\varepsilon _2+m_2+\mu _2}\) for an animal. \(R_0^t\) is the product of \(R_{2t}\times R_{t2}\), where \(R_{2t}\) is the number of new infections in livestock from one infected attached tick and is given by

$$\begin{aligned} R_{2t}=\displaystyle \frac{\beta _{2t}\alpha S_2^0}{d_t(1+S_a^0)}+\frac{\beta _{2t}}{\delta }, \end{aligned}$$

which is the product of the probability of transmission \(\beta _{2t}\) from an infectious tick, number of ticks attached to the host \(\frac{\alpha S_2^0}{1+S_a^0}\), and the infectious lifespan of ticks \(1/d_t\) together with new infections \(\beta _{2t}/\delta\) during the attached period. \(R_{t2}\) represents the number of new infections in ticks from one infected animal and is given by

$$\begin{aligned} R_{t2}=\displaystyle \frac{\beta _{t2}S_a^0}{\varepsilon _2+m_2+\mu _2}, \end{aligned}$$

which is the product of the probability of transmission \(\beta _{t2}\) from an infected animal to attached susceptible ticks, number of susceptible ticks \(S_a^0\) and the animal infective period \(1/(\varepsilon _2+m_2+\mu _2)\).

The square root in the expression for \(R_0\) comes from the ’two generations’ required for an infected vector or host to reproduce itself.

If \(q_1>0\), \(R_0\) increases because vertical transmission directly increases the number of infectious mosquitoes and indirectly increases the transmission from livestock to mosquitoes and back to livestock. Therefore, from Driessche and Watmough (2002) (Theorem 2), the following result holds;

Lemma 1

The disease-free equilibrium \(X^0\), of the RVF model with ticks, given by (1–4) exists for \(\alpha >\delta\) and is locally asymptotically stable if \(R_0 < 1\), and unstable if \(R_0 > 1\).

Note that, applying Theorem 2 in Driessche and Watmough (2002) we state that for \(R_0 >1\) there exists an endemic equilibrium (EE), a solution where the disease persists in the community. However, we will not be investigating the stability of the EE for the following reasons: the model is too complex to analytically derive expressions for the EE, and it will be a daunting task to analyse its stability. This will be considered in a subsequent study that will extend the model and then employ techniques of numerical simulations.

3.1.3 Global Asymptotic Stability of the DFE, \(X^0\)

Following the approach and results obtained by Kamgang and Sallet (2005) and successfully applied in Dumont et al. (2008) and using the properties of DFE, we write the system (1–4) in the following form

$$\begin{aligned} \dot{x}_{S}&=A_1(x)(x_S-x_{DFE,S})+A_{12}(x)x_I\\ \dot{x}_I&=A_2(x)x_I \end{aligned}$$
(21)

where \(x_S\) is the vector representing disease-free compartments (susceptible and immune individuals) and the vector \(x_I\) represents the state of infected compartments (exposed and infectious individuals). Hence, we have \(x_S=(P_1,S_1,S_2,R_2,P_3,S_3,S_a,S_d)^T\), \(x_I=(U_1,E_1,I_1,I_2,E_3,I_3,I_a,I_d)^T\) and \(x_{DFE,S}=(P_1^0,S_1^0,S_2^0,R_2^0,P_3^0,S_3^0,S_a^0,S_d^0)^T\). Then we rewrite some equations of system (1–4) as follows:

$$\begin{aligned} \dot{P_1}(t)&=-\theta _1 (P_1-P_1^0)-\mu _1q_1A_0\frac{I_1}{N_1},\end{aligned}$$
(22)
$$\begin{aligned} \dot{S_1}(t)&=\theta _1(P_1-P_1^0)-g_1I_2S_1-\mu _1(S_1-S_1^0),\end{aligned}$$
(23)
$$\begin{aligned} \dot{S_2}(t)&=-\mu _2(S_2-S_2^0)-g_3I_1S_2-g_4I_3S_2-\beta _{2t}I_aS_2,\end{aligned}$$
(24)
$$\begin{aligned} \dot{P_3}(t)&=-\theta _3(P_3-P_3^0),\end{aligned}$$
(25)
$$\begin{aligned} \dot{S_3}(t)&=\theta _3(P_3-P_3^0)-g_5I_2S_3-\mu _3(S_3-S_3^0),\end{aligned}$$
(26)
$$\begin{aligned} \dot{S_a}(t)&=\frac{\alpha N_2S_d}{1+S_a+I_a}-(\beta _{t2}I_2+\delta N_2)(S_a-S_a^0)-(\beta _{t2}I_2+\delta N_2)S_a^0,\end{aligned}$$
(27)
$$\begin{aligned} \dot{S_d}(t)&=b_t(S_a+I_a)+\delta N_2 S_a-\left( \frac{\alpha N_2}{1+S_a+I_a}+d_t\right) (S_d-S_d^0)-\left( \frac{\alpha N_2}{1+S_a+I_a}+d_t\right) S_d^0. \end{aligned}$$
(28)

Thus, we obtain the following matrices for \(A_1(x),A_{12}(x)\,\text {and}\,A_2(x)\)

$$\begin{aligned} A_1(x)= & {} \begin{pmatrix} -\theta _1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \theta _1 &{} -\mu _1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -\mu _2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -\mu _2 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} -\theta _3 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} \theta _3 &{} -\mu _3 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -(\beta _{t2}I_2+\delta N_2) &{} \frac{\alpha N_2}{1+S_a^0}\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \delta N_2 &{} -(\frac{\alpha N_2}{1+S_a^0}+d_t) \end{pmatrix} \end{aligned}$$
(29)
$$\begin{aligned} A_{12}(x)= & {} \begin{pmatrix} 0 &{} 0 &{} -\mu _1q_1A_0\frac{I_1}{N_1} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -g_1S_1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -g_3S_2 &{} 0 &{} 0 &{} -g_4S_2 &{} -\beta _{2t}S_2 &{} 0\\ 0 &{} 0 &{} 0 &{} \varepsilon _2 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -g_3S_3 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} -\beta _{t2}S_a^0 &{} 0 &{} 0 &{} -\frac{\alpha N_2S_d}{(1+S_a+I_a)^2} &{} \frac{\alpha N_2}{1+S_a+I_a}\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} b_t- \frac{\alpha N_2S_d}{(1+S_a+I_a)^2} &{} 0 \end{pmatrix} \end{aligned}$$
(30)

and

$$\begin{aligned} \begin{array}{ll} A_2(x)=\\ \begin{pmatrix} -\theta _1 &{} 0 &{} \mu _1q_1A_0\frac{I_1}{N_1} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} -(\gamma _1+\mu _1) &{} 0 &{} g_1S_1 &{} 0 &{} 0 &{} 0 &{} 0\\ \theta _1 &{} \gamma _1 &{} -\mu _1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} g_3S_2 &{} -(\varepsilon _2+m_2+\mu _2) &{} 0 &{} g_4S_2 &{} \beta _{2t}S_2 &{} 0\\ 0 &{} 0 &{} 0 &{} g_5S_3 &{} -(\gamma _3+\mu _3) &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} \gamma _3 &{} -\mu _3 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \beta _{t2}S_a &{} 0 &{} 0 &{} -(\frac{\alpha N_2I_d}{(1+S_a+I_a)^2}+\delta N_2) &{} \frac{\alpha N_2}{1+S_a+I_a}\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{\alpha N_2I_d}{(1+S_a+I_a)^2}+\delta N_2 &{} -(\frac{\alpha N_2}{1+S_a+I_a}+d_t) \end{pmatrix}. \end{array} \end{aligned}$$
(31)

Both \(A_1(x)\) and \(A_2(x)\) are Metzler matrices and the eigenvalues of matrix \(A_1(x)\) are real and negative. Thus, the system \(\dot{x}_{S}=A_1(x)(x_S-x_{DFE,S})\) is globally asymptotically stable (GAS) at \(x_{DFE,S}\). In order to proceed with the investigation of the GAS of the disease-free equilibrium, \(X^0\) we base our results on the Theorem in Kamgang and Sallet (2005), which was successfully applied in Dumont et al. (2008).

Theorem 1

Let \(\varPhi \subset {\mathcal {U}}={\mathbb {R}}_+^8\times {\mathbb {R}}_+^8\). The system (21) is of class \(C^1\), defined on \({\mathcal {U}}\) if

  1. 1.

    \({\mathcal {U}}\) is positively invariant relative to (21),

  2. 2.

    The system \(\dot{x}_{S}=A_1(x)(x_S-x_{DFE,S})\) is GAS at \(x_{DFE,S}\),

  3. 3.

    For any \(x\in \varPhi\), matrix \(A_2(x)\) is Metzler irreducible,

  4. 4.

    There exists a matrix \(\bar{A}_2\), which is an upper bound of the set \({\mathcal {M}}=\left\{ A_2(x)\in {\mathcal {M}}_8(\mathbb {R})|x\in \bar{\varPhi }\right\}\), with the property that if \(\bar{A}_2\in {\mathcal {M}}\), for any \(\bar{x}\in \bar{\varPhi }\), such that \(A_2(\bar{x})=\bar{A}_2\), then \(\bar{x}\in {\mathcal {R}}^8\times \{0\}\),

  5. 5.

    The stability modulus of \(\bar{A}_2,\alpha (\bar{A}_2)=\text {max}_{\lambda \in S_p(A_2)}{\mathbin {Re}}(\lambda )\), satisfies \(\alpha (\bar{A}_2)\le 0\).

Then the DFE is GAS in \(\bar{\varPhi }\)

Proof

The proof of the theorem requires verification of its underlying assumptions: it is obvious that condition (1–3) are satisfied. In particular for all \(x\in \varPhi\), \(A_2(x)\) is irreducible if and only if \((I+|A_2(x)|)^7>0\). An upper bound of the set of matrices \({\mathcal {M}}\), which is the matrix \(\bar{A}_2\) is given by matrix \(A_2(\bar{x})\), where \(\bar{x}=(\bar{P}_1,\bar{S}_1,\bar{S}_2,0,\bar{P}_3,\bar{S}_3,\bar{S}_a,\bar{S}_d,0,0,0,0,0,0,0,0)\in {\mathbb {R}}^8\times \{0\}\). Similarly matrix \(\bar{A}_2\) is irreducible. Recall that the Perron-Frobenius theorem for an irreducible matrix states that one of the matrix eigenvalues is positive and greater than or equal to all others, that is, the dominant eigenvalue. Thus, matrix \(A_2\) is exactly the matrix used to compute the basic reproductive number, i.e., the dominant eigenvalue. For more details or proof in general settings see Kamgang and Sallet (2005).

Conditions (1–4) are now verified. The proof of the last condition is based on the following Lemma in Kamgang and Sallet (2005), also successfully applied in Dumont et al. (2008).

Lemma 2

Let H be a square Metzler matrix written in block form \(H=\left( {\begin{matrix} A&{}B\\ C&{}D \end{matrix}} \right)\), with A and D squares matrices. H is Metzler stable if and only if matrices A and \(D-CA^{-1}B\) are Metzler stable.

Thus, matrix \(A_2(x)\) can be written in the following block form:

$$\begin{aligned} A&= \begin{pmatrix} -\theta _1 &{} 0 &{} \mu _1q_1A_0\frac{I_1}{N_1} &{} 0 \\ 0 &{} -(\gamma _1+\mu _1) &{} 0 &{} g_1S_1 \\ \theta _1 &{} \gamma _1 &{} -\mu _1 &{} 0 \\ 0 &{} 0 &{} g_3S_2 &{} -(\varepsilon _2+m_2+\mu _2) \end{pmatrix},\quad B= \begin{pmatrix} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} g_4S_2 &{} \beta _{2t}S_2 &{} 0 \end{pmatrix},\\ C&= \begin{pmatrix} 0 &{} 0 &{} 0 &{} g_4S_2\\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \beta _{t2}S_a\\ 0 &{} 0 &{} 0 &{} 0 \end{pmatrix},\quad D= \begin{pmatrix} -(\gamma _3+\mu _3) &{} 0 &{} 0 &{} 0\\ \gamma _3 &{} -\mu _3 &{} 0 &{} 0\\ 0 &{} 0 &{} -(\frac{\alpha N_2I_d}{(1+S_a+I_a)^2}+\delta N_2) &{} \frac{\alpha N_2}{1+S_a+I_a}\\ 0 &{} 0 &{}\frac{\alpha N_2I_d}{(1+S_a+I_a)^2}+\delta N_2 &{} -(\frac{\alpha N_2}{1+S_a+I_a}+d_t) \end{pmatrix}. \end{aligned}$$

A is a stable Metzler matrix and

$$\begin{aligned} D-CA^{-1}B= \begin{pmatrix} -(\gamma _3+\mu _3) &{} \frac{(\mu _1-\mu _1q_1)(\gamma _1+\mu _1)g_5\bar{S}_3g_4\bar{S}_2}{1-R_0^1} &{} \frac{(\mu _1-\mu _1q_1)(\gamma _1+\mu _1)g_5\bar{S}_3\beta _{2t}\bar{S}_2}{1-R_0^1} &{} 0\\ 0 &{} -\mu _3 &{} 0 &{} 0\\ 0 &{} \frac{(\mu _1-\mu _1q_1)(\gamma _1+\mu _1)\beta _{t2}\bar{S}_ag_4\bar{S}_2}{1-R_0^1} &{} -\delta N_2+\frac{(\mu _1-\mu _1q_1)(\gamma _1+\mu _1)\beta _{t2}\bar{S}_a\beta _{2t}\bar{S}_2}{1-R_0^1} &{} 0\\ 0 &{} 0 &{} \delta N_2 &{} -\frac{\delta N_2}{1+\bar{S}_a}-d_t \end{pmatrix} \end{aligned}$$

is a stable Metzler matrix if \(\frac{\beta _{2t}\bar{S}_2}{\delta N_2}\frac{\beta _{t2}\bar{S}_a}{\varepsilon _2+m_2+\mu _2}<\frac{1-R_0^1}{(\mu _1-\mu _1q_1)(\gamma _1+\mu _1)(\varepsilon _2+m_2+\mu _2)}\) (see Appendix 2). It is worth noting that at disease-free equilibrium \(N_1 = A_0\). Finally, from Theorem 1 and Lemma 2, we deduce the following:

Theorem 2

If \(b_t\alpha > d_t\delta\), then the disease-free equilibrium of the system (1–4), \((P_1^0,0,S_1^0,0,0,S_2^0,0,0,P_3^0,S_3^0,0,0,S_a^0,S_d^0,0,0)\) exists and is globally asymptotically stable if \(R_0^1<1\) and \(\frac{\beta _{2t}\bar{S}_2}{\delta N_2}\frac{\beta _{t2}\bar{S}_a}{\varepsilon _2+m_2+\mu _2}<\frac{1-R_0^1}{(\mu _1-\mu _1q_1)(\gamma _1+\mu _1)(\varepsilon _2+m_2+\mu _2)}\).

The above result is epidemiologically relevant, because it shows that even though disease vectors such as Culex and H. truncatum and others, may play a significant role on amplifying the disease, the initial spread of RVF outbreaks depends on the competence of the Aedes species. From Theorem 2 we observe that \(R_{0,V}\) should be kept below unity. This is an indication that if at any time, through appropriate interventions (e.g. destruction of breading sites, herd vaccination, etc), we are able to lower \(R_0^1\) below unity and \(R_0^t\) below its critical value, then the disease will die out. From \(R_0^1=\frac{1}{\mu _1-\mu _1q_1}\bar{R}_0^1\), which depends on both Aedes mosquitoes vertical and horizontal transmission, the inequality, \(\mu _1q_1<\mu _1\), shows that vertical infection efficiency should be kept below the threshold, which can be accomplished by destroying possible locations that may allow Aedes eggs to dessicate. \(\bar{R}_0^1<1\) shows that horizontal transmission can be controlled for instance through herd immunization. Another important relation is \(\frac{\beta _{t2}\bar{S}_2}{\delta N_2}\frac{\beta _{2t}\bar{S}_a}{\varepsilon _2+m_2+\mu _2}<\frac{1-R_0^1}{(\mu _1-\mu _1q_1)(\gamma _1+\mu _1)(\varepsilon _2+m_2+\mu _2)}\), the left hand side is related to \(R_0^t\) describing the host-ticks interactions while the right side describes the Aedes ability to intermediate disease invasion. From this we learn that if herd immunity is not attained and ticks are capable of transmitting the disease, host-ticks interactions may serve as disease reservoirs or possibly disease amplifiers. Therefore, we argue that if ticks are capable of carrying and transmitting RVFV, ticks may play an important role in the spread of the disease and may also be more responsible for RVF inter-epidemic activities.

4 Numerical Simulation

Fig. 2
figure 2

Time series plot of both animal and vector populations against time of a model with and without ticks. The initial conditions are \(S_1=5000, P_1=1000, U_1=400, E_1=0, I_1=1, S_2=1000, I_2=1, R_2=0, S_3=5000, P_3=1000, E_3=0\) and \(I_3=1, S_a=1000, S_d=0, I_a=I_d=1\)

Full size image

To explore the behaviour of RVF when introduced into a naive population taking into account ticks as RVF competent vectors, we conducted numerical simulations of an isolated system (i.e. no immigration or emigration). The model uses a daily time step and high rainfall and moderate temperature (wet season) parameter values see Table 3. Figure 2 (a) depicts the time series plot of susceptible, infectious and recovered livestock for a model with ticks while (b) shows the dynamics of disease transmission in livestock within a model without ticks. The epidemic is very brief lasting for only 20 and 30 days and reaching its peak at about 10 and 15 days for a model with and without ticks respectively. This results indicate that ticks not only increase the size of an epidemic but also accelerate it and reduce its duration. Data about cases of infected livestock are very difficult to compile for several reasons: (1) RVF cases are not clinically specific and laboratory confirmation is necessary; (2) virus isolation techniques are costly and time consuming, and require high biocontainment level facilities (Pepin et al. 2010; 3) RVF outbreaks occur in rural areas with low accessibility to various needed services and during this period most of the areas are flooded and cannot be accessed by road. However, information about human cases is available although it is not representative as it only captures clinical cases reported in some district level hospitals (Munyua et al. 2010). Human cases appear about one month after infection in livestock and they reach a peak between the fourth and sixth week after the onset of the epidemic (Munyua et al. 2010; Anyamba et al. 2010; CDC 2007; Nguku et al. 2010). Figure2c, d depict the time series plot of infectious Aedes, Culex mosquitoes and infectious detached ticks for a model with and without ticks respectively. Their population densities reach their peak at approximately 25 and 30 days respectively. This results suggest that by time mosquito population reaches its peak the peak of the disease in the mammalian host is already passed, meaning that the presence of mosquitoes may not be appropriate for timing the peak of the outbreak, highlighting the necessity of continuous surveillance and communications with communities in endemic areas as has been pointed out by empirical studies (Munyua et al. 2010; Anyamba et al. 2010; Sumaye et al. 2013).

5 Sensitivity Analysis

To assess the impact of the parameters and decision rules within the model, uncertainty in input parameters and their sensitivity analysis are performed to determine how sensitive the model is to changes/shifts in the values of the parameters (Helton et al. 2006; Saltelli et al. 2000). Many of the parameters in disease epidemiological models can be found in the literature, not necessarily as constants but as approximate values or intervals. These intervals describe the range of values a parameter may assume with the evolution of the disease (Samsuzzoha et al. 2013). In order to measure the sensitivity of the model, two different approaches are considered. One is based on perturbation of the model parameters, useful for determining the impact of local changes, while the other is based on uncertainty in the model parameter estimation, which allows for determining the impact of global changes (Saltelli et al. 2000).

5.1 Global Sensitivity Analysis

Here, we employed the Latin Hypercube Sampling (LHS) technique, which belongs to the Monte Carlo class of sampling methods (McKay et al. 1979). LHS technique is a stratified sampling without replacement, where each parameter distribution is divided in N equal probable intervals, which are then sampled (Marino et al. 2008). For each input parameter we assumed uniform distribution (Gaff et al. 2007) across the range listed in Table 3 (see Appendix 3). We calculated \(R_0\) using \(N=1000\) sets of sampled parameters as the model output. In the LHS scheme, all model parameters are independent and varied simultaneously such that the Partial Rank Correlation Coefficient (PRCC) is used to evaluate statistical relationships (Saltelli et al. 2000; Marino et al. 2008), in order to assess the significance of each parameter with respect to \(R_0\). PRCC indicates the qualitative relationship, in particular the degree of monotonicity between specific input parameter and model output (Marino et al. 2008) and it has been successfully applied in other epidemic models (Gaff et al. 2007; Samsuzzoha et al. 2013). The results are shown in Fig. 3. The PRCCs for the parameters \(b_2,q_1,\alpha ,\beta _{t2},\beta _{2t},\beta _{12},\beta _{21},\beta _{23},\beta _{32},1/\varepsilon _2,S_a^0,S_1^0,S_2^0,\sigma _1,\sigma _2,\sigma _3\) are all positive indicating an increase in \(R_0\) with an increase in livestock birth/death rates, Aedes vertical transmission, ticks attachment, probabilities of transmission, mosquito incubation periods, length of infection in livestock, initial number of susceptible ticks, Aedes, Culex and animal host populations, number of times a mosquito would like to bite host and number of bites a host can sustain respectively.

It is expected that at an early stage of an epidemic in a system without ticks, RVF outbreaks may be highly influenced by numerous factors including: the initial number of susceptible Aedes mosquitoes; competence of Aedes mosquitoes in transmitting the disease transovarially and the initial number of available susceptible hosts and mosquito death rates (Gaff et al. 2007; Mpeshe et al. 2011; Chitnis et al. 2013). However, we observe that in the presence of ticks \((b_t\alpha > d_t\delta )\) the situation changes. The ticks attachment rate \(\alpha\), probability of transmission from ticks to host \(\beta _{2t}\) and from host to ticks \(\beta _{t2}\), length of infection in livestock, ticks detachment rate \(\delta\) and ticks death rate have greater impact on \(R_0\). This shows that if ticks are capable of transmitting RVF, they may be playing a major role in RVF outbreaks and endemicity. Ticks spend long periods feeding on the host, disease-bearing ticks may survive long periods of dessication and an infected fully fed female H. truncatum tick can continue to harbour RVFV post oviposition (Linthicum et al. 1989) enhancing virus circulation and maintenance. As observed in our model analysis, the RVF tick-system only exists if the number of ticks that attach to a host is greater than those that detach. This emphasizes that the time ticks spend attached to a particular host is a critical factor in the dynamics of the disease. Therefore, this calls for more attention to research with strength to establish the role that H. truncatum, other ticks and biting insects play in the transmission of RVFV in nature.

Fig. 3
figure 3

PRCC results and (asterisk) denotes PRCCs that have P value \({<}0.01\)

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Fig. 4
figure 4

Distribution of the basic reproductive number, \(R_0\) from a pool of 1000 sets of model parameters for \(R_0 <= 1\) and for \(R_0 >1\)

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In addition to the PRCCs results we show the distribution of \(R_0\) when it is less and above unity see Fig. 4. Very few parameter combinations would yield \(R_0 <1\) as the disease is intermediated by more than one disease vector. In this situation it is generally possible to have the overall \(R_0\) greater than unity even if either the vector or host reproductive number is less than unity (Massad et al. 2010). Averaging \(R_0\) when it is above unit over all parameter sets gives a mean of 4.8497 and it ranges between 0.5822 and 26.5939.

Moreover, varying Aedes mosquito biting preference parameter, \(\sigma _1\) and ticks attachment frequency, \(\alpha\) in the host, results in significant changes in the values of the basic reproductive number, \(R_0\). This is shown in Fig. 6 (left) in which we note that higher efficacy of \(\sigma _1\) and \(\alpha\) result in higher probability of occurrence of RVF outbreaks as expected. Given that the vectorial capacity of the vector is a critical component of its ability to transmit the infection (Lardeux et al. 2008). Hence, an increase in the parameter \(\sigma _1\) results in an exponential increase in the basic reproduction number. Another important observation is that a minimum of \(\delta = 0.3\) is necessary to keep \(R_0\) around the average value, highlighting the role of ticks questing behaviour. In addition, analysis of the parameter \(\sigma _1\), suggests that Aedes vertical transmission efficiency may play a significant role at an early stage of the epidemic, which eventually leads to a rapid spread of the disease as witnessed by Fig. 2.

5.2 Local Sensitivity Analysis

Using uncertainty and Partial Rank Correlation Coefficient (PRCC) analysis we were able to identify which parameters are important in contributing to variability in the outcome of the basic reproductive number. However, in order to reduce disease mortality and morbidity in livestock, focus should be oriented to disease prevalence (Mpeshe et al. 2011; Samsuzzoha et al. 2013). Therefore, in this section we use localized sensitivity analysis to determine the relative importance of some chosen parameters with respect to the state variable \(I_2\). Many parameters are directly related to disease prevalence, but for our analysis we focus on parameters related to the time ticks spend attached to a host and the number of times an Aedes mosquito would want to bite a host. These parameters are: \(\sigma _1,\alpha\) and \(\delta\). We simulate the system at a fixed value of the parameter then plot \(I_2\) versus the entire range of values that the parameter assumes along the evolution of the disease. The dynamical behaviour of disease prevalence in livestock, \(I_2\) along the range of these parameters is depicted in Fig. 5. Clearly, Fig. 5 shows the zone where changes should be made for an input parameter to determine the desired value of a predictor parameter.

Fig. 5
figure 5

Simulation of the system where maximum number of infected livestock \(I_2\) is selected at each point of the ranges of the number of bites an Aedes mosquito would want to bite an animal \(\sigma _1\), ticks attachment rate \(\alpha\) and the ticks detachment rate \(\delta\). a–c illustrate the changes in the local maximums of the state variable \((I_2)\) with respect to model parameters \(\sigma _1,\alpha\) and \(\delta\). d The contour plot of max\((I_2)\) in the \((\delta ,\alpha )\) plane

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The maximum density of infected livestock increases with any increment in the number of bites an Aedes mosquito would bite a host and with the rate at which ticks attach to a host (see Fig. 5a, b); however, it decreases exponentially with an increase in the rate at which ticks detach from a host (Fig. 5c). In Fig. 5d we observe that even at very low values of ticks attachment rate, the maximum number of infected hosts will rise to very high values. This suggests that any amount of time ticks spend on the host represents a high risk factor in the transmission of RVF. In addition we observe that there should be a balance between the attachment and the detachment rates for the disease to persist. This finding further confirms the analytical results in Sect. 3.1 obtained from Eq. (16) which says that persistence of ticks on the host population is subject to the condition \(\alpha >\delta\). Although this result is a direct consequence of the model structure, its underlying form of tick-host force of infection does not affect the results rather the ticks questing behaviour.

Furthermore, we compute sensitivity indices for some chosen parameters previously identified to contribute to the basic reproductive number variability. Thus we derive an analytical expression for each \(R_0\) sensitivity index based on the concept of the normalized sensitivity index (Samsuzzoha et al. 2013; Chitnis et al. 2013), given by

$$\begin{aligned} \Upsilon _{\psi }^{R_0}=\frac{\partial R_0}{\partial \psi }\frac{ \psi }{ R_0} \end{aligned}$$

for any parameter \(\psi\).

Fig. 6
figure 6

Left This shows \(R_0\) as a function of the number of times an Aedes mosquito would want to bite an animal, \(\sigma _1\) and the ticks attachment rate, \(\alpha\). The curves are contours in the \((\sigma _1,\alpha )\) plane along which \(R_0\) at early stage of disease following an introduction of a single infectious animal or vector (mosquito or tick), is constant. Right Sensitivity indices of the model outcome \(R_0\) with respect to some model parameters

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The sensitivity index results [see Fig. 6 (right)] agree strongly with the above uncertainty analysis. Increasing transmission rates of infection from livestock to ticks \(\beta _{t2}\) and from ticks to livestock \(\beta _{2t}\), and initial number of susceptible livestock, \(S_2^0\), increases the basic reproductive number, \(R_0\). While decreasing livestock recovery and disease-induced death rates as well as ticks death rate increase \(R_0\). The former results from the fact that ticks may die before transmitting the infection, thereby reducing \(R_0\).

5.3 Comparison of Mean Values of \(R_0\)

In order to quantify the extent at which ticks may be contributing in the transmission and spread of RVF, we review the overall mean of the basic reproductive number, \(R_0\) obtained from different studies which did not include additional vectors such as ticks. Figure 7 (left) represents the distribution of \(R_0\) derived from our model without ticks. Figure 7 (right) is a table describing the review of \(R_0\) from previous studies, which did not include ticks in their models. From these studies we observe that \(R_0\) varies from 1.19 to 3.5 on average while our model with ticks predicts \(R_0=4.8497\) on average. However, it is worth noting that some of the studies mentioned in the table used a different approach to compute the expression of \(R_0\), by taking the sum of \(R_0\) from vertical transmission and \(R_0\) from horizontal transmission (Gaff et al. 2007; Niu et al. 2012). These differences on the computation of \(R_0\) imply different values of \(R_0\) but do not influence the trend we observe in this comparison of different values of \(R_0\). From these comparisons it appears that ticks may have a significant contribution in the dynamics of the disease. Further, \(R_0\) computed in the basis of the next-generation matrix for vector-borne disease such as RVF does reflect the actual average number of secondary infections but rather the geometric mean (Li et al. 2011). This leads to low values of \(R_0\) as this method estimates \(R_0\) at each generation regardless of whether the generation is of mammalian host or vector. Consider that at the first generation or infection event, a single Aedes infects two animals, each of whom subsequently infect two Aedes, two Culex and two animals, then, a single infected Aedes will result in four Aedes, four Culex and four animals. This leads to an exponential increase of \(R_0\), resulting in a rapid spread of the virus if the target host is vulnerable to the disease, resulting in the so-called exponential phase of the outbreak (Keeling and Rohani 2007; Massad et al. 2010).

Fig. 7
figure 7

Left Distribution of \(R_0\) of the model without ticks from a pool of 1000 sets of model parameters. Right Mean values of \(R_0\) in previous models without ticks

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6 Discussion and Conclusion

To the best of our knowledge this is the first time, compartments representing epidemiological states of ticks are included in models of the evolution of RVF epizootics. Using a mathematical epidemiological model via a system of nonlinear ordinary differential equations we have formulated and analysed a RVF model that includes ticks as disease vectors other than Aedes and Culex mosquitoes. Based on the basic reproductive number, \(R_0\) Theorems in Driessche and Watmough (2002), analytical results establish that when \(R_0<1\), the disease-free equilibrium (DFE) is locally asymptotically stable while for \(R_0>1\) the endemic equilibrium (EE) is locally asymptotically stable. This is a very important result in epidemiology if one seeks to control vector-borne diseases via the control of the vector population, which remains an alternative of RVF control strategies (Munyua et al. 2010; Mpeshe and Luboobi 2014; Pedro et al. 2014). The above results suggest that Rift Valley fever virus (RVFV) is endemic if \(R_0 >1\) and and more likely remains at a very low level after an outbreak or between outbreaks. These findings are in line with empirical studies in many endemic areas which have shown that 1-3 % of domesticated animals are being infected with RVFV in endemic areas during the inter-epidemic period (Davies et al. 1992; Zeller et al. 1997; Sumaye et al. 2013). In addition to the overall model basic reproduction number \(R_0\) other epidemiological thresholds have been derived and interpreted. An example of this is the number of new infections resulting from an introduction of an Aedes mosquito, \(R_0^1\) in the absence of Culex and ticks. The derivation of such type reproductive numbers is of great epidemiological significance since it is possible for the disease to persist even if the overall \(R_0\) is greater than unity as long as either the vector or host type reproductive number is beyond unity (Massad et al. 2010). Global stability of the DFE was determined following the approach and results obtained by Kamgang and Sallet (2005). The results showed that although host-ticks interactions may serve as disease reservoirs or disease amplifiers, the values of \(R_0^1\) should be kept under unity if disease post epizootics activities are to be controlled. \(R_0^1\) encompasses two important informations: Aedes vertical transmission efficiency and their competence for initial spread and endemicity of the disease. This results from the fact that female Aedes mosquitoes can transmit the virus transovarially to their eggs (Pepin et al. 2010), which can survive from months to years before hatching at the next flood event (Horsfall et al. 1973; Becker et al. 2010). Newly emerged transovarially infected mosquitoes will then infect nearby vulnerable ruminants leading to persistence of the virus during the endemic cycle (Linthicum et al. 1985; Anyamba et al. 2010). However, for this transmission cycle to take place after large outbreaks resulting in sporadic disease activities in ruminants new pools of susceptible animals should be recruited into the population (Chamchod et al. 2014), given that after the outbreak surviving animals may still be immune to the disease (Pepin et al. 2010). These conclusions suggest that to prevent these low level transmission cycles appropriate control interventions (for instance destruction of mosquito breading sites and herd vaccination of newly recruited animals) or integrated vector control programs (Munyua et al. 2010) should be employed. However, herd vaccination is almost impossible to sustain in RVF affected countries for economic reasons (FAO 2011; Mpeshe and Luboobi 2014; Pepin et al. 2010). Therefore, affordable and effective integrated control interventions need to be developed in support of those vulnerable African communities with low resilience to economic and environmental challenges (LaBeaud et al. 2007; Pepin et al. 2010; Murithi et al. 2011).

Global and local sensitivity analysis of \(R_0\) have been carried out to determine the relative importance of each parameter in the disease transmission and prevalence. Through PRCCs analysis we found that \(R_0\) increases with an increase in the following model parameters, \(b_2,q_1,\alpha ,\beta _{t2},\beta _{2t},\beta _{12},\beta _{21},\beta _{23},\beta _{32},1/\varepsilon _2,S_a^0,S_1^0,S_2^0,\sigma _1,\sigma _2,\sigma _3\). The fact that an increase in livestock birth rate \(b_2\) implies an increase in the magnitude of \(R_0\) indicates that this parameter may be an important predictor for determining the total size of livestock population and the prevalence of RVFV after an outbreak. This result further confirms our findings through global stability analysis of the DFE highlighting the relative importance of recruitment of new susceptible animals after an outbreak and a study by Chamchod et al. (2014) have arrived to a similar conclusion. Moreover, results of sensitivity analysis have indicated that ticks attachment and detachment rates \(\alpha\) and \(\delta\), probability of transmission from ticks to host \(\beta _{2t}\) and from host to ticks \(\beta _{t2}\), length of infection in livestock and ticks death rate \(d_t\) have a greater impact on \(R_0\). Recall that \(R_0\) is a measure of initial disease transmission. Hence, decreasing the time and the proportion of ticks that attach to a host and reducing host-ticks interactions, disease transmission probabilities reduce the magnitude of \(R_0\). The parameters \(\alpha\) and \(\delta\) are related to ticks questing behaviour while the parameters \(\beta _{2t}\) and \(\beta _{t2}\) correspond to ticks feeding behaviour. Therefore, these results highlight the role that ticks questing and feeding behaviour play in the transmission of the virus. These findings are in good agreement with empirical studies indicating that ticks spend long periods feeding on the host, disease-bearing ticks may survive long periods of desiccation and an infected fully fed female H. Truncatum tick can continue to harbour RVF virus post-oviposition (Linthicum et al. 1989). Another important result is that increasing ticks death rate \(d_t\) substantially decreases \(R_0\). Hence, control strategies aiming to reduce the tick population may help reduce the disease burden. Although our assumption of constant tick death rate suggests that control of ticks implies eliminating their population, Gaff et al. (2011b) discuss several scenarios of optimal application of tick-killing treatment in controlling risk of tick-borne diseases without completely eliminating their population. Despite that little is known about what mainly regulates tick populations in the field, several studies demonstrate that host’s immune status strongly affects tick survival and fecundity (Rosà and Pugliese 2007). This deserves, on itself, further investigation for determining specific features that regulates ticks population that can then be targeted for disease management. Additionally, controlling the lifespan and biting rates of the vectors will help control both the initial spread of disease and ongoing infections, hence, reduce disease prevalence during an outbreak and after. These results are in agreement with findings from other theoretical studies (Gaff et al. 2007; Mpeshe et al. 2011; Chitnis et al. 2013; Chamchod et al. 2014), which have highlighted the relative importance of both RVF related death and natural deaths.

Munyua et al. (2010) discussed several RVF vector control programs and strategies with different applications such as interrupting epidemic cycles, preventing emergence of new cohorts of infected vectors and much more. Ticks questing and feeding behaviour is a critical component of the vector’s ability to transmit diseases. Therefore, control strategies targeting these features would be of great epidemiological significance. In addition, other studies (Gaff et al. 2011b) have suggested that use of pesticides (acaricides) to control tick population would enhance the effort made by hosts to avoid ticks attaching. Moreover, results from sensitivity analysis have suggested that decreasing the length of infection in hosts and decreasing the number of susceptible hosts significantly reduces \(R_0\). Contours curves of the maximum of \(I_2\) in the \((\delta ,\alpha )\) plane suggested that even at very minimum values of the attachment rate, \(\alpha\) we obtain high values of max\((I_2)\) if the detachment rate is very small, which is equivalent to long periods of time ticks spend attached to the host. This result further confirm the epidemiological implications of the time ticks spend attached to a host as highlighted by empirical studies (Linthicum et al. 1989; Nchu and Rand 2013).

Using the model parameter ranges to capture the entire history of disease evolution and running 1000 stochastic simulations, we computed the mean value of \(R_0\) which was found to exceed 4.5. This result is far beyond findings from other epidemic models without ticks (Gaff et al. 2007; Mpeshe et al. 2011; Chitnis et al. 2013; Niu et al. 2012), including our own model. These results stem from the fact that ticks not only increase the size of an epidemic but also accelerate the exponential phase of the outbreak. This calls for attention in designing preventive measures to curtail and stop the epidemic in the event of an outbreak (Mpeshe et al. 2011). Surveillance studies (Munyua et al. 2010; Sumaye et al. 2013) have highlighted that effective and continuous surveillance in livestock is a critical factor in detecting and responding to both RVF outbreaks and inter-epidemic activities. These results are important for controlling transient epidemics which are likely to take off even when \(R_0 <1\) (Chitnis et al. 2013; Massad et al. 2010). This is important in preventing the virus eventually building up from disease inter-epidemic activities and halt any future outbreak (Pedro et al. 2014; Sumaye et al. 2013), as a result of vertical transmission efficiency which was found to linearly increase \(R_0\). These results further confirm our findings through analytical analysis of the global asymptotic stability of the DFE, which indicated that the global stability of this equilibrium can only be attained if the Aedes mosquito reproductive number, \(R_0^1\) with vertical transmission is kept under unity. This is the first time that global stability of the DFE of a complex realistic RVF model is thoroughly analysed by means of analytical methods. This analysis has enabled us to obtain useful qualitative insights about the necessary conditions for long-term stability of the DFE which have shown dependence between tick-host interactions thresholds and Aedes mosquito reproductive number. This highlights the role of vertical transmission in some female Aedes mosquitoes and mammalian hosts in sustaining low levels of disease activities between outbreaks as virus reservoirs. However, the reservoirs and means of inter-epizootic maintenance are still subjects of further investigations.

Entomological studies have suggested that biting insects other than Aedes and Culex mosquitoes are also involved in the transmission of RVF (Fontenille et al. 1998; Chevalier et al. 2004; Pepin et al. 2010). In this paper we have formulated a deterministic model of RVF transmission which accounts for ticks as additional RVF potential vectors. The ticks species taken as an example in this model have been indicated to be potential vectors of RVFV (Linthicum et al. 1989; Fontenille et al. 1998; Pepin et al. 2010). Similar to Nchu and Rand (2013), Gora et al. (2000), Davies (2006) and other theoretical studies investigating the hypothetical introduction of RVF in temperate countries (Fischer et al. 2013; Xue et al. 2013; Hartley et al. 2011), this paper does not by any means aim to demonstrate that ticks are actually involved in the spread of RVF in endemic areas. Rather, we aimed by means of an epidemic model to yield qualitative information for enhancing our understanding in the case when ticks are assumed to be potential vectors of disease transmission. The analyses of this model have provided critical insights on the mechanisms underlying the possible role of ticks in the transmission of RVF. This was possible by structuring the ticks population following their questing and feeding behaviour which is a critical component of their ability to transmit and further disperse the virus (Rosà and Pugliese 2007). Our results indicate that ticks not only increase the size of an epidemic but also accelerate it and reduce its duration. These results implicate ticks as one of the contributors to the exponential phase of the outbreak.

The results of all model analysis presented in this paper should be interpreted as qualitative and relative, as opposed to quantitative, until future data, obtained by further interdisciplinary research studies focusing on RVF molecular epidemiology and tick chemical ecology, can be used to parametrize, calibrate and validate the model. Nevertheless, the current model framework and analysis enables us to gain valuable insights regarding the epidemiology of the disease and its implications and the model remains an important step towards the theoretical study of the role of ticks on the dynamics of RVF. The value of the present research study is not limited to only providing qualitative understanding of the systems underlying processes but it is also useful in pointing out relevant model parameters that require further attention from experimental ecologists and modellers. Such estimates would be of great importance for parametrizing more refined predictive models which would yield specific informative indicators useful for improving disease control strategies (Mwambi 2002; Chamchod et al. 2014), by providing effective guidance to public health policy makers. Furthermore, this model framework can be of great use to theoretical ecologists and epidemiologists working on vector-borne diseases in which ticks are secondary vectors or additional potential vectors.