Journal of Applied Mathematics

Volume 2018, Article ID 2618985, 16 pages

https://doi.org/10.1155/2018/2618985

## Basic Properties and Qualitative Dynamics of a Vector-Borne Disease Model with Vector Stages and Vertical Transmission

Departamento de Matemática e Informática, Universidade Eduardo Mondlane, 257, Maputo, Mozambique

Correspondence should be addressed to Sansao A. Pedro; moc.liamg@ordepoasnas

Received 7 July 2018; Revised 30 September 2018; Accepted 10 October 2018; Published 1 November 2018

Academic Editor: Zhidong Teng

Copyright © 2018 Sansao A. Pedro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work systematically discusses basic properties and qualitative dynamics of vector-borne disease models, particularly those with vertical transmission in the vector population. Examples of disease include Dengue and Rift Valley fever which are endemic in Sub-Saharan Africa, and understanding of the dynamics underlying their transmission is central for providing critical informative indicators useful for guiding control strategies. Of particular interest is the applicability and derivation of relevant population and epidemic thresholds and their relationships with vertical infection. This study demonstrates how the failure of derived using the next-generation method compounds itself when varying vertical transmission efficiency, and it shows that the host type reproductive number gives the correct . Further, novel relationships between the host type reproductive number, vertical infection, and ratio of female mosquitoes to host are established and discussed. Analytical results of the model with vector stages show that the quantities , , and , which represent the vector colonization threshold, the average number of female mosquitoes produced by a single infected mosquito, and effective reproductive number, respectively, provide threshold conditions that determine the establishment of the vector population and invasion of the disease. Numerical simulations are also conducted to confirm and extend the analytical results. The findings imply that while vertical infection increases the size of an epidemic, it reduces its duration, and control efforts aimed at reducing the critical thresholds , , and to below unity are viable control strategies.

#### 1. Introduction

Vector-borne diseases have been the scourge of man and animals since the beginning of time [1]. Today, vector-borne diseases account for over 17% of all infectious diseases causing more than 1 million deaths annually, and their distribution is mainly determined by a complex dynamic of environmental and social factors [2]. In spite of all these inherent complexities, mathematical models have been used to translate assumptions concerning biological, environmental, and social aspects into mathematical structures, linking biological processes of transmission and dynamics of infection at population level. Such dynamic models have impacted both our understanding of epidemic spread and public health planning (for more details see [3–5] and references therein).

In this study our particular interest is in investigating qualitative properties of epidemic models of mosquito-borne diseases in which the vector is of genera* Aedes*. These mosquito species are known to transmit many vector-borne diseases of vast epidemiological importance including Dengue fever and Rift Valley fever (RVF), just to mention a few. These diseases are endemic in Sub-Saharan Africa with pronounced health and economic impacts on domestic animals and humans. In fact, losses due to RVF can reach millions of dollars during periods of disease outbreaks [6, 7]. An interesting phenomenon underlying many vector-borne diseases is their ability to persist year round, fluctuating seasonally but only falling to zero during some months. Hence, an important question is, how does the virus maintain itself in nature? For RVF it has been hypothesized that RVF virus (RVFV) is maintained through transovarial transmission in* Aedes* mosquito eggs [8].* Aedes* eggs need be dry for several days before they can mature. After maturing, they hatch during the next flooding event large enough to cover them with water [9–11]. The eggs have high desiccation resistance and can survive dry conditions in a dormant form for months to years [12–15]. Thus, the emergence of adult* Aedes* mosquitoes from infected eggs can reintroduce RVF in livestock at the beginning of the rainy season, before other mosquitoes species amplify it further [16]. For instance, in eastern and southern Africa there is more and more evidence of disease activities between outbreaks [7, 17–20] highlighting the role of vertical transmission for initial disease spread and endemicity.

In epidemiology disease spread and persistence are measured through quantities known as epidemic thresholds. Their derivation and characterization are one of the most important results of mathematical epidemic models. The basic reproductive number, , is the most critical epidemic threshold given its applicability and suitability for deciding whether an outbreak will occur or fade out, making it essential for guiding disease control efforts. However, the derivation of this epidemic threshold in vector-borne disease models in particular suffers from a lack of uniqueness and it fails to give the correct average number of expected secondary infections produced by one infected individual [21]. This failure is more likely to compound itself when vertical transmission mode is included in the transmission model, since the resulting comes as the sum of the vertical and horizontal transmission components, if the next-generation method is used. Previous mathematical models have made a significant attempt in including vertical infection in modelling vector-borne diseases [22–28], but none of them discuss how the failure of compounds itself in the presence of vertical transmission. Therefore, the present work aims to discuss some relevant basic properties of vector-borne disease models when vertical infection is taken into account and their implications for disease control efforts. Further, our goal is to derive new epidemic thresholds useful for guiding control efforts in the settings of vector-borne disease models with vector stages that include vertical transmission mode.

We formulate two models, one simple but realistic and the other more complex with vector stages. The first is an extension of the one proposed by Ross [29] and popularized by Macdonald [30] and Anderson [31]. The model is used to discuss system properties such as the asymmetric relationship between the host-to-vector and vector-to-host reproductive numbers. In addition, we highlight how to derive epidemic thresholds useful for guiding disease control efforts and discuss their relationships with vertical transmission efficiency. It is shown that the model has two model equilibria, namely, the disease-free and the endemic, and Lyapunov function theory is used to establish their global qualitative dynamics. The second is an extension of the basic model, where the dynamics of both aquatic and adult mosquitoes are modelled explicitly. In this model we let the populations of aquatic and adults vary with time but be limited by their respective carrying capacity. The inclusion of the explicit vector submodel allows for derivation of critical thresholds such as the reproductive number for both the vector population and the disease system. Then, these thresholds are used to determine global qualitative dynamics of both the disease-free and endemic equilibria.

The paper is set out as follows. In Section 2 we formulate and discuss the two model systems. In Section 3 we provide the epidemic threshold theorems regarding both the vector population and disease equilibria. We also discuss important model properties and how to derive and identify model epidemic thresholds useful for guiding disease control efforts. Furthermore, numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease (taking RVF as disease example), to support analytical analyses and conclusions and illustrate possible behavioural scenarios of the model with vector stages. Finally, in Section 4 we present a short discussion of the results and their biological implications.

#### 2. Materials and Methods

For human and animal diseases, horizontal transmission typically occurs through direct or indirect physical contact with infectious hosts, or through disease vectors such as mosquitoes, ticks, or other biting insects. Among mosquito vectors vertical transmission is often through eggs. Of particular interest are female mosquitoes of genera* Aedes* which transmit the virus to their eggs. These eggs have some adaptive behaviour which allows them to stay dormant in nature for relatively long periods. Although vertical transmission also occurs among vectors involved in the transmission of Dengue disease, RVF is the disease for which the model is a good approximation. In particular, parameter values related to RVF are used to illustrate the dynamics of the disease numerically.

##### 2.1. Host-Vector Basic Model with Vertical Transmission

Let and denote the total host and vector populations sizes, respectively. We assume that individuals at each compartment mix homogeneously and each mosquito bites each individual host at a constant rate , where is the biting rate per unit time. Let be a probability of successful infection transmission from an infected mosquito to a susceptible host and be a probability of successful infection transmission from an infected host to a susceptible mosquito per bite. Thus, the forces of infection are as follows, and . Hosts are recruited into the population at per capita rate which is proportional to the total population and leave each compartment through death. Noninfected mosquitoes join the susceptible compartment at rate while vertically infected mosquitoes join the infected class at rate . Assuming constant population sizes, that is, births equal to deaths, can be obtained when both and are known. In the same way, can be obtained when is known. Thus, the expressions for both and can be omitted and the system can be written in terms of proportions:where is the probability of vertical infection, the rate at which infected hosts recover from infection, and denotes the ratio of female mosquitoes to hosts.

##### 2.2. Model with Vector Stages and Vertical Transmission

Here we extend the basic model to include vector stages. Partial results of the resulting model without vertical infection have been obtained in [32]. Our aim is to extend their model and analysis by investigating global dynamics of all model equilibria and examine the extent to which vertical infection alters the dynamics of the system. The mosquito population is divided into aquatic (eggs, larvae, and pupae) and terrestrial (adults) subpopulations with and being their carrying capacity, respectively. The parameter represents the larval maximal capacity limited by the availability of breeding sites while is the maximal capacity of adult mosquitoes limited by factors conditioning their survival such as high altitudes and high temperatures. Further, the aquatic subpopulation is divided into epidemiological classes, susceptible (), and infected () while adults are divided also into susceptible () and infected (). The per capita oviposition rate is , where is the intrinsic oviposition rate and . Aquatic mosquitoes emerge as adults at a per capita rate where the proportion emerge noninfected while the remainder are infected. Disease transmission dynamics between vector and host populations remain the same as in the basic model. As a result the following nondimensional system of ordinary differential equations represents the model that governs the temporal evolution of the disease: where , , , , , and .

Let be the initial conditions of system (2). It is easy to check that the feasible region for (2) is the positive orthant of and that the closed setis positively invariant for system (2).

###### 2.2.1. Positivity of Solutions

Lemma 1 (see [33]). *Let us denote and consider the function continuous with respect to and Lipschitz with respect to If for , with , then, for every , there exists such that the solution to exists and is unique and positive with value in and defined on some interval . If , then *

Theorem 2. *The solution set of the model (2) exists and is unique and positive for .*

*Proof. *Let and denote the function such thatSince the function is continuous and Lipschitz continuous with respect to , according to Picard’s theorem, there exists such that the solution to (2) exists locally at least on an interval of this form . Further, considering the initial condition at and using Picard’s theorem, it follows that there exists such that the solution to (2) exists and is unique on . Since is continuous and differentiable, the solution of (2) with a given initial condition is unique. Therefore, the solutions of (2) obtained on and on form the unique solution of (2) on with the initial condition at . Repeating this process again and again, we end up with the maximal forward interval of existence for the solutions of (2), say with . Furthermore, for , Therefore, the solutions of (2) on are positive, according to Lemma 1.

Finally, according to Theorem 2, the solutions to (2) are bounded on . In other words, they do not blow up on any finite interval of . It follows that, according to Lemma 1, the solution of (2) exists for all time. Hence for any initial condition in , system (2) possesses a unique and positive solution in .

#### 3. Results

##### 3.1. Analysis of the Basic Host-Vector Model

###### 3.1.1. Model Equilibria and Stability Analysis

The basic host-vector model with vertical transmission exhibits two equilibria, namely, the disease-free and the endemic , respectively. At the disease-free equilibrium,both vector and host populations persist but with no disease. The prevalence of the disease is denoted bywhere , , , , and

The Jacobian matrix of system (1) at is given bysuch that the characteristic polynomial of matrix (11) is then given aswith , , and . The coefficient and both are nonnegative if and only if . Hence, all Routh stability criteria are satisfied; that is, the three eigenvalues of matrix (11) are negative or have negative real parts. Furthermore, for , (9) becomes the disease-free equilibrium. Therefore, the following result holds.

Theorem 3. *The disease-free equilibrium exists and it is globally asymptotically stable if .*

Alternatively, the global stability of the disease-free equilibrium can be established using the following Lyapunov function:where are some positive constants. Calculating the derivative of along the solutions of system (1), we obtainChoosing and , (16) becomes Thus, is negative for . Note also that if and only if and . Therefore, the largest invariant set for (1) is the singleton . Hence, by LaSalle’s invariance principle [34], is globally asymptotically stable when and Theorem 3 is valid.

*Remark 4. *Clearly, the endemic equilibrium exists and is unique for . This excludes the possibility of occurrence of backward bifurcation. This result is of great epidemiological significance in guiding efforts for disease control as it indicates that is the critical epidemic threshold.

To establish the local stability of the endemic equilibrium we evaluate the Jacobian of the system at , which givesThe characteristic polynomial of matrix (18) is then given bywhere The coefficient and both are nonnegative if and only if . Hence, all Routh criteria are satisfied; that is, the three eigenvalues of matrix (11) are negative or have negative real parts. Therefore, the following results holds.

Theorem 5. *The endemic equilibrium exists and it is locally asymptotically stable if .*

A global stability result for the endemic equilibrium of system (1) is given below.

Theorem 6. *If , the endemic equilibrium is globally asymptotically stable.*

*Proof. *Let , , and be the components of the Lyapunov functionwhere are some positive parameters to be chosen later. Differentiating along the solutions of system (1), we obtain for and at equilibrium. for and at equilibrium. for and at equilibrium. Now all together, Using the inequality for with equality holding if and only if and the fact that the arithmetic mean is greater than or equal to the geometric mean, we obtain for all . Furthermore, we obtain that holds only when and that is the only equilibrium state of these systems on this plane (line). Therefore, by LaSalle’s invariance principle [34], the positive equilibrium is globally asymptotically stable.

###### 3.1.2. Epidemic Thresholds, Vertical Infection, and Basic Properties

One of the most important critical thresholds in epidemic models is the basic reproductive number, , which is usually found using the next-generation method, as the dominant eigenvalue of the* next-generation matrix* [35, 36]. Following the method in [36] we write system (1) consisting only of infectious compartments as the difference between new infection and transfer rates and the resulting Jacobian matrices evaluated at the disease-free equilibrium . For system (1) we have two infected classes, namely, and . It follows that the transmission and the transfer matrices and , respectively, are defined as Unlike in host-vector models without vertical transmission, the diagonal elements of the transmission matrix are nonzero. This stems from the fact that in the presence of vertical transmission there is vector to vector transmission, which completely changes the nature of the basic reproductive number. Thus, the next-generation matrix, , is then given by and the resulting dominant eigenvalue of the spectral radius , which is the basic reproductive number, , is given byWhen there is no vertical transmission, , as in the case of malaria, is simply the geometric mean of the product of the number of new infections in hosts from one infected vector and the number of new infections in vectors from one infected host, in the limiting case that both populations are fully susceptible. The interpretation and epidemiological significance of are well established. It is easy to see that the transmission of infection is increased with efficiency of vector biting and probabilities of successful infection transmission, but it is hindered by high mosquito death rates and faster host recovery. The biting rate appears as because it enters twice in the transmission cycle [31]. An important parameter is the ratio of female mosquitoes to hosts , which is central for disease spread according to model settings. increases with the number (or density) of mosquitoes but decreases with the number (or density) of host population. This results from the asymmetry in the dependence of the vector’s biting rate on the sizes of the host and vector populations such that when there are many more hosts compared to mosquitoes, sustained transmission may be impossible. Therefore, in the absence of vertical transmission, for the infection to successfully spread and invade, the ratio of mosquitoes to hosts needs to be sufficiently large so that double bites are common [37]: where the critical ratio is given by . Note that each mosquito could infect less than one host on average, and yet could still be more than unity. To elucidate this fact we write as a product of each host type single-step reproductive number; that is, where represents the number of new infections in mosquitoes from a single infected host while represents the number of new infections in hosts from a single infected mosquito. Clearly, can be greater than unity even when one of these reproduction numbers is less than unity, and it can also be less than unity even if one of its components is greater than unity. In Figure 1(a) we depict contours plots corresponding to the overall in (31) along the plane in the presence of vertical transmission. Its effects in this asymmetric relationship between and its components is not very pronounced and the ratio of mosquitoes to hosts remains one of the leading factors when there is a large disparity between the sizes of the host and vector populations.