Delays and Exposed Populations in Infection Models

Abstract

Most biological processes take time to occur. In infectious diseases, such as malaria or chikungunya, there is a period of time between when a susceptible individual gets bitten by the vector, and when the individual develops the disease. These times are commonly modeled by introducing delays or by adding exposed as a new population class. Given a model based on differential equations, delays can be introduced in different forms. In this paper, we study different ways of introducing the delays and, alternatively, using exposed populations. We also analyze their steady solutions and stability, and establish the conditions under which the studied models predict an epidemic. Results and conclusions are presented.

1. Introduction

Many processes related to biology, chemistry, even physics and economics, take time to happen. Newborns take time to mature, signals take time to reach their destination, immune cells take time to eliminate bacteria or debris, and many others. These times are delays in the processes. Of particular importance are the delays in epidemics and infectious diseases. For example, in a disease transmitted by direct contact such as the flu or a sexually transmitted disease, there is an incubation period from the time the contact with an infectious individual happens and the disease develops in the previously susceptible individual. Furthermore, after the individual develops the disease, there is a time before it recovers. In diseases transmitted by a vector such as malaria or chikungunya, there are also delays. The same is true for many diseases in animals and plants. Mathematical models can be used to understand the dynamics of many biological processes, including epidemics. Many of these models consider that all populations involved are well mixed and independent of the spatial position. Models based on ordinary differential equations are commonly used in these situations. However, the presence of delays may change the dynamics [1,2,3]. Among the possible changes are the appearance of oscillations, introduction of discontinuities in the time derivatives, non-uniqueness and different stability. There is a wide literature about models of epidemics or disease transmission using delay differential equations (DDE’s). See for example [3,4,5,6]. Epidemic models may involve several delays: the time between a susceptible having contact with an infective and actually getting the disease, maturation times, the time it takes a vaccine to become effective, the time it takes an infected individual to recover after getting the disease, etc. In this paper, we will study only discrete delays, that is, delays that have a constant value. The topic of delay differential equations is just mentioned in classical Mathematical Biology books such as [7,8]. In reference [9], the coverage is more extensive. Books on epidemic mathematical models also contain sparse information on delay differential equations [10,11,12]. This last book has a chapter [13] that provides a good introduction to epidemic models with delays. On the other side, books about delay differential equations, for example [1,14,15], usually do not deal with disease models. Two exceptions are [3,16]. The main objective of this paper is to study how to introduce discrete delays in ordinary differential models of disease transmission so that the delay differential equation makes biological sense. The use of exposed or latent populations is also considered. This is done in Section 2. In this section, the equilibrium solutions and their stability are also analyzed. The conditions under which a model with delays or exposed populations predicts an epidemic are determined. In Section 3, results are given, including numerical simulations to illustrate the theoretical results. Finally, in Section 4 a discussion of the results is presented.

2. Materials and Methods

Many processes including disease transmission involve delays and there are different ways of introducing them. We will introduce several well-known models based on ordinary differential equations, see how delays can be introduced and whether the delay differential equations make sense. We will also analyze their steady solutions and stability. For the epidemic models considered, we will also present conditions for the existence of an epidemic. Note that all the parameters in the presented models are non-negative, except when explicitly stated that they are negative.

2.1. One-Population Models

For simplicity we will start with models involving only one population $N\left(t\right)$. Malthus model is $\frac{dN}{dt}=rN$, with r the growth rate, which is equal to the birth rate $b>0$ minus the death rate $\mu >0$. For $r\ne 0$ it has a unique equilibrium solution $N\left(t\right)=0$. Since the exact solution is $N\left(t\right)=N\left(0\right)exp\left(rt\right)$, the equilibrium solution is unstable for $r>0$ and stable for $r<0$. Depending on the population, there is a significant delay in the maturation time $\tau$, but the number of deaths depends on the current population. Putting the delay as $\frac{dN}{dt}=(b-\mu )N(t-\tau )$ does not make biological sense since the deaths are also delayed. However, the delayed model

$$\frac{dN}{dt}=bN(t-\tau )-\mu N\left(t\right)$$

(1)

is biologically consistent. Of course the model $\frac{dN}{dt}=bN\left(t\right)-\mu N(t-\tau )$ is not correct from the biological perspective.

For $b-\mu \ne 0$, the model given by Equation (1) has a unique equilibrium solution $N\left(t\right)=0$. Since the equation is linear, it has the general solution $N\left(t\right)=N\left(0\right)exp\left(\lambda t\right)$ with $\lambda$ satisfying the characteristic equation $\lambda =bexp(-\lambda \tau )-\mu$. For $b>\mu$, the left-hand side is 0 at $\lambda =0$ and an increasing function of $\lambda$. The right-hand side is positive at $\lambda =0$ and a decreasing function of $\lambda$. Therefore, the characteristic equation has a solution $\lambda >0$ and $N\left(t\right)=0$ is asymptotically unstable. In a similar way, it can be shown that for $b<\mu$, this solution is asymptotically stable.

Furthermore, since the model $\frac{dN}{dt}=-\mu N(t-\tau )$ is linear, there is a solution of the form $N\left(t\right)=exp\left(\lambda t\right)$ with $\lambda$ satisfying the characteristic equation. For $\mu =\frac{\pi }{2\tau }$, a solution is $Acos\frac{\pi t}{2\tau }$ for arbitrary A [9]. As there are only deaths with no births, this solution does not make sense.

Next, consider the logistic growth model which is more realistic as its solutions are bounded

$$\frac{dN}{dt}=rN\left(t\right)(1-\frac{N\left(t\right)}{K}),$$

(2)

where $N\left(t\right)$ is the population, r the growth rate, and $K>0$ the carrying capacity. The new term $r{N}^2/K$ says that the interaction between members of the population reduces its growth due to competition for resources. It has two steady solutions, $N\left(t\right)=K$ which is stable and $N\left(t\right)=0$ which is unstable, as can be easily checked by looking at the sign of $\frac{dN}{dt}$ for different values of N. From the exact solution, it can be shown that it has no oscillatory solutions.

An alternative perspective of logistic equation can be obtained by considering the processes as biochemical reactions:

$$\begin{array}{cc}\hfill N& \stackrel{rN}{\to }2N\hfill \\ \hfill N& \stackrel{r{N}^2/K}{\to }.\hfill \end{array}$$

(3)

The first reaction says that the population growth is given by $rN$ and the second that the population decreases at the rate $r{N}^2/K$. Using chemical terminology, an individual turns into two with a propensity $rN$ and is eliminated from the system with a propensity $r{N}^2/K$. Population models given in terms of ordinary differential equations can be written in terms of biochemical reactions and vice versa. Scientists working with biochemical reactions usually convert them into differential equations for ease of solving, but going in the opposite direction helps understand the processes involved in population models. In the well-known experiments by Nicholson’s [17,18] studying Australian sheep blowflies under controlled conditions, he found that the population oscillated with a period between 35 and 40 days. May, in 1975, (reprinted in [19]) modified the logistic equation by introducing a delay to obtain oscillatory solutions. This delay is assumed to be the maturation time.

Delays can be introduced in different ways. The first way of introducing the delay, as done in [19] and denoted by Delay Model 1, is

$$\frac{dN}{dt}=rN\left(t\right)(1-\frac{N(t-\tau )}{K}).$$

(4)

The second, denoted by Delay Model 2, is

$$\frac{dN}{dt}=bN(t-\tau )-r\frac{N{\left(t\right)}^2}{K}-\mu N\left(t\right).$$

(5)

The third, Delay Model 3, is

$$\frac{dN}{dt}=bN(t-\tau )-r\frac{N{(t-\tau )}^2}{K}-\mu N\left(t\right).$$

(6)

And finally, Delay Model 4, is

$$\frac{dN}{dt}=rN(t-\tau )(1-\frac{N(t-\tau )}{K}).$$

(7)

Here, $r=b-\mu$ as before. Delay Model 1 has competition between populations at times $t-\tau$ and t, but not competition of the population with itself at time t or at time $t-\tau$, which does not make biological sense. In Nicholson’s [17,18] experiments, he had both adult blowflies and larva, but if $N\left(t\right)$ is the adult population at time t, $N(t-\tau )$ is not the population of larva. Furthermore, as Nicholson states, the adults and larva were fed limited amounts of food; however, the food for the adults was different from the food for the larva. So there was no competition between the adults and the larva for food (or any other resources). Delay Model 2 has the birth rate depending on the population at the previous time $t-\tau$ and the competition and death terms on the current time t, which is more realistic since the number of individuals at time t depends on the number of non-mature individuals (larva for insect populations) at time $t-\tau$. However, the competition for resources and the death terms depend on the current population N(t). Delay Model 3 has the growth depending on the present population and not on the maturing population, which does not make biological sense, and the competition is of the population at the previous time $t-\tau$. While it can be argued that the competition at time $t-\tau$ weakens the populations and cause death at time t, the delay is not the maturation time as suggested by [19]. So Delay Model 3 is unrealistic. In Delay Model 4, none of the terms depend on the current population, which is also unrealistic.

For steady solutions of the delay models, $N\left(t\right)=N(t-\tau )$, so $N\left(t\right)=0$ and $N\left(t\right)=K$ are also equilibrium solutions of all the presented delay models. To determine the asymptotic stability of the delay models, we linearize the equations about the equilibrium solutions. For Delay Model 1, linearizing about $N\left(t\right)=0$, that is, letting $N\left(t\right)=v\left(t\right),v\left(t\right)\ll 1$, Equation (4) reduces to $\frac{dv}{dt}=rv$ and since $r>0$, $N\left(t\right)=0$ is asymptotically unstable. Linearizing about $N\left(t\right)=K$, let $N\left(t\right)=K+v\left(t\right),v\left(t\right)\ll 1$ we get the equation $\frac{dv}{dt}=-v(t-\tau$, and the characteristic equation is $\lambda =-rexp(-\lambda \tau )$. For $r>0$ the right-hand side is always negative so $\lambda <0$ and the equilibrium $N\left(t\right)=K$ is asymptotically stable.

For Delay Model 2, the linearized equation about $N\left(t\right)=0$ is $\frac{dv}{dt}=b(t-\tau )-\mu v\left(t\right)$ which is Equation (1), and, therefore, has the same characteristic equation and the equilibrium $N\left(t\right)=0$ is asymptotically unstable for $b>\mu$. Linearizing about $N\left(t\right)=K$, the characteristic equation is $\lambda =bexp(-\lambda \tau )-2(b-\mu )-\mu$. For $b-\mu >0$ the left-hand side is 0 at $\lambda =0$ and an increasing function but the right-hand side is negative at $\lambda =0$ and a decreasing function. So there is no solution $\lambda >0$ and $N\left(t\right)=K$ is asymptotically stable.

By linearizing Delay Model 1 about $N\left(t\right)=K$, finding the characteristic transcendental equation for the eigenvalue $\lambda$, separating its real and imaginary parts, and assuming that the real part is zero, it can be shown that there is Hopf bifurcation to a periodic solution when $\tau r=\pi /2$ as done in [6,9,15] and in others.

For Delay Model 2, linearizing about $N\left(t\right)=K$ gives the characteristic transcendental equation $\lambda =bexp(-\lambda \tau )-(2r+\mu )$. Writing $\lambda$ in complex form, $\lambda ={\lambda }_R+i{\lambda }_I$, and assuming ${\lambda }_R=0$ gives $cos\left({\lambda }_{I}t\right)=\frac{2r+\mu }{r+\mu }>1$. Therefore, there is no periodic solution. In summary, Delay Model 1 makes no biological sense and Delay Model 2 does make sense but contains no periodic solutions. Therefore, the oscillations observed by [19] must not be due to the introduction of discrete delays.

2.2. SIS Model

A simple model consisting of two population compartments, susceptibles S and infectives I is the SIS model which considers that a susceptible becomes infected with probability $\beta /N>0$ via contact with an infective, and then recovers with no immunity to susceptible. The flu is an example of a disease that can be modeled by an SIS model. The total population is usually considered constant which assumes that the disease is fast compared with demographic effects or the number of births is equal to the number of deaths with no migration. The SIS model with no demographics is

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}SI+\gamma I\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}SI-\gamma I,\hfill \end{array}$$

(8)

where $\gamma >0$ is the recovery rate and $N>0$ is the total population. Since the population is constant, $S\left(t\right)=N-I\left(t\right)$ and we can work with only one equation

$$\frac{dI}{dt}=\frac{\beta }{N}(N-I)I-\gamma I.$$

(9)

The SIS model has two equilibrium states, $I\left(t\right)=0$ and $I\left(t\right)=(1-\frac{\gamma }{\beta })N$. The last one is physical only for $\frac{\gamma }{\beta }\le 1$. The first equilibrium is the disease-free equilibrium (DFE) and the second one is the endemic equilibrium. An easy way to check the stability is to rewrite Equation (9) as a logistic equation

$$\frac{dI}{dt}={\beta }_1(1-\frac{1}{N_1})I,$$

(10)

with ${\beta }_1=\beta -\gamma$ and $N_1=\frac{\beta 1N}{\beta }$.

Depending on the disease, two delays may be added to the SIS model, ${\tau }_1$ the time it takes the disease to develop in a susceptible individual after being infected by contact with an infective individual, and ${\tau }_2$ the time it takes for a newly infected individual to recover. This second delay is the minimum time that the individual has the disease. Since $\gamma$ is the inverse of the average time an individual has the disease, this coefficient needs to be adjusted if the delay is introduced. For consistency in the delayed models, the $\gamma$ needs to be replaced by ${\gamma }_1=\frac{\gamma }{\gamma {\tau }_2+1}$. Of course, if the model parameters are fitted from real data, the fitted $\gamma$ already has this correction. There are two ways to introduce the first delay but only one to introduce the second. The first way is SIS Delay Model 1:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}S(t-{\tau }_1)I(t-{\tau }_1)+\frac{\gamma }{\gamma {\tau }_2+1}I(t-{\tau }_2)\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S(t-{\tau }_1)I(t-{\tau }_1)-\frac{\gamma }{\gamma {\tau }_2+1}I(t-{\tau }_2).\hfill \end{array}$$

(11)

The assumption is that a contact at $t-{\tau }_1$ produces a new infective at time t, and an individual infected at time $t-{\tau }_2$ recovers at time t. The second way is SIS Delay Model 2:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}S\left(t\right)I(t-{\tau }_1)+\frac{\gamma }{\gamma {\tau }_2+1}I(t-{\tau }_2)\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S\left(t\right)I(t-{\tau }_1)-\frac{\gamma }{\gamma {\tau }_2+1}I(t-{\tau }_2).\hfill \end{array}$$

(12)

The assumption now is that an infective takes a period of time ${\tau }_1$ to actually contract the disease and be able to infect. For the recovery, the assumption is the same as for SIS Model 1. Both models conserve the total population and both make biologically sense. For simplicity, we will call the recovery rate just $\gamma$.

Now adding births and deaths but no migration, assuming all new births are susceptible and keeping the population constant

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mu N-\frac{\beta }{N}SI+\gamma I-\mu S\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}SI-\gamma I-\mu I,\hfill \end{array}$$

(13)

where $\mu >0$ is the death rate and is equal to the birth rate. Equation (13) has two steady states $I\left(t\right)=0,S\left(t\right)=N$ and $I\left(t\right)=(1-\frac{\gamma +\mu }{\beta })N,S\left(t\right)=N-I\left(t\right)$. The second one makes sense only when $\frac{\gamma +\mu }{\beta }\le 1$.

Now there is a third delay, the maturation time ${\tau }_3$. So the term involving the births can be evaluated at time $t-{\tau }_3$, while the death terms are evaluated at t. However, since the total population is constant this third delay will not appear in the equations. Again, there are two ways to introduce the first delay but only one to introduce the second. The first way is SIS Delay Model 3:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mu N-\frac{\beta }{N}S(t-{\tau }_1)I(t-{\tau }_1)+\gamma I(t-{\tau }_2)-\mu S\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S(t-{\tau }_1)I(t-{\tau }_1)-\gamma I(t-{\tau }_2)-\mu I.\hfill \end{array}$$

(14)

The assumption for Equation (14) is that a contact at $t-{\tau }_1$ produces a new infective at time t, and a new susceptible can recover after been ill for a time.

The second way is SIS Delay Model 4:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mu N-\frac{\beta }{N}S\left(t\right)I(t-{\tau }_1)+\gamma I(t-{\tau }_2)-\mu S\hfill \\ \hfill \frac{dI}{dt}& =exp(-\mu {\tau }_1)\frac{\beta }{N}S\left(t\right)I(t-{\tau }_1)-\gamma I(t-{\tau }_2)-\mu I.\hfill \end{array}$$

(15)

The assumption for Equation (15) is that an infective takes a period of time ${\tau }_1$ to actually contract the disease and be able to infect. For the recovery, the assumption is the same as for SIS Model 1. The factor $exp(-\mu {\tau }_1)$ is added to the equation for I to account for the number of susceptible individuals that had contact with an infective, represented by $I(t-{\tau }_1)$, but die before they actually become infective. This exponential term cannot be added to the equation for S since the contact between susceptible and infective is at the current time t and until then the individual is still susceptible and its death rate is included in the term $\mu S$. For the model given by Equation (14), this exponential term is not added since the change from a susceptible to a new infective due to a contact at time $t-{\tau }_1$ happens at time t, and so in the equation for S they are counted as susceptibles until time t so they form part of the S in the term $\mu S$. Now, only model SIS Delay Model 3 conserves the total population. Model SIS Delay Model 4 with small $\mu {\tau }_1$ will almost conserve the total population and can then be considered to make biological sense. For example, since for humans the average life span is about 70 years and the duration of a typical disease a few days, the term $exp(-\mu {\tau }_1)$ is very close to 1.

The equilibrium solutions of Equation (9) are also equilibrium solutions of all the delayed models. Since $0\le I\left(t\right)\le N$ and that solutions of ODE’s are unique, the fact that the DFE equilibrium is asymptotically unstable means that the endemic equilibrium is asymptotically stable.

Solutions of DDE’s are also unique as can be seen by solving them using the method of steps [3,14]. Furthermore, since solutions are bounded, the asymptotic stability of the endemic equilibrium can be determined from the asymptotic stability of the DFE. For SIS Delay Model 1 Equation (11), linearizing about $I\left(t\right)=0$ and proposing solutions of the form $exp\left(\lambda t\right)$, the following characteristic equation is obtained

$$\lambda =\beta exp(-{\tau }_1\lambda )-\gamma exp(-{\tau }_2\lambda ).$$

The left-hand side is zero at $\lambda =0$ and an increasing function of $\lambda$. For the case when the endemic equilibrium exists, $\beta >\gamma$, the right-hand side is positive at $\lambda =0$ and a decreasing function if $\beta >\gamma exp(-\lambda ({\tau }_2-{\tau }_1))$. This is satisfied for $\beta >\gamma$ and ${\tau }_2>{\tau }_1$. Under these conditions, there is a solution $\lambda >0$ and the DFE is asymptotically unstable. Therefore, to have a positive eigenvalue $\lambda$ it is necessary that $\beta >\gamma exp(-\lambda ({\tau }_2-{\tau }_1))$ and then the endemic equilibrium is asymptotically stable. For SIS Delay Model 2 Equation (12), the linearization about the DFE equilibrium is the same as for SIS Delay Model 1 so the stability results are the same. Note that the basic reproductive number is $R_0=\beta /\gamma$. It can easily be calculated using the new generation matrix method [20,21].

2.3. SIRS Model

Next we study the SIRS model for epidemics with no demographics [8]. It consists of three compartments: susceptibles (S), infectives (I) and recovered (R), all functions of time t. A susceptible individual converts into an infective with a given probability after a contact with an infective. An infective recovers with a rate $\gamma$. A recovered loses immunity at a rate $\nu$ and turns into a susceptible. The total population is assumed constant, $S\left(t\right)+I\left(t\right)+R\left(t\right)=N$. It can also be applied to plant virus transmission, by assuming that the transmission occurs via mechanical means since there are no vectors. The infection coefficient will be denoted by $\beta /N$. We present the SIRS model to illustrate different ways of introducing a delay. The system of ordinary differential equations (ODE’s) describing the model is

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}SI+\nu R\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}SI-\gamma I\hfill \\ \hfill \frac{dR}{dt}& =\gamma I-\nu R.\hfill \end{array}$$

(16)

It can be simplified since the total population is constant: $R\left(t\right)=N-S\left(t\right)-I\left(t\right)$:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}SI+\nu (N-S-I)\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}SI-\gamma I.\hfill \end{array}$$

(17)

Equation (17) have two equilibrium solutions. The DFE, $S\left(t\right)=N,I\left(t\right)=0$ and the endemic equilibrium $S\left(t\right)=\frac{\gamma N}{\beta },I\left(t\right)=\frac{(\beta -\gamma )N\nu }{\beta (\gamma +\nu )}$. The endemic equilibrium exists only for $\beta >\gamma$. The eigenvalues of the linearized system at the DFE are ${\lambda }_1=\beta -\gamma ,{\lambda }_2=-\nu$. So the DFE is asymptotically unstable for $R_0=\beta /\gamma >1$. This is the $R_0$ given by the method of the next generation matrix.

There are three common methods of introducing delays in the infection: SIRS Model 1 [4,22,23] given by:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}S(t-\tau )I(t-\tau )+\nu (N-S\left(t\right)-I\left(t\right))\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S(t-\tau )I(t-\tau )-\gamma I\left(t\right),\hfill \end{array}$$

(18)

where the hypothesis is that an infective has contact with a susceptible and it takes a period of time $\tau$ for this susceptible to turn infective. A second model is the SIRS Model 2 [24] described by:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}S\left(t\right)I(t-\tau )+\nu (N-S\left(t\right)-I\left(t\right))\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S\left(t\right)I(t-\tau )-\gamma I\left(t\right),\hfill \end{array}$$

(19)

where the assumption is that a susceptible after a contact with an infective, a period of time $\tau$ has to pass before it is able to infect another susceptible. Thus, there is a new infective after contact between the susceptible and an individual infected a period of time ago equal to the delay.

A third model is the SIRS Model 3, described by:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}S\left(t\right)I\left(t\right)+\nu (N-S\left(t\right)-I\left(t\right))\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S(t-\tau )I(t-\tau )-\gamma I\left(t\right),\hfill \end{array}$$

(20)

where the susceptible after the contact with an infective leaves the susceptible population but takes a period of time $\tau$ to be included in the infective population. It does conserve the total population and the infection contact is between individuals at the same time.

There are other ways of introducing the delay in the transmission term, but they do not make sense biologically. For example, consider

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}S(t-\tau )I\left(t\right)+\nu (N-S\left(t\right)-I\left(t\right))\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S(t-\tau )I\left(t\right)-\gamma I\left(t\right).\hfill \end{array}$$

Here a susceptible in the past time $t-\tau$ becomes infected through contact with an infective $I\left(t\right)$ and the contact occurs at time t when that susceptible has already turned infective, so it will be a contact between two infectives.

As has been already mentioned, epidemic models can also be interpreted as biochemical reactions and this formulation helps understand the processes involved. For example, an infective reacts with a susceptible and the result are two infectives. This approach may help determine the right way to introduce the delay. The SIRS model with no delays can be written in terms of biochemical reactions as

$$\begin{array}{cc}\hfill S& \stackrel{SI\beta /N}{\to }\hfill \\ \hfill & \stackrel{SI\beta /N}{\to }I\hfill \\ \hfill & \stackrel{\nu N}{\to }S\hfill \\ \hfill S& \stackrel{\nu I}{\to }\hfill \\ \hfill S& \stackrel{\nu S}{\to }\hfill \\ \hfill I& \stackrel{\gamma I}{\to }.\hfill \end{array}$$

It is important to note that the system of reactions is not unique as has been established, for example, by [25]. In this case, a simpler system is

$$\begin{array}{cc}\hfill I+S& \stackrel{SI\beta /N}{\to }2I\hfill \\ \hfill & \stackrel{\nu N}{\to }S\hfill \\ \hfill S& \stackrel{\nu I}{\to }\hfill \\ \hfill S& \stackrel{\nu S}{\to }\hfill \\ \hfill I& \stackrel{\gamma I}{\to }.\hfill \end{array}$$

SIRS Model 1 with the delay can be represented by biochemical reactions by delaying the following reaction

$$I+S\stackrel{SI\beta /N}{\to }2I,$$

and SIRS Model 2 by delaying

$$\stackrel{SI\beta /N}{\to }I.$$

The concern with the reaction for Model 2 is that the propensity $S\left(t\right)I(t-\tau )\beta /N$ has terms evaluated at different times. It can be fixed by adding a reaction that takes an infective and produces an infective and delaying it, but for a model with no delays one would never add a reaction saying that an infective converts to an infective.

As shown by [26], temporary immunity can be included in the SIRS model. There they used a distributed delay but introducing the delay after exposure ${\tau }_1$, the delay due to the minimum duration of the disease ${\tau }_2$ and the delay due to the minimum time with immunity ${\tau }_3$. We then get the model

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =-\frac{\beta }{N}S(t-{\tau }_1)I(t-{\tau }_1)+\nu R(t-{\tau }_3)\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta }{N}S(t-{\tau }_1)I(t-{\tau }_1)-\gamma I\left({\tau }_2\right)\hfill \\ \hfill \frac{dR}{dt}& =\gamma I(t-{\tau }_2)-\nu R(t-{\tau }_3).\hfill \end{array}$$

Here are the equations for SIRS Model 1 but for SIRS Models 2 or 3 they are similar.

All three delayed models presented, Equations (18)–(20), have the same equilibrium points as the non-delayed model, Equations (17). To study the stability of the DFE of Equations (18), they are linearized about the DFE, $I\left(t\right)=0,S\left(t\right)=N$. The characteristic equation is

$${\lambda }^2+\lambda (\gamma -\beta exp(-\lambda \tau )+\nu )=\nu (\beta exp(-\lambda \tau )-\gamma ).$$

(21)

To determine if Equation (21) has a solution $\lambda >0$ we follow the method used by [27,28] and others. Let the left-hand side be $F\left(\lambda \right)$ and the right-hand side be $G\left(\lambda \right)$. F(0) = 0 and $F\to \infty$ as $\lambda \to \infty$. $G\left(0\right)>0$ for $R_0>1$ and is a decreasing function of $\lambda$. Therefore there is a positive value of $\lambda$ where both functions intersect. So if $R_0>1$ the DFE is unstable and there exists an epidemic. The same result is also true for Equations (19) and (20) since the characteristic equations are the same.

2.4. Plant Virus with Vector Transmission Model

2.4.1. Model A

The processes of a virus disease transmitted to humans, animals or plants by a vector are the same. For simplicity, we will consider models for plants. We consider two different models of the transmission of a virus due to vectors in plants. Both are simple and have different assumptions. Our objective is to observe the effects and validity of introducing the delay in different ways in the two models. In the first one, Model A there are two populations of plants, susceptible, $S\left(t\right)$, healthy and subject to be infected, and infective, $I\left(t\right)$, already infected. We consider that plants do not recover so there is no recovered class. Some species may recover from certain viruses but that is not the rule. There are also two populations of vectors, susceptible, $X\left(t\right)$ and infective, $Y\left(t\right)$. Vectors do not contract the disease so they do not recover. We will not consider the case of the vectors shedding the virus. This model is a simplified version of the models presented, for example, in [23,29].

The assumptions of Model A are: all new plants and vectors are susceptible, the total population of plants is a constant K since a farmer can plant new healthy plants to replace any dead ones, the interaction between vector and plant is of mass action type, the viruses kill plants but not vectors, and neither plants nor vectors recover from the disease.

The system of ODE’s for Model A is

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mu (K-S)+dI-\beta YS\hfill \\ \hfill \frac{dI}{dt}& =\beta YS-(d+\mu )I\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}IX-mX\hfill \\ \hfill \frac{dY}{dt}& ={\beta }_{1}IX-mY.\hfill \end{array}$$

(22)

The parameters of the model are: $\beta$ the infection rate of plants due to infective vectors feeding on the plant, ${\beta }_1$ the infection rate of vectors due to them feeding on an infected plant, $\mu$ the natural death rate of plants, d the additional death rate of plants due to the disease, m the natural death rate of vectors, and $\Lambda$ the replenishing rate of vectors (due to birth and/or migration).

The equilibrium solutions of Equation (22) are: DFE, $S\left(t\right)=K,I\left(t\right)=0,X\left(t\right)=\Lambda /m,Y\left(t\right)=0$, and the endemic equilibrium,

$$\begin{array}{cc}\hfill S\left(t\right)& =\frac{m({\beta }_{1}K+m)(d+\mu )}{{\beta }_1(\beta \Lambda +m(d+\mu ))}\hfill \\ \hfill I\left(t\right)& =\frac{\beta {\beta }_{1}K\Lambda -m^2(d+\mu )}{{\beta }_1(\beta \Lambda +m(d+mu))}\hfill \\ \hfill X\left(t\right)& =\frac{\beta \Lambda +dm+m\mu }{\beta {\beta }_{1}K+\beta m}\hfill \\ \hfill Y\left(t\right)& =\frac{\beta {\beta }_{1}K\Lambda -m^2(d+\mu )}{\beta m({\beta }_{1}K+m)}.\hfill \end{array}$$

(23)

The basic reproduction number $R_0$ can be calculated using the new generation matrix method, $R_0=\frac{\sqrt{\beta {\beta }_{1}K\Lambda /m}}{\sqrt{m(d+\mu )}}$. So for $R_0>1$. the DFE is asymptotically unstable and the endemic equilibrium asymptotically stable. Note that the endemic equilibrium exists only for $R_0>1$.

In virus transmission via a vector there are two delays. The first one is the time it takes the virus to spread in the plant after contact with an infected vector. The second is the time it takes the virus to spread in the vector after feeding from an infected plant. This second delay is much smaller than the first since the virus is not replicating in the vector and many times stays around the mandibles of the vector. For simplicity, we will consider only the first delay.

As was done for the SIS and SIRS models, we will introduce the delay in the transmission and will do so in two different ways. The first one is using the assumption that a susceptible after contact with an infective takes the delay time to become infective itself [22,23]. That is, a newly exposed susceptible stays as a susceptible until a time equal to the delay elapses and only then it turns into an infective. This will be Model A1:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mu (K-S(t\left)\right)+dI\left(t\right)-\beta Y(t-\tau )S(t-\tau )\hfill \\ \hfill \frac{dI}{dt}& =\beta Y(t-\tau )S(t-\tau )-(d+\mu )I\left(t\right)\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}I\left(t\right)X\left(t\right)-mX\left(t\right)\hfill \\ \hfill \frac{dY}{dt}& ={\beta }_{1}I\left(t\right)X\left(t\right)-mY\left(t\right).\hfill \end{array}$$

(24)

The second model with transmission delay uses the assumption that after a contagion the susceptible immediately stops being susceptible but it takes the delay time to become infective. It also takes into account that the newly infected plant may die before becoming infective by introducing the term $exp(-\mu \tau )$, which is the average survival percentage in time $\tau$. This is Model A2:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mu (K-S(t\left)\right)+dI\left(t\right)-\beta Y\left(t\right)S\left(t\right)\hfill \\ \hfill \frac{dI}{dt}& =exp(-\mu \tau )\beta Y(t-\tau )S(t-\tau )-(d+\mu )I\left(t\right)\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}I\left(t\right)X\left(t\right)-mX\left(t\right)\hfill \\ \hfill \frac{dY}{dt}& ={\beta }_{1}I\left(t\right)X\left(t\right)-mY\left(t\right).\hfill \end{array}$$

(25)

Again, presenting the biochemical reactions corresponding to each model helps decide whether it is a realistic model. For Model A, the reactions are

$$\begin{array}{cc}\hfill S& \stackrel{SY\beta }{\to }I\hfill \\ \hfill I& \stackrel{(\mu +d)I}{\to }S\hfill \\ \hfill X& \stackrel{{\beta }_{1}IX}{\to }Y\hfill \\ \hfill & \stackrel{\Lambda }{\to }X\hfill \\ \hfill X& \stackrel{mX}{\to }\hfill \\ \hfill Y& \stackrel{mY}{\to }.\hfill \end{array}$$

The delay model, Equation (24), has the same equilibrium solutions as the non-delayed system (22). To study the stability, it is convenient to assume initial conditions $S\left(0\right)+I\left(0\right)=K,X\left(0\right)+Y\left(0\right)=\Lambda /m$. Then using $S\left(t\right)=K-I\left(t\right),X\left(t\right)=\Lambda /m-Y\left(t\right)$, Equation (24) reduce to

$$\begin{array}{cc}\hfill \frac{dI}{dt}& =\beta Y(t-\tau )(K-I(t-\tau \left)\right)-(d+\mu )I\left(t\right)\hfill \\ \hfill \frac{dY}{dt}& ={\beta }_{1}I\left(t\right)(\Lambda /m-Y\left(t\right))-mY\left(t\right).\hfill \end{array}$$

(26)

Linearizing about the DFE and letting

$$\left(\begin{array}{c}I\\ Y\end{array}\right)=\mathbf{c}exp\left(\lambda t\right),$$

with $\mathbf{c}$ a constant vector, we obtain the resulting characteristic equation,

$${\lambda }^2+\lambda (m+d+\mu )=exp(-\lambda \tau )\beta {\beta }_{1}K\Lambda /m-m(d+\mu )=(exp(-\lambda \tau )R_0^2-1)m(d+\mu ).$$

Letting the left-hand side be $F\left(\lambda \right)$, then $F\left(0\right)=0$ and F is an increasing function of $\lambda$. Let the right-hand side be $G\left(\lambda \right)$. If $R_0>1$ then $G\left(0\right)>0$ and a decreasing function of $\lambda$. Therefore, the characteristic equation has a solution $\lambda >0$ and the DFE is unstable.

The second delayed model, Equation (25), has a different endemic equilibrium due to the $exp(-\mu \tau )$ term:

$$\begin{array}{cc}\hfill S\left(t\right)& =\frac{m(d+\mu )(d(-1+exp(-\mu \tau ))m-({\beta }_{1}exp(-\mu \tau )K+m)\mu )}{(bet{a}_{1}exp(-\mu \tau )(\beta \Lambda (d(-1+exp(-\mu \tau ))-\mu )-m\mu (d+\mu ))}\hfill \\ \hfill I\left(t\right)& =\frac{\mu (-\beta {\beta }_{1}exp(-\mu \tau )K\Lambda +m^2(d+\mu ))}{{\beta }_1(\beta \Lambda (d(-1+exp(-\mu \tau ))-mu)-m\mu (d+\mu ))}\hfill \\ \hfill X\left(t\right)& =\frac{\beta \Lambda \left(d\right(-1+exp(-\mu \tau ))-\mu )-m\mu (d+\mu )}{\beta d(-1+exp(-\mu \tau ))m-beta({\beta }_{1}exp(-\mu \tau )pK+m)\mu }\hfill \\ \hfill Y\left(t\right)& =\frac{\mu (-\beta {\beta }_{1}exp(-\mu \tau )K\Lambda +m^2(d+\mu ))}{\beta m(d(-1+exp(-\mu \tau ))m-({\beta }_{1}exp(-\mu \tau )K+m)\mu )}.\hfill \end{array}$$

(27)

The $R_0$ calculated using the next generation matrix is also different

$$R_0=\frac{\sqrt{\beta {\beta }_{1}K\Lambda /mexp(-\mu \tau )}}{\sqrt{m(d+\mu )}}.$$

The characteristic equation is

$$\begin{array}{cc}\hfill {\lambda }^2+\lambda (m+d+\mu )& =exp(-\mu \tau )exp(-\lambda \tau )\beta {\beta }_{1}K\Lambda /m-m(d+\mu )\hfill \\ \hfill & =(exp(-\lambda \tau )R_0^2-1)m(d+\mu ).\hfill \end{array}$$

so the DFE is unstable for $R_0>1$. Delay differential equations may be more realistic but they have some drawbacks. For one, it is necessary to give initial conditions on an interval instead of just at a point as for ordinary differential equation. These initial conditions may be unknown, and, if that is the case, they are usually taken to be constant. A second one is that the analysis is much harder due to the infinite number of eigenvalues [1,14]. An alternative for accounting for the delay in the transmission term is to introduce instead another compartment, the exposed population (E). After contact with an infective, a susceptible is converted into an exposed or latent, that cannot yet infect. The exposed turns into an infective at a rate $ϵ=1/\tau$:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =\mu (K-S(t\left)\right)+dI\left(t\right)-\beta Y\left(t\right)S\left(t\right)+\mu E\hfill \\ \hfill \frac{dE}{dt}& =\beta Y\left(t\right)S\left(t\right)-\mu E-ϵE\hfill \\ \hfill \frac{dI}{dt}& =ϵE-(d+\mu )I\left(t\right)\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}I\left(t\right)X\left(t\right)-mX\left(t\right)\hfill \\ \hfill \frac{dY}{dt}& ={\beta }_{1}I\left(t\right)X\left(t\right)-mY\left(t\right).\hfill \end{array}$$

(28)

Epidemic models with an exposed class are very common. For plant virus propagation, see, for example, [30,31]. The advantages of using models with exposed populations is that they do not require the initial conditions to be given in an interval of length equal to the delay. Furthermore, ordinary differential equations are simpler to analyze and the methods are better known. However, there is the complication of an additional equation for the exposed population, and, from a biological point of view, the fact that only in the average it takes the delay time for an exposed to become infective. Just looking at the transition from exposed to infective in the exposed equation $\frac{dE}{dt}=-ϵE$, it is possible for the transition to even take an infinite time. This term is the same as for Malthus model with only deaths $\frac{dN}{dt}=-\mu N$ and it is considered an acceptable simplification since it is realistic in the average.

Equation (28) have the same DFE as Equation (22) but a different endemic equilibrium. The $R_0$ is also different

$$R_0=\frac{\sqrt{\beta {\beta }_1ϵK\Lambda /m}}{\sqrt{m(d+\mu })ϵ(ϵ+\mu )}.$$

2.4.2. Model B

As a second plant virus transmission model, we construct a model based on that presented in [28,32], but modified to include susceptible vectors. It considers four populations, susceptible plants, S, infective plants, I, susceptible vectors, X, and infective vectors, Y. The plants grow logistically, so the total plant population is not constant. This is the main difference with Model A. There is also a saturation effect on the insect plant interaction. All new vectors are susceptible and their growth rate due to births and migration is constant. As in Model A, plants do not recover and insects do not contract the disease.

Without delays, the system of equations for the plant virus propagation Model B is

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =rS(1-\frac{S+I}{k})-\frac{\beta SY}{1+\alpha S+cY}\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta SY}{1+\alpha S+cY}-(\mu +d)I\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}XI-mX\hfill \\ \hfill \frac{dY}{dt}& ={\beta }_{1}XI-mY.\hfill \end{array}$$

(29)

Here, r is the growth rate of the plants, k their carrying capacity and $\beta$ the infection rate from an infective vector to a susceptible plant, $\mu$ is the natural death rate of the plants and d the additional death rate of the plants due to the virus. $\Lambda$ is the recruitment rate of susceptible vectors, ${\beta }_1$ the infection rate from infective plants to susceptible vectors, and m the natural death rate of the vectors. $\alpha$ and c are the parameters in the Beddington–DeAngelis saturation function [33]. There is no additional death rate for vectors. Finally, there is no recovery for either plants or vectors.

Equation (29) have three equilibrium solutions, $S\left(t\right)=0,I\left(t\right)=0,X\left(t\right)=\Lambda /m$, $Y\left(t\right)=0$ which are not of interest, $S\left(t\right)=K,I\left(t\right)=0,X\left(t\right)=\Lambda /m,Y\left(t\right)=0$ the DFE, and an endemic equilibrium. The basic reproduction number is $R_0=\frac{\sqrt{\beta {\beta }_{1}K\Lambda /m}}{\sqrt{m(d+\mu )}}$.

Even though the delay in the transmission from vector to plant is much larger than the delay in the transmission from plant to vector, we will consider both delays. Furthermore, since the delays in the transmission are the only ones that can be introduced in several forms, they are the only ones studied. The first way of introducing the delays in the plant becoming infected is using the assumption that a susceptible plant after a contact with an infective vector takes the a delay time to become infective itself and that after feeding on an infected plant the vector takes a second delay time to become infected, that is, to be able to spread the virus [22,23]. This will be Model B1:

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =rS\left(t\right)(1-\frac{S\left(t\right)+I\left(t\right)}{k})-\frac{\beta S(t-{\tau }_1)Y(t-{\tau }_1)}{1+\alpha S(t-{\tau }_1)+cY(t-{\tau }_1)}\hfill \\ \hfill \frac{dI}{dt}& =\frac{\beta S(t-{\tau }_1)Y(t-{\tau }_1)}{1+\alpha S(t-{\tau }_1)+cY(t-{\tau }_1)}-(-\mu +d)I\left(t\right)\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}I(t-{\tau }_2)X(t-{\tau }_2)-mX\left(t\right)\hfill \\ \hfill \frac{dY}{dt}& ={\beta }_{1}I(t-{\tau }_2)X(t-{\tau }_2)--mY\left(t\right),\hfill \end{array}$$

(30)

where ${\tau }_1$ is the delay in the plant becoming infected and ${\tau }_2$ the delay in the vector becoming infected.

In the second approach, we follow [28,32] in that after a contact between an susceptible plant and an infective insect, the plant immediately stops being susceptible, but it takes the delay time for it to become infective. Furthermore, the plant may die before becoming infective, so there is survival rate proportional to $exp(-\mu {\tau }_1)$, where $\mu$ is the death rate of the plant and ${\tau }_1$ the delay. The delay in the vector becoming infected, ${\tau }_2$ is similarly introduced. This is Model B2.

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =rS\left(t\right)(1-\frac{S\left(t\right)+I\left(t\right)}{k})-\frac{\beta SY}{1+\alpha S\left(t\right)+cY\left(t\right)}\hfill \\ \hfill \frac{dI}{dt}& =exp(-\mu {\tau }_1)\frac{\beta S(t-{\tau }_1)Y(t-{\tau }_1)}{1+\alpha S(t-{\tau }_1)+cY(t-{\tau }_1)}-(-\mu +d)I\left(t\right)\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}I\left(t\right)X\left(t\right)-mX\left(t\right)\hfill \\ \hfill \frac{dY}{dt}& =exp(-m{\tau }_2){\beta }_{1}I(t-{\tau }_2)X(t-{\tau }_2)--mY\left(t\right).\hfill \end{array}$$

(31)

Note that in Model B1, an infected vector infects a plant at time $t-{\tau }_1$ and the susceptible plant turns infected at time t. Furthermore, the vector that feeds on an infected plant at time $t-{\tau }_2$ turns infected at time t. In Model B2, an infected vector infects a susceptible plant that takes a time ${\tau }_1$ to become infective and the term $exp(-{\tau }_1\mu$) is the average percentage of infected susceptibles that do not die in time ${\tau }_1$. Similarly for the infection from plant to vector. Only Model B1 has the same steady state as the non-delayed Model B; however, both models are correct from the biochemical reactions point of view since the propensities have terms evaluated at the same time. That is, only populations at time t interact with populations at time t, and populations at time $t-{\tau }_1$ with populations at time $t-{\tau }_1$ and the same for populations at time $t-{\tau }_2$.

Model Plant B1, Equation (30), has the same equilibrium solutions as the non-delayed model Equation (29). The basic reproduction is $R_0=\frac{\sqrt{\beta {\beta }_{1}K\Lambda /m}}{\sqrt{m(d+\mu )}}$, the same as for the non-delayed model. Linearizing the system around the DFE equilibrium, the equations for S and X decouple and the reduced system has the characteristic equation

$$\begin{array}{cc}\hfill {\lambda }^2+\lambda (m+d+\mu )& =exp(-\lambda {\tau }_1)exp(-\lambda {\tau }_2)\beta {\beta }_{1}K\Lambda /m-m(d+\mu )\hfill \\ & =(exp(-\lambda {\tau }_1)exp(-\lambda {\tau }_2)R_0^2-1)m(d+\mu ).\hfill \end{array}$$

The left-hand side $F\left(\lambda \right)$ is 0 at $\lambda =0$ and increases with $\lambda$. The right-hand side $G\left(\lambda \right)$ is positive at $\lambda =0$ and decreases with $\lambda$. Therefore, there is a solution $\lambda >0$ and the DFE equilibrium is unstable for $R_0$ > 1.

Model Plant B2 has the same DFE equilibrium as the non-delayed model but a different endemic one due to the presence of $exp(-\mu {\tau }_1)$. The basic reproduction number also has an extra $exp\left(\mu {\tau }_1\right)$, $R_0=\frac{\sqrt{exp(-\mu {\tau }_1)\beta {\beta }_{1}K\Lambda /m}}{\sqrt{m(d+\mu )}}$. The characteristic equation now is

$$\begin{array}{cc}\hfill {\lambda }^2+\lambda (m+d+\mu )& =exp(-\mu {\tau }_1)exp(-\lambda {\tau }_1)exp(-\lambda {\tau }_2)\beta {\beta }_{1}K\Lambda /m-m(d+\mu )\hfill \\ \hfill & =(exp(-\lambda {\tau }_1)exp(-\lambda {\tau }_2)R_0^2-1)m(d+\mu ),\hfill \end{array}$$

with the corresponding $R_0$.

As before, an alternative is to add an exposed or latent compartment E for the plants and Z for the vectors. The Model B with the exposed compartment is

$$\begin{array}{cc}\hfill \frac{dS}{dt}& =rS(1-\frac{S+I+E}{k})-\frac{\beta SY}{1+\alpha S+cY}\hfill \\ \hfill \frac{dE}{dt}& =\frac{\beta SY}{1+\alpha S+cY}-\mu E-ϵE\hfill \\ \hfill \frac{dI}{dt}& =ϵE-(\mu +d)I\hfill \\ \hfill \frac{dX}{dt}& =\Lambda -{\beta }_{1}IX-m\hfill \\ \hfill \frac{dZ}{dt}& ={\beta }_{1}IX--mZ-{ϵ}_{1}Z\hfill \\ \hfill \frac{dY}{dt}& ={ϵ}_{1}Z-mY,\hfill \end{array}$$

(32)

with $ϵ=1/{\tau }_1$ the rate at which the exposed plants turn into infected plants, and ${ϵ}_1=1/{\tau }_2$ the rate at which the exposed vectors turn into infectives.

The DFE is the same as for the non-delayed model. The endemic one is different due to the presence of the exposed class. The $R_0$ is also different

$$R_0=\frac{\sqrt{\beta {\beta }_1ϵ{ϵ}_{1}k\Lambda /m}}{m({ϵ}_1+m)(d+\mu )(ϵ+\mu )(1+\alpha k)}.$$

2.5. Numerical Methods

One of the criteria used to determine the applicability of a way of introducing the delays is to assess whether the delayed model has the same steady solutions as the non-delayed model. Mathematica [34] or maxima [35] can be used to solve the algebraic equations exactly. However, the steady solutions, for given values of the parameters, can also be found by carrying out simulations and solving the systems of ordinary or delayed differential equations. Numerical methods for ordinary differential equations are well known and we used the Runke–Kutta implementation in the routine ode45 from Matlab. To determine the steady solutions of any of the models, the right-hand sides of the systems of equations are equaled to 0 [36]. Numerical methods for solving ODE’s can be modified for DDE’s, for example, by using the step-by-step method [15]. We used the Matlab routine dde23 [37].

3. Results

In this section, we present the results of simulations for the models presented in the previous section. The simulations are conducted to validate the theoretical results presented in Section 2 and to illustrate the differences in the ways of introducing the delay. For the logistic population models, the parameters are taken from [1] and are intended to replicate the experiments by [17]. For the epidemic models, the parameters do no relate to any particular infectious disease. They were chosen so that an endemic equilibrium would exist. For the Plant virus Model A, they were taken from [23,29]. For Plant virus Model B, they were taken mostly from [28,38].

3.1. One-Population Models

For the logistic population model, Equation (2), different ways of introducing the delay are presented. The delays have to be in the birth terms and not in the death or competition terms to make sense biologically. Furthermore, the populations in the competition term should be evaluated at the same time. The Delay Model 1 Equation (4) is the one given by [17] to explain the oscillations in the experiments by [19] but it does not make sense biologically. The only model that makes biological sense is Equation (5). Figure 1 shows the results for a simulation using the non-delayed model and Delay Models 1 and 2. The parameters used are $K=1,r=0.15,m=0.01,b=0.16$, and $\tau =14$. They were chosen similar to those used by [39]. Only the Delay Model 1 has oscillations but the period is not that of the experiments and it is not realistic.

3.2. SIS Model

The SIS population model is used to illustrate how to introduce the delay in the transmission term. Figure 2 has the solution for the SIS model with no delay Equation (8), and SIS Models 1 and 2 Equations (11) and (12). In this case there are no demographics, so both delay models conserve the total population, have the same steady states as the non-delayed model, and make biological sense. The parameters used in the simulations are $\beta =1,N=100,\gamma =0.4$ and $\tau =2$.

Introducing demographics in the SIS models by adding births and deaths but no migration, and with constant total population N, we see from Figure 3 that SIS Model 1 Equation (14) conserves the total population but it does have the same equilibrium solution as the non-delayed model Equation (13), but that SIS Model 2 Equation (15) does not conserve the total population and has different steady states than the non-delayed model Equation (13). The parameters used in the simulations are $\beta =1,N=100,\gamma =0.4,\mu =0.1$ and ${\tau }_1=2,{\tau }_2=1$.

3.3. SIRS Model

In Section 2.3, a common SIRS model with constant total population was introduced given by Equation (17). First, three different ways of introducing a delay ${\tau }_1$ in the infection term were presented. Next, two additional delays were added, ${\tau }_2$ which is the minimum time an infective individual has the disease and ${\tau }_3$ which is the minimum time a recovered individual keeps its immunity. There is only one way to introduce these last two delays. Of course, their presence changes the solutions. Figure 4 shows the simulations with the following values of the parameters: $\beta =1,N=100,\gamma =0.4,\nu =0.4$ and ${\tau }_1=2,{\tau }_2=1$ and ${\tau }_3=1$. All three delayed models have the same steady solutions which are not the same as for the model with no delays. Comparing with the models having only the delay ${\tau }_1$ in the infection, Figure 5, now the three delayed models have the same steady states as the non-delayed model. Thus, the two delays ${\tau }_2$ and ${\tau }_3$ change the steady solutions.

3.4. Plant Virus with Vector Transmission Model

Now we present the results for two distinct epidemic models with vector transmission. In models with vector transmission, the delays are introduced in a different manner. The first model, Equation (22), has demographics for both the plants and the vectors, constant population for the plants and mass action interaction for the transmission of the disease. The plants become sick and do not recover. The vectors do not contract the disease and do not recover. There is an additional death rate for the plants due to the disease. Two models with delays are used, Equations (24) and (25), and a model with an exposed population instead of delay Equation (28). The parameters used are $\tau =2,K=100,\beta =0.01,\mu =0.1,d=0.1,{\beta }_1=0.01,m=0.2,ϵ=1/\tau ,\Lambda =10$. Figure 6 shows the simulations for the Model A with no delay, with delay Models A1 and A2, and with exposed population. Model A2 does not conserve the total population while all three other models do. Model A2 has the additional concern that looking at the equivalent reactions, there is a reaction with propensity having terms evaluated at two different times.

The second model for virus propagation in plants is given by Equation (29). The model now has a non-constant total plant population with the plants growing in a logistic form. Neither plants nor vectors recover, and the virus does not affect the vectors. There are two delays in the transmission of the virus to the plants and to the vectors. The time to infection of the vectors is much smaller than the time to infection of the plants. Two ways of introducing the infection delays ${\tau }_1$ and ${\tau }_2$ are given by Models B1 and B2, Equations (30) and (31), respectively. The infection terms were considered to be mass action with no saturation ($\alpha =c=0$). The other parameters used in the simulations given in Figure 7 are $r=5,k=100,\beta =0.01,{\beta }_1=0.01,\mu =0.1,d=0.025,\Lambda =10;{\beta }_1=0.01,m=0.1$, ${\tau }_1=2$, ${\tau }_2=1,ϵ=1/{\tau }_1,{ϵ}_1=1/{\tau }_2$.

In Model B, the total plant population is not constant so in principle Model B2 is satisfactory. However, if Model B was modified to consider constant plant population, then Model 2 could not satisfy this constraint.

4. Discussion

Many processes including disease spread do not occur instantaneously. Thus, more realistic models include these delays. There are different ways of introducing delays into epidemic models described by systems of ordinary differential equations. We presented different possibilities using common disease spread models and interpreted whether the delay was introduced in a realistic way. First, from using two simple population growth models, Malthus and logistic growth, the delay can be introduced in the growth terms related to birth but not in those related to death. The experimental result [19] may have oscillations and delay differential equations may have oscillatory solutions, but the delay as introduced by [39] (Equation (4)) is non-biological. Oscillations also occur in predator–prey models [8] or by introducing oscillatory terms or coefficients in the model. However, the first consideration is that the model should be realistic.

The SIS Model, Equation (8), was used to show the introduction of a delay in the infection term under the assumption that after a contact with an infective, the susceptible takes a period of time equal to the delay to actually become infective. First, in the model with no demographics Equation (8), the delay in the infection ${\tau }_1$ is introduced in two ways. In the first one, the contact between the susceptible and an infective happens at time $t-{\tau }_1$ and the susceptible stays in the susceptible class until time t, Equation (11). In the second, an infective infected at time $t-{\tau }_1$ infects a susceptible at time t Equation (12). Since there are no demographics, both models conserve the total population. However, interpreting the SIS model in terms of biochemical reactions, for the second delay model, the contact involves populations at two distinct times. The second delay ${\tau }_2$, which is the minimum time an infective stays infected, can only be introduced in one consistent way. Next, to study what happens with delays in a model with demographics, births and death were added to the SIS model but the total population was kept constant Equation (13). The infection delay was introduced in two ways, Equations (14) and (15) in similar ways as was done in Equations (11) and (12), respectively. Now Equation (15) have the additional problem that the total population is not conserved since the term $exp(-\mu {\tau }_1)$ is introduced in the equation for the infectives to take into account the deaths between the time a susceptible is exposed and actually becomes infected.

The SIRS model based on the SIR (susceptible, infective, recovered) first proposed by Kermack and McKendrick [40] is applied to many different diseases in plants, humans and animals. We first consider the model with no demographics Equation (17). The delay in the infection is introduced in three different ways. In the first, the hypothesis is that an infective has contact with a susceptible and it takes a time $\tau$ for this susceptible to turn infective Equation (18). The total population is conserved and the infection contact happens between individuals at the same time. In the second way, the assumption is that after a susceptible makes contact with an infective, a period of time $\tau$ has to pass before it is able to infect another susceptible Equation (19). The total population is conserved but the infectious contact is between populations at different times. The third way is where the susceptible after the contact with an infective leaves the susceptible population but takes a time $\tau$ to be included in the infective population. It does conserve the total population, and the infection contact is between individuals at the same time. The main difference in the methods is the time it takes to achieve the steady state. Another delay which gives the minimum time a recovered individual has temporary immunity can be introduced to the three delayed models in the same way for all three.

For vector-transmitted diseases, there are two possible delays in the transmission, the delay in a susceptible plant becoming infected after a bite by an infected vector, and the delay in a susceptible vector becoming infected after biting and infected plant. We consider two models. In Model A, Equations (22), the total plant population is constant, and neither plants nor vectors recover. We consider that the delay in the vector becoming infected is very small and can be neglected. Two delayed models are presented. In Model A1 Equation (24), the susceptible plant is bitten by an infected vector at time $t-\tau$ but turns infective at time t. In Model A2, after being bitten by an infected vector the plant immediately leaves the susceptible compartment but takes a time $\tau$ to become infective. In this time, a fraction of the new infected plants dies. Both models have the infection between populations at the same time, but only the first conserves the total population. An explicit exposed compartment can also be added to take into account the delay Equation (28). The total population of plants is conserved, since there are no delays all interactions occur at the same time, and the system of ODE’s is easier to work with than the DDE’s for Models A1 and A2. Regarding the second vector-transmitted disease model, Model B Equation (29), the plant population now grows in logistic form, so the total population is not conserved. Thus, both delay models, Models B1 and B2, work well in this aspect. The delay in the infection of the vector ${\tau }_2$ is not neglected. Model B is written with saturation in the plant infection term but not used in the simulations since the effects of the delays were of primary interest. A third model with an explicit exposed population was also used with the advantages mentioned above.

For epidemic models based on ordinary differential equations, the new generation matrix method [20,21] can be used to calculate the basic reproductive number $R_0$, and to show that for $R_0>1$ there is an epidemic. For models based on delay differential equations, there is no corresponding general theorem. The $R_0$ can be calculated from the new generation matrix, but it has to be shown in each case whether for $R_0>1$ there is an epidemic. All the epidemic models studied herein state that there is an epidemic for $R_0>1$. The $R_0$ values are different depending on how the delay is introduced and also different for the models with exposed populations. However, their values are not very different and in real epidemics the $R_0$ is very difficult to determine. Thus, all the models considered are all right from their results about the stability of the DFE equilibrium and their ability to predict an epidemic.

The main question is whether delays should be introduced in the infection terms in such a way that they also work for models, whether the only change is that the total population is taken as constant. Furthermore, the fact that different ways of introducing the delay have been used in the literature implies that there is no consensus. For delays that appear in linear terms, this is not an issue. The use of terms such as $\frac{\beta }{N}S(t-\tau )I(t-\tau )$ also works for models with constant population and this presents an advantage. Introducing exposed populations removes the objections with the infection delay models such as when to stop counting an individual as susceptible, or reactions between terms at different times. There is the objection that individuals may stay infected for a very long time, but this may happen for a very small percentage, and the same can be argued about the use of death terms in a Malthusian way. However, exposed populations cannot be used in other population models, such as predator–prey, instead of delays.

In summary, for epidemic models, adding an exposed population presents a very good alternative to using delay differential equations when the delay appears in the infection terms. For the delay models considered, the stability and conditions for an epidemic to exist are similar, but not all of them make biological sense, such as not conserving the total population, or evaluating death and competition terms at the previous time. Therefore, not all the models are realistic.