## Abstract

Vector-borne disease transmission is a common dissemination mode used by many pathogens to spread in a host population. Similar to directly transmitted diseases, the within-host interaction of a vector-borne pathogen and a host’s immune system influences the pathogen’s transmission potential between hosts via vectors. Yet there are few theoretical studies on virulence–transmission trade-offs and evolution in vector-borne pathogen–host systems. Here, we consider an immuno-epidemiological model that links the within-host dynamics to between-host circulation of a vector-borne disease. On the immunological scale, the model mimics antibody-pathogen dynamics for arbovirus diseases, such as Rift Valley fever and West Nile virus. The within-host dynamics govern transmission and host mortality and recovery in an age-since-infection structured host-vector-borne pathogen epidemic model. By considering multiple pathogen strains and multiple competing host populations differing in their within-host replication rate and immune response parameters, respectively, we derive evolutionary optimization principles for both pathogen and host. Invasion analysis shows that the \({\mathcal {R}}_0\) maximization principle holds for the vector-borne pathogen. For the host, we prove that evolution favors minimizing case fatality ratio (CFR). These results are utilized to compute host and pathogen evolutionary trajectories and to determine how model parameters affect evolution outcomes. We find that increasing the vector inoculum size increases the pathogen \({\mathcal {R}}_0\), but can either increase or decrease the pathogen virulence (the host CFR), suggesting that vector inoculum size can contribute to virulence of vector-borne diseases in distinct ways.

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## Acknowledgements

The authors H. Gulbudak and V. Cannataro acknowledge partial support from IGERT Grant NSF DGE-0801544 in the Quantitative Spatial Ecology, Evolution and Environment Program at the University of Florida. Authors N. Tuncer and M. Martcheva would also like to acknowledge support from the National Science Foundation (NSF) under Grants DMS-1515661/DMS-1515442.

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## Appendix

### Appendix

### Analysis of Immunological Model: Virus Ultimately Dies Out

Notice that in the system (1), in the absence of immune response (\(M_0=0, G_0=0\)), the pathogen grows exponentially, in which case it is expected that the infected host dies since the parasite damages the host. However, when immune system is active, we establish the following result:

### Theorem 4.1

If initial immune response is active (\(M_0>0 \text { or } G_0>0\)), then the pathogen eventually dies out (\(\lim _{\tau \rightarrow \infty }P(\tau )=0\)), the IgM immune response antibodies decay to zero after viral clearance, and subsequently, the IgG immune memory antibodies reach a steady state, i.e., \(\ \lim _{\tau \rightarrow \infty }M(\tau )=0 \ \text {and} \ \lim _{\tau \rightarrow \infty }G(\tau )=G^*,\) where \(G^*>0\).

### Proof

Let \(P(0)>0\). By the first equation in the immunological model (1), we obtain the following inequality:

Also note that if \(M_0>0 \text { or } G_0>0\), then by comparison principle (similar argument to above), after some time, namely \(\tilde{\epsilon },\) we obtain

By the way of contradiction assume that \(\limsup _{\tau \rightarrow \infty }P(\tau )\ne 0\). Then, the right-hand side of the inequality (9) goes to infinity. Therefore, \(\lim _{\tau \rightarrow \infty }G(\tau )= \infty \). Then, \(\exists \tau ^*: \forall \tau >\tau ^*,\) \(G(\tau )>\frac{r}{\delta }\). Then, as \(\tau \rightarrow \infty \), the right-hand side of the inequality (8) goes to zero. It is a contradiction. Then, \(\lim _{\tau \rightarrow \infty }P(\tau )=0,\) subsequently \(\lim _{\tau \rightarrow \infty }M(\tau )=0\) and \(\lim _{\tau \rightarrow \infty }G(\tau )=G^*\), for some \(G^*>0\). \(\square \)

In summary, for any solution \((P(\tau ),M(\tau ),G(\tau ))\) of the immunological model (1), with nonzero initial condition (*P*(0), *M*(0), *G*(0)), we have \(\lim _{\tau \rightarrow \infty } P(\tau )=0,\ \lim _{\tau \rightarrow \infty } M(\tau )=0, \ \lim _{\tau \rightarrow \infty } G(\tau )= G^*,\) where \(G^*\) is a positive real number and depends on the initial condition.

### Existence and Uniqueness of Equilibrium

### Proof

(Proof of Theorem (2.1)) The proof is omitted. However, a sketch of the proof is as follows. It can be shown that the system is dissipative and asymptotically compact. In addition, the system is uniformly persistent when \({\mathcal {R}}_0>1\) (it is not hard to show that DFE is unstable when \({\mathcal {R}}_0>1;\) then by a similar approach to Proposition 4.4 in Yang et al. (2012), it can be shown that the system is uniform persistent). Then by Zhao (2013), there exists at least one positive steady state. Subsequently, by the argument of Proposition 3.1 in Martcheva et al. (2016), it can be shown that this positive equilibrium has to be unique. The proof of local stability under the condition (5) is contained in the proof of \({\mathcal {R}}_0\) maximization in the next subsection. \(\square \)

### Virus Evolution: \({\mathcal {R}}_0\) Maximization

### Proof

(Proof of Theorem (3.1)) By taking \(S_H(t)=S^*_H+x_H(t), \ i_{H_1}(\tau ,t)=i^*_{H_1}(\tau )+y_{H_1}(\tau ,t), \ i_{H_2}(\tau ,t)=y_{H_2}(\tau ,t), R_H(t)=R^*_H+z_H(t),S_V(t)=S^*_V+x_V(t), I_{V_1}(t)=I^*_{V_1}+y_{V_1}(t),\) and \(I_{V_2}(t)=y_{V_2}(t),\) we linearize the one-host two-strain model (6) about the equilibrium \({\mathcal {E}}_1=(S^*_H, i^*_{H_1}(\tau ),0,R^*_H, S^*_V, I^*_{V_1},0)\) and look for eigenvalues of the linear operator, that is, we look for solutions of the form \(x_H(t)=\overline{x}_H e^{\lambda t}, y_{H_1}(\tau ,t)=\overline{y}_{H_1}(\tau ) e^{\lambda t}, y_{H_2}(\tau ,t)=\overline{y}_{H_2}(\tau )e^{\lambda t}, z_H(t)=\overline{z}_H e^{\lambda t}, x_V(t) = \overline{x}_V e^{\lambda t}, y_{V_1}(t)=\overline{y}_{V_1}e^{\lambda t},\) and \(y_{V_2}(t)=\overline{y}_{V_2} e^{\lambda t},\) where \(\overline{x}_H, \overline{y}_{H_1},\overline{y}_{H_2},\overline{z}_H,\overline{x}_V,\overline{y}_{V_1}\) and \(\overline{y}_{V_2}\) are arbitrary nonzero constants (a function of \(\tau \) in the case of \(y_{H_i}\)), but the eigenvalue \(\lambda \) is common. This process results in the following system (the bars have been omitted):

Note that by fourth and fifth equation in (10), we have \(y_{H_2}=y_{H_2}(0)e^{-\lambda \tau }\pi _2(\tau ),\) where \(\pi _2(\tau )=e^{-\displaystyle \int _0^\tau {(\alpha _2(s)+\kappa _2(s)+\gamma _2(s)+d)}}\). Then, rearranging the last equation in (10), we obtain

Substituting (11) into the fifth equation (boundary condition for strain type 2), we obtain

Suppose that \(y_{V_2} \ne 0\), then the eigenvalues of the system will be determined by the characteristic equation \(G(\lambda ) =1\), where \(G(\lambda )\) is the right-hand side of the equation (12).

Note by the equilibrium condition, we have

where \(\pi _1(\tau )=e^{-\displaystyle \displaystyle \int _0^\tau {(\alpha _1(s)+\kappa _1(s)+\gamma _1(s)+d)ds}}\). Substituting (13) into (12), we obtain

Notice that \(G(0)=\displaystyle \frac{{\mathcal {R}}^2_0}{{\mathcal {R}}^1_0},\) where \({\mathcal {R}}^i_0\) is reproduction number for strain type *i*. If \({\mathcal {R}}^2_0>{\mathcal {R}}^1_0,\) then \(G(0)>1\). Since \(G(\lambda )\) is a decreasing function of \(\lambda \), where \(\lambda \) is restricted to real numbers, and \(\lim _{\lambda \rightarrow \infty } G(\lambda )=0,\) by intermediate value theorem, there exists a positive real number \(\lambda ^*\) such that \(G(\lambda ^*)=1\). Therefore, if \({\mathcal {R}}^2_0>{\mathcal {R}}^1_0,\) then the strain-one equilibrium \({\mathcal {E}}_1\) is unstable.

Assume \({\mathcal {R}}^2_0<{\mathcal {R}}^1_0,\) that is \(G(0)<1\). Suppose that the system (10) has a solution \(\lambda =a+ib\) such that \(a\ge 0\). Then, by the equation (12), we have

Hence, the characteristic equation \(G(\lambda )=1\) does not have solutions with nonnegative real part.

Now assume that \(y_{V_2}= 0\). Then, the stability of \({\mathcal {E}}_1\) depends on the eigenvalues of the following system:

Since \(y_{H_1}(\tau ) = y_{H_1}(0) e^{-\lambda \tau } \pi _1(\tau )\), we have

Then, the first equation in (16) is an equality in the terms of \(x_H, \ y_{V_1}\), and \(\lambda \) obtained as follows:

Also from the last two equations in (16), we obtain an equation in the terms of \(x_H, \ y_{V_1}\), and \(\lambda \).

where \(T_1=\displaystyle \int _0^\infty {\beta _{h_1}(\tau )i^*_{H_1}(\tau )d\tau } (>0)\). The second equality is also in the terms of \(x_H, \ y_{V_1}\), and \(\lambda \). Then, the characteristic equation is as follows:

Suppose that \(\lambda =a+bi\) with \(a\ge 0\) is a solution of the characteristic equation. Taking the absolute value of both sides of the equality above, we get

Moreover,

Note that for \(\lambda \) with nonnegative real part, when \(\mathfrak {R}\,\hat{T}\ge 0\) and \(\mathfrak {I}\,\hat{T}\ge 0\) the left-hand side remains strictly greater than one, while the right-hand side is strictly smaller than one. Thus, such \(\lambda \)’s cannot satisfy the characteristic equation (17). Hence the one strain endemic equilibrium \({\mathcal {E}}_1\) is locally asymptotically stable whenever it exists given the assumptions on \(\hat{T}\). We notice that the conditions \(\mathfrak {R}\,\hat{T}\ge 0\) and \(\mathfrak {I}\,\hat{T}\ge 0\) for any \(\lambda \) with \(\mathfrak {R}\, \lambda \ge 0\) may not always hold. \(\square \)

### Host Evolution: Case Fatality Ratio (\({\mathcal {F}}\)) Minimization

### Proof

(Proof of Theorem (3.2))

Let \((x_1(t), y_1(\tau ,t),z_1(t),x_2(t), y_2(\tau ,t),z_2(t),u(t),v(t))\) be the small perturbations around the mutant-free equilibrium \({\mathcal {E}}_R\). Linearizing the system (7) (with two-host one-strain population) at the mutant-free equilibrium \({\mathcal {E}}_R \), we get:

where \(\eta _j=x_j+\displaystyle \displaystyle \int _0^\infty y_j(\tau ) d\tau +z_j\).

Solving the differential equations, we obtain

and

Note that any eigenvalue of the system (18) is also an eigenvalue of the following decoupled subsystem

Rearranging the first equation in (21) and substituting \(\displaystyle \displaystyle \int _0^\infty y_2(\tau ) d\tau \) and \(z_2 = \displaystyle \frac{1}{\lambda +d}\displaystyle \int _0^\infty \gamma _2(\tau ) y_2(\tau ) d\tau \), we obtain

Substituting the equation (20) into (22), we get

(assuming \(x_2 \ne 0\)). Susceptible host type 1 when at equilibrium satisfies

where \(N_1^* = S_1^* + i_1^*(0)\displaystyle \int _0^\infty \pi _{H_1}(\tau ) d\tau + i_1^*(0)\displaystyle \int _0^\infty \displaystyle \frac{\gamma _1(\tau )}{d}\pi _{H_1}(\tau ) d\tau \).

Substituting the right-hand side of the equation into (23), we obtain the characteristic equation

### Claim 4.1

Assume that \({\mathcal {F}}_1<{\mathcal {F}}_2\). Then, all eigenvalues of the system (18) have negative real part.

### Proof

By the way of contradiction, suppose that the system (18) has an eigenvalue \(\lambda \) with nonnegative real part. Then, \(\lambda \) also is an eigenvalue of the decoupled subsystem (21). Then,

Also for the right-hand side of the equation (26), we get

Therefore,

Subtracting and adding \(\nu _1(\tau )\) on the left-hand side of the equation above and \(\nu _2(\tau )\) on the right side of the equation, we obtain

Then,

which implies that \({\mathcal {F}}_1 \ge {\mathcal {F}}_2\). It is a contradiction. This completes the proof for the case \(x_2 \ne 0\). Now assume that \(x_2=0\). Then, \(\eta _2=0,\) in which case we obtain the subsystem

This is exactly the linearized system determining the local stability of \({\mathcal {E}}\), the endemic equilibrium for the single-host single-strain model. Thus, by Theorem 2.1, the conditions (5) imply that the eigenvalues of subsystem (31) have negative real part. Then, the case \(x_2=0\) also contradicts the assumption that the system (18) has an eigenvalue with nonnegative real part. \(\square \)

Then, when \({\mathcal {F}}_1<{\mathcal {F}}_2\) and boundary conditions hold, the mutant-free equilibrium is locally asymptotically stable.

### Claim 4.2

(Local invasion) If \({\mathcal {F}}_2 < {\mathcal {F}}_1\). Then, the mutant-free equilibrium \({\mathcal {E}}^*_R\) is unstable; i.e., mutant population invades resident population.

### Proof

Let define the left-hand side of the equation (26) as \(g(\lambda )\) and the right-hand side as \(f(\lambda )\). Note that any \(\lambda \) solution of this system must be an eigenvalue of the decoupled subsystem (31). Now we want to show that the equation (26) has a positive real root \(\lambda ,\) whenever \({\mathcal {F}}_2 < {\mathcal {F}}_1\). First note that \(g(\lambda )\) is an increasing function of \(\lambda \) and \(f(\lambda )\) is a decreasing function of \(\lambda \) for \(\lambda \in \mathbb R\). Therefore, if \(g(0)< f(0),\) then the equality (26) has a positive real root \(\lambda \). Next we want to show that \(g(0)< f(0) \Leftrightarrow {\mathcal {F}}_2 < {\mathcal {F}}_1\). Note that

where \(i^*_{1}(0)=\beta _1 I^*_VS^*_{1}\). Then, the rest of the proof follows similar steps to the proof of Claim (4.1) after the inequality (29). \(\square \)

\(\square \)

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### Cite this article

Gulbudak, H., Cannataro, V.L., Tuncer, N. *et al.* Vector-Borne Pathogen and Host Evolution in a Structured Immuno-Epidemiological System.
*Bull Math Biol* **79, **325–355 (2017). https://doi.org/10.1007/s11538-016-0239-0

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### Keywords

- Immuno-epidemiological modeling
- Rift Valley fever
- West Nile virus
- Vector-borne pathogen
- Differential equations
- Reproduction number
- Vector inoculum size
- Age-since-infection
- Within-host dynamics
- Coevolutionary attractor
- Trade-offs

### Mathematics Subject Classification

- 92D30
- 92D40